Tom Bridges

Professor Thomas Bridges


Professor
PhD, MSc, MA, BSc

Biography

My publications

Publications

Thomas J Bridges, Daniel J Ratliff (2020)Nonlinear theory for coalescing characteristics in multiphase Whitham modulation theory, In: Journal of Nonlinear Science31(7)
The multiphase Whitham modulation equations with N phases have 2N characteristics which may be of hyperbolic or elliptic type. In this paper, a nonlinear theory is developed for coalescence, where two characteristics change from hyperbolic to elliptic via collision. Firstly, a linear theory develops the structure of colliding characteristics involving the topological sign of characteristics and multiple Jordan chains, and secondly, a nonlinear modulation theory is developed for transitions. The nonlinear theory shows that coalescing characteristics morph the Whitham equations into an asymptotically valid geometric form of the two-way Boussinesq equation, that is, coalescing characteristics generate dispersion, nonlinearity and complex wave fields. For illustration, the theory is applied to coalescing characteristics associated with the modulation of two-phase travelling wave solutions of coupled nonlinear Schrödinger equations, highlighting how collisions can be identified and the relevant dispersive dynamics constructed.
HA Ardakani, TJ Bridges, MR Turner (2015)Adaptation of f-wave finite volume methods to the two-layer shallow-water equations in a moving vessel with a rigid-lid, In: Journal of Computational and Applied Mathematics296pp. 462-479 ELSEVIER SCIENCE BV
A numerical method is proposed to solve the two-layer inviscid, incompressible and immiscible 1D shallow-water equations in a moving vessel with a rigid-lid with different boundary conditions based on the high-resolution f-wave finite volume methods due to Bale et al. (2002). The method splits the jump in the fluxes and source terms including the pressure gradient at the rigid-lid into waves propagating away from each grid cell interface. For the influx–efflux boundary conditions the time dependent source terms are handled via a fractional step approach. In the linear case the numerical solutions are validated by comparison with the exact analytical solutions. Numerical solutions presented for the nonlinear case include shallow-water sloshing waves due to prescribed surge motion of the vessel.
In this thesis we investigate the dynamics of coupled liquid sloshing systems, which consist of a one-dimensional array of vessels, partially filled with fluid, being connected together by nonlinear springs. The fluid motion induces a hydrodynamic force on the side walls of the vessel, which induces the vessel to move. The vessel movement is also controlled via a restoring force of the attached springs, which in turn causes the fluid to alter its motion. The simplest coupled liquid sloshing system consists of one vessel connected to a side wall via a spring. We review this example and investigate the techniques documented in the literature to study the linear problem of such a system. Then we extend this linear theory to multivessel systems, in particular focussing on the 2-vessel system which introduces the notion of modes being in-phase or out-of-phase with each other. In the general N-vessel system we also identify the existence in parameter space of internal resonances, where different modes oscillates at the same frequency. Such a resonance provides a mechanism for energy exchange between modes in the nonlinear system. The nonlinear dynamics of the coupled liquid sloshing system are studied by employing a symplectic integration scheme based on a variational principle, to the shallow-water form of the governing equations. We present results for the shallow-water scheme in the linear, weakly nonlinear and fully nonlinear regimes. The shallow-water numerical scheme has trouble dealing with breaking waves, so we perform a feasibility study where we attempt to use a similar approach for a non-shallow fluid system. The governing equations are derived and results presented in the linear amplitude regime for the 1-vessel system. Finally, we develop variational principles for the coupled liquid sloshing system in the Eulerian framework based on the principle of constrained variations to derive the governing fluid equations and free-surface boundary conditions, from a natural Lagrangian functional. We use this constrained variational approach to derive the fully rotational 2D Euler equations and its stream function formulation.
Thomas J Bridges, Anna Kostianko, Sergey Zelik (2021)Validity of the hyperbolic Whitham modulation equations in Sobolev spaces, In: Journal of Differential Equations274pp. 971-995 Elsevier Inc
It is proved that modulation in time and space of periodic wave trains, of the defocussing nonlinear Schrödinger equation, can be approximated by solutions of the Whitham modulation equations, in the hyperbolic case, on a natural time scale. The error estimates are based on existence, uniqueness, and energy arguments, in Sobolev spaces on the real line. An essential part of the proof is the inclusion of higher-order corrections to Whitham theory, and concomitant higher-order energy estimates.
Andreas Foiniotis (2020)Evolution of resistance in ecological and epidemiological contexts. University of Surrey
Understanding the dynamics of an infectious disease, such as malaria, helps us to reduce the number of deaths and to achieve better control of its spread. Mathematical models can help to predict the outcomes of new ideas for containing disease spread. This thesis constitutes an extension of previous attempts to understand, via mathematical modelling, the dynamics of mosquito populations and malaria. We propose 5 distinct non-linear age dependent mathematical models. The first model uses, as a starting point, an approach to modelling nonlinear effects in agestructured models due to Gurtin and MacCamy [31]. However, the model we derive (which takes the form of a system of delay differential equations) is much more complex. We demonstrate analytical results on linear stability of both zero and positive equilibria in various cases. We then examine a more complex equation which incorporates competition among larval mosquitoes. Furthermore, results on boundedness of solutions and on the existence of positive equilibria are proved. Numerical simulations show that for specific values of several parameters three equilibrium points can be achieved as well as that the equilibria decrease as we increase the larval competition coefficient. In the second mathematical model, we examine a neutral delay differential equation. This specific type of equation is consequent upon the assumption that an adult mosquito lays a batch of eggs immediately upon maturation, followed possibly by further batches (not necessarily containing the same number of eggs) on reaching the particular ages τi + nτ, n = 1,2,..., with no egg laying in between these ages. This models the idea that egg laying follows blood meals. A particular case is the case when adult mosquito lays all of its eggs immediately on maturation, and none at all later in life. In that case the non-trivial equilibrium is locally stable but the roots of the characteristic equation are not bounded away from the imaginary axis. More generally, with adults laying eggs at ages τi +nτ, we may show under some conditions that the unique positive equilibria is linearly stable. Results on the existence of positive equilibria and boundedness of solutions are proved in this general case. Moving to the third mathematical model of the thesis, in Chapter 4, we examine two strains for the mosquito population, the vulnerable and the resistant strain. This model is based on the assumption that mosquitoes may become resistant to insecticides. One particular idea that we examine is the possibility that the parameter values such as the per-capita death rate, maturation time and the kernel g(a) which describes the adult mosquito egg laying activity are different in the two strains. We present analytical results on the global stability of the zero equilibrium and the linear stability of the boundary equilibrium. Numerical simulations show that for several parameter values either of the two strains can win the competition and drive the other one to extinction. In Chapter 5, the fourth mathematical model that we propose has similarities to the model of Chapter 4 but also allows the possibility of an adult vulnerable mosquito to die due to the insecticide. We propose a model for the case of an insecticide that attacks a mosquito with increasing potency as it ages, eventually giving us a system of four-integral equations. We compare two kinds of insecticides, late-life acting (LLA) insecticides and conventional insecticides, and try to find under what circumstances the LLA insecticide will slow down the evolution of insecticide resistance. The final part of the thesis examines the interaction between the host (human) and the vector (mosquito). Our fifth model provides analytical and numerical results for an eight-dimensional system of equations, consisting of two differential equations and six integral equations. For this model we find a set of conditions sufficient for the eradication of malaria.
Hamid Alemi Ardakani, Thomas J. Bridges, Matthew R. Turner (2016)Shallow-water sloshing in a moving vessel with variable cross-section and wetting-drying using an extension of George's well-balanced finite volume solver, In: Journal of Computational Physics314pp. 590-617 Elsevier
A class of augmented approximate Riemann solvers due to George (2008) [12] is extended to solve the shallow-water equations in a moving vessel with variable bottom topography and variable cross-section with wetting and drying. A class of Roe-type upwind solvers for the system of balance laws is derived which respects the steady-state solutions. The numerical solutions of the new adapted augmented f-wave solvers are validated against the Roe-type solvers. The theory is extended to solve the shallow-water flows in moving vessels with arbitrary cross-section with influx–efflux boundary conditions motivated by the shallow-water sloshing in the ocean wave energy converter (WEC) proposed by Offshore Wave Energy Ltd. (OWEL) [1]. A fractional step approach is used to handle the time-dependent forcing functions. The numerical solutions are compared to an extended new Roe-type solver for the system of balance laws with a time-dependent source function. The shallow-water sloshing finite volume solver can be coupled to a Runge–Kutta integrator for the vessel motion.
TJ Bridges (2012)GEOMETRIC LIFT OF PATHS OF HAMILTONIAN EQUILIBRIA AND HOMOCLINIC BIFURCATION, In: INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS22(12)ARTN 12503 WORLD SCIENTIFIC PUBL CO PTE LTD
T J Bridges, N M Donaldson (2006)Secondary Criticality of Water Waves. Part 1.
Definition, Bifurcation and Solitary Waves
, In: Journal of Fluid Mechanics565pp. 381-417

A generalization of criticality - called secondary criticality - is introduced and applied to finite-amplitude Stokes waves. The theory shows that secondary criticality signals a bifurcation to a class of steady dark solitary waves which are biasymptotic to a Stokes wave with a phase jump in between, and synchronized with the Stokes wave. We find the that the bifurcation to these new solitary waves - from Stokes gravity waves in shallow water - is pervasive, even at low amplitude. The theory proceeds by generalizing concepts from hydraulics: three additional functionals are introduced which represent non-uniformity and extend the familiar mass flux, total head and flow force, the most important of which is the wave action flux. The theory works because the hydraulic quantities can be related to the governing equations in a precise way using the multi-symplectic Hamiltonian formulation of water waves. In this setting, uniform flows and Stokes waves coupled to a uniform flow are relative equilibria which have an attendant geometric theory using symmetry and conservation laws. A flow is then 'critical' if the relative equilibrium representation is degenerate. By characterizing successively non-uniform flows and unsteady flows as relative equilibria, a generalization of criticality is immediate. Recent results on the local nonlinear behaviour near a degenerate relative equilibrium are used to predict all the qualitative properties of the bifurcating dark solitary waves, including the phase shift. The theory of secondary criticality provides new insight into unsteady waves in shallow water as well. A new interpretation of the Benjamin-Feir instability from the viewpoint of hydraulics, and the connection with the creation of unsteady dark solitary waves, is given in Part 2.

TJ Bridges (2013)A universal form for the emergence of the Korteweg-de Vries equation, In: PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES469(2153)ARTN 20120 ROYAL SOC
Leonid Brevdo, Patrice Laure, Frederic Dias, Thomas J. Bridges (1999)Linear pulse structure and signalling in a film flow on an inclined plane, In: Journal of Fluid Mechanics

The film flow down an inclined plane has several features that make it an interesting prototype for studying transition in a shear flow: the basic parallel state is an exact explicit solution of the Navier–Stokes equations; the experimentally observed transition of this flow shows many properties in common with boundary-layer transition; and it has a free surface, leading to more than one class of modes. In this paper, unstable wavepackets – associated with the full Navier–Stokes equations with viscous free-surface boundary conditions – are analysed by using the formalism of absolute and convective instabilities based on the exact Briggs collision criterion for multiple k-roots of D(k, omega) = 0, where k is a wavenumber, omega is a frequency and D(k, omega) is the dispersion relation function.

The main results of this paper are threefold. First, we work with the full Navier–Stokes equations with viscous free-surface boundary conditions, rather than a model partial differential equation, and, guided by experiments, explore a large region of the parameter space to see if absolute instability—as predicted by some model equations—is possible. Secondly, our numerical results find only convective instability, in complete agreement with experiments. Thirdly, we find a curious saddle-point bifurcation which affects dramatically the interpretation of the convective instability. This is the first finding of this type of bifurcation in a fluids problem and it may have implications for the analysis of wavepackets in other flows, in particular for three-dimensional instabilities. The numerical results of the wavepacket analysis compare well with the available experimental data, confirming the importance of convective instability for this problem.

The numerical results on the position of a dominant saddle point obtained by using the exact collision criterion are also compared to the results based on a steepest-descent method coupled with a continuation procedure for tracking convective instability that until now was considered as reliable. While for two-dimensional instabilities a numerical implementation of the collision criterion is readily available, the only existing numerical procedure for studying three-dimensional wavepackets is based on the tracking technique. For the present flow, the comparison shows a failure of the tracking treatment to recover a subinterval of the interval of unstable ray velocities V whose length constitutes 29% of the length of the entire unstable interval of V. The failure occurs due to a bifurcation of the saddle point, where V is a bifurcation parameter. We argue that this bifurcation of unstable ray velocities should be observable in experiments because of the abrupt increase by a factor of about 5.3 of the wavelength across the wavepacket associated with the appearance of the bifurcating branch. Further implications for experiments including the effect on spatial amplification rate are also discussed.

T. J. Bridges (2004)Superharmonic instability, homoclinic torus bifurcation and water-wave breaking, In: Journal of Fluid Mechanics505pp. 153-162

The superharmonic instability is pervasive in large-amplitude water-wave problems and numerical simulations have predicted a close connection between it and crest instabilities and wave breaking. In this paper we present a nonlinear theory, which is a generic nonlinear consequence of superharmonic instability. The theory predicts the nonlinear behaviour witnessed in numerics, and gives new information about the nonlinear structure of large-amplitude water waves, including a mechanism for noisy wave breaking.

The theory for criticality presented in Part 1 is extended to the unsteady problem, and a new formulation of the Benjamin–Feir instability for Stokes waves in finite depth coupled to a mean flow, which takes the criticality matrix as an organizing centre, is presented. The generation of unsteady dark solitary waves at points of stability changes and their connection with the steady dark solitary waves of Part 1 are also discussed.

T. J. Bridges, N. M. Donaldson (2006)Secondary criticality of water waves. Part 1. Definition, bifurcation and solitary waves, In: Journal of Fluid Mechanics565pp. 381-417

A generalization of criticality – called secondary criticality – is introduced and applied to finite-amplitude Stokes waves. The theory shows that secondary criticality signals a bifurcation to a class of steady dark solitary waves which are biasymptotic to a Stokes wave with a phase jump in between, and synchronized with the Stokes wave. We find the that the bifurcation to these new solitary waves – from Stokes gravity waves in shallow water – is pervasive, even at low amplitude. The theory proceeds by generalizing concepts from hydraulics: three additional functionals are introduced which represent non-uniformity and extend the familiar mass flux, total head and flow force, the most important of which is the wave action flux. The theory works because the hydraulic quantities can be related to the governing equations in a precise way using the multi-symplectic Hamiltonian formulation of water waves. In this setting, uniform flows and Stokes waves coupled to a uniform flow are relative equilibria which have an attendant geometric theory using symmetry and conservation laws. A flow is then ‘critical’ if the relative equilibrium representation is degenerate. By characterizing successively non-uniform flows and unsteady flows as relative equilibria, a generalization of criticality is immediate. Recent results on the local nonlinear behaviour near a degenerate relative equilibrium are used to predict all the qualitative properties of the bifurcating dark solitary waves, including the phase shift. The theory of secondary criticality provides new insight into unsteady waves in shallow water as well. A new interpretation of the Benjamin–Feir instability from the viewpoint of hydraulics, and the connection with the creation of unsteady dark solitary waves, is given in Part 2.

CJ Grudzien, TJ Bridges, CKRT Jones (2016)Geometric phase in the Hopf bundle and the stability of non-linear waves, In: Physica D: Nonlinear Phenomena334pp. 4-18 Elsevier
We develop a stability index for the travelling waves of non-linear reaction diffusion equations using the geometric phase induced on the Hopf bundle S2n−1⊂CnS2n−1⊂Cn. This can be viewed as an alternative formulation of the winding number calculation of the Evans function, whose zeroes correspond to the eigenvalues of the linearization of reaction diffusion operators about the wave. The stability of a travelling wave can be determined by the existence of eigenvalues of positive real part for the linear operator. Our method of geometric phase for locating and counting eigenvalues is inspired by the numerical results in Way’s Dynamics in the Hopf bundle, the geometric phase and implications for dynamical systems Way (2009). We provide a detailed proof of the relationship between the phase and eigenvalues for dynamical systems defined on C2C2 and sketch the proof of the method of geometric phase for CnCn and its generalization to boundary-value problems. Implementing the numerical method, modified from Way (2009), we conclude with open questions inspired from the results.
Daniel Ratliff, Thomas Bridges (2016)Multiphase wavetrains, singular wave interactions and the emergence of the Korteweg-de Vries equation, In: Proceedings of the Royal Society A472(2196)0160456 The Royal Society
Multiphase wavetrains are multiperiodic travelling waves with a set of distinct wavenumbers and distinct frequencies. In conservative systems such families are associated with the conservation of wave action or other conservation law. At generic points (where the Jacobian of the wave action flux is nondegenerate) modulation of the wavetrain leads to the dispersionless multiphase conservation of wave action. The main result of this paper is that modulation of the multiphase wavetrain, when the Jacobian of the wave action flux vector is singular, morphs the vector-valued conservation law into the scalar KdV equation. The coefficients in the emergent KdV equation have a geometric interpretation in terms of projection of the vector components of the conservation law. The theory herein is restricted to two phases to simplify presentation, with extensions to any finite dimension discussed in the concluding remarks. Two applications of the theory are presented: a coupled nonlinear Schrödinger equation and two-layer shallow water hydrodynamics with a free surface. Both have two-phase solutions where criticality and the properties of the emergent KdV equation can be determined analytically.
T. J. Bridges (2005)Steady dark solitary waves emerging from wave-generated meanflow: The role of modulation equations, In: Chaos: An Interdisciplinary Journal of Nonlinear Science037113(2005)

Various classes of steady and unsteady dark solitary waves (DSWs) are known to exist in modulation equations for water waves in finite depth. However, there is a class of steady DSWS of the full water-wave problem which are missed by the classical modulation equations such as the Hasimoto–Ono, Benney–Roskes, and Davey–Stewartson. These steady DSWs, recently discovered by Bridges and Donaldson, are pervasive in finite depth, arise through secondary criticality of Stokes gravity waves, and are synchronized with the Stokes wave. In this paper, the role of DSWs in modulation equations for water waves is reappraised. The intrinsic unsteady nature of existing modulation equations filters out some interesting solutions. On the other hand, the geometry of DSWs in modulation equations is very similar to the full water wave problem and these geometrical properties are developed. A model equation is proposed which illustrates the general nature of the emergence of steady DSWs due to wave-generated mean flow coupled to a periodic wave. Although the existing modulation equations are intrinsically unsteady, it is shown that there are also important shortcomings when one wants to use them for stability analysis of DSWs.

Matthew Turner, Thomas Bridges (2015)Time-dependent conformal mapping of doubly-connected regions, In: Advances in Computational Mathematics42(4)pp. 947-972 Springer
This paper examines two key features of time-dependent conformal mappings in doubly-connected regions, the evolution of the conformal modulus Q(t) and the boundary transformation generalizing the Hilbert transform. It also applies the theory to an unsteady free surface flow. Focusing on inviscid, incompressible, irrotational fluid sloshing in a rectangular vessel, it is shown that the explicit calculation of the conformal modulus is essential to correctly predict features of the flow. Results are also presented for fully dynamic simulations which use a time-dependent conformal mapping and the Garrick generalization of the Hilbert transform to map the physical domain to a time-dependent rectangle in the computational domain. The results of this new approach are compared to the complementary numerical scheme of Frandsen (2004) (J. Comp. Phys. 196, 53-87) and it is shown that correct calculation of the conformal modulus is essential in order to obtain agreement between the two methods.
Daniel Ratliff, Tom Bridges (2018)Reduction to modified KdV and its KP-like generalization via phase modulation, In: Nonlinearity31(8)3794 IOP Publishing
The main observation of this paper it that the modified Korteweg-de Vries equation has its natural origin in phase modulation of a basic state such as a periodic travelling wave or more generally a family of relative equilibria. Extension to 2+1 suggests that a modified Kadomtsev-Petviashvili (or a Konopelchenko-Dubrovsky) equation should emerge, but our result shows that there is an additional term which has gone heretofore unnoticed. Thus through the novel application of phase modulation a new equation appears as the 2+1 extension to a previously known one. To demonstrate the theory it is applied to the cubic-quintic Nonlinear Schrodinger (CQNLS) equation, showing that there are relevant parameter values where a modified KP equation bifurcates from periodic travelling wave solutions of the 2+1 CQNLS equation.
T. J. Bridges (2004)Superharmonic Instability, Homoclinic Torus Bifurcation and Water-Wave Breaking, In: Journal of Fluid Mechanics505pp. 153-162

The superharmonic instability is pervasive in large-amplitude water-wave problems and numerical simulations have predicted a close connection between it and crest instabilities and wave breaking. In this paper we present a nonlinear theory, which is a generic nonlinear consequence of superharmonic instability. The theory predicts the nonlinear behaviour witnessed in numerics, and gives new information about the nonlinear structure of large-amplitude water waves, including a mechanism for noisy wave breaking.

T.J. Bridges, Gianne Derks (2002)Linear Instability of Solitary Wave Solutions of the Kawahara Equation and Its Generalizations, In: the SIAM Journal on Mathematical Analysis33pp. 1356-1378

The linear stability problem for solitary wave states of the Kawahara---or fifth-order KdV-type---equation and its generalizations is considered. A new formulation of the stability problem in terms of the symplectic Evans matrix is presented. The formulation is based on a multisymplectification of the Kawahara equation, and leads to a new characterization of the basic solitary wave, including changes in the state at infinity represented by embedding the solitary wave in a multiparameter family. The theory is used to give a rigorous geometric sufficient condition for instability. The theory is abstract and applies to a wide range of solitary wave states. For example, the theory is applied to the families of solitary waves found by Kichenassamy--Olver and Levandosky.

F Chardard, F Dias, TJ Bridges (2011)Computing the Maslov index of solitary waves, Part 2: Phase space with dimension greater than four, In: Physica D: Nonlinear Phenomena240(17)pp. 1334-1344 Elsevier
This paper extends the theory of the Maslov index of solitary waves in Part 1 to the case where the phase space is of dimension greater than four. The starting point is Hamiltonian PDEs, in one space dimension and time, whose steady part is a Hamiltonian ODE with a phase space of dimension six or greater. This steady Hamiltonian ODE is the main focus of the paper. Homoclinic orbits of the steady ODE represent solitary waves of the PDE, and one of the properties of the homoclinic orbits is the Maslov index. We develop formulae for the Maslov index, the Maslov angle and its subangles, in an exterior algebra framework, and develop numerical algorithms to compute them. In addition, a new numerical approach based on a discrete QR algorithm is proposed. The Maslov index is of interest for classifying solitary waves and as an indicator of stability or instability of the solitary wave in the time-dependent problem. The theory is applied to a class of reaction–diffusion equations, the longwave–shortwave resonance equations and the seventh-order KdV equation.
Thomas J. Bridges, Daniel J. Ratliff (2019)Krein signature and Whitham modulation theory: the sign of characteristics and the “sign characteristic”, In: Studies in Applied Mathematics142(3)pp. pp 314-335 Wiley
In classical Whitham modulation theory, the transition of the dispersionless Whitham equations from hyperbolic to elliptic is associated with a pair of nonzero purely imaginary eigenvalues coalescing and becoming a complex quartet, suggesting that a Krein signature is operational. However, there is no natural symplectic structure. Instead, we find that the operational signature is the “sign characteristic” of real eigenvalues of Hermitian matrix pencils. Its role in classical Whitham single‐phase theory is elaborated for illustration. However, the main setting where the sign characteristic becomes important is in multiphase modulation. It is shown that a necessary condition for two coalescing characteristics to become unstable (the generalization of the hyperbolic to elliptic transition) is that the characteristics have opposite sign characteristic. For example the theory is applied to multiphase modulation of the two‐phase traveling wave solutions of coupled nonlinear Schrödinger equation.
Thomas J. Bridges (1997)Multi-symplectic structures and wave propagation, In: the Mathematical Proceedings of the Cambridge Philosophical Society121(01)pp. 147-190

A Hamiltonian structure is presented, which generalizes classical Hamiltonian structure, by assigning a distinct symplectic operator for each unbounded space direction and time, of a Hamiltonian evolution equation on one or more space dimensions. This generalization, called multi-symplectic structures, is shown to be natural for dispersive wave propagation problems. Application of the abstract properties of the multi-symplectic structures framework leads to a new variational principle for space-time periodic states reminiscent of the variational principle for invariant tori, a geometric reformulation of the concepts of action and action flux, a rigorous proof of the instability criterion predicted by the Whitham modulation equations, a new symplectic decomposition of the Noether theory, generalization of the concept of reversibility to space-time and a proof of Lighthill's geometric criterion for instability of periodic waves travelling in one space dimension. The nonlinear Schrödinger equation and the water-wave problem are characterized as Hamiltonian systems on a multi-symplectic structure for example. Further ramifications of the generalized symplectic structure of theoretical and practical interest are also discussed.

Thomas J. Bridges, Gianne Derks (1999)Unstable eigenvalues and the linearization about solitary waves and fronts with symmetry, In: Proceedings of the Royal Society of London A455pp. 2427-2469 The Royal Society

The linear stability of solitary-wave or front solutions of Hamiltonian evolutionary equations, which are equivariant with respect to a Lie group, is studied. The organizing centre for the analysis is a multisymplectic formulation of Hamiltonian PDEs where a distinct symplectic operator is assigned for time and space. This separation of symplectic structures leads to new characterizations of the following components of the analysis. The states at infinity are characterized as manifolds of relative equilibria associated with the spatial symplectic structure. The momentum of the connecting orbit, or shape of the solitary wave, considered as a heteroclinic orbit in a phase-space representation, is given a new characterization as a one-form on the tangent space to the heteroclinic manifold and this one-form is a restriction of the temporal symplectic structure. For the linear stability analysis, a new symplectic characterization of the Evans function and its derivatives are obtained, leading to an abstract geometric proof of instability for a large class of solitary-wave states of equivariant Hamiltonian evolutionary PDEs. The theory sheds new light on several well-known models: the gKdV equation, a Boussinesq system and a nonlinear wave equation. The generalization to solitary waves associated with multidimensional heteroclinic manifolds and the implications for solitary waves or fronts which are biasymptotic to invariant manifolds such as periodic states are also discussed.

F Chardard, F Dias, TJ Bridges (2009)On the Maslov index of multi-pulse homoclinic orbits, In: PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES465(2109)pp. 2897-2910 ROYAL SOC
F Chardard, TJ Bridges (2015)Transversality of homoclinic orbits, the Maslov index and the symplectic Evans function, In: NONLINEARITY28(1)pp. 77-102 IOP PUBLISHING LTD
Partial differential equations in one space dimension and time, which are gradient-like in time with Hamiltonian steady part, are considered. The interest is in the case where the steady equation has a homoclinic orbit, representing a solitarywave. Such homoclinic orbits have two important geometric invariants: a Maslov index and a Lazutkin-Treschev invariant. A new relation between the two has been discovered and is moreover linked to transversal construction of homoclinic orbits: the sign of the Lazutkin-Treschev invariant determines the parity of the Maslov index. A key tool is the geometry of Lagrangian planes. All this geometry feeds into linearization about the homoclinic orbit in the time-dependent system, which is studied using the Evans function. A new formula for the symplectification of the Evans function is presented, and it is proven that the derivative of the Evans function is proportional to the Lazutkin-Treschev invariant. A corollary is that the Evans function has a simple zero if, and only if, the homoclinic orbit of the steady problem is transversely constructed. Examples from the theory of gradient reaction-diffusion equations and pattern formation are presented.
T. J. Bridges, P. E. Hydon, Sebastien Reich (2005)Vorticity and symplecticity in Lagrangian fluid dynamics, In: Journal of Physics A: Mathematical and General38(6)pp. 1403-1418

The relationship between potential vorticity (PV) and the symplectic form is explored, for the shallow-water equations governing Lagrangian particle paths. Starting with the symplectic form, the PV is found by the pullback operation to the reference space. At first sight, the encoding of PV in the symplectic form appears to be independent of the particle relabelling symmetry. The analysis is carried a step further in two ways. Using the ‘conservation of symplecticity’ as a starting point, the fluxes of symplecticity arise as differential forms, and a complete pull back of the flux forms leads to a geometric description of PV conservation. Secondly, symmetry methods are used to give a rigorous connection between particle relabelling, symplecticity and PV conservation. Generalizations of these issues to semi-geostrophic flow and three-dimensional Lagrangian fluid flows, and the connection with Ertel’s theorem are also discussed.

Thomas J. Bridges, F. Dias, D. Menasce (2001)Steady three-dimensional water-wave patterns on a finite-depth fluid, In: Journal of Fluid Mechanics

The formation of doubly-periodic patterns on the surface of a fluid layer with a uniform velocity field and constant depth is considered. The fluid is assumed to be inviscid and the flow irrotational. The problem of steady patterns is shown to have a novel variational formulation, which includes a new characterization of steady uniform mean flow, and steady uniform flow coupled with steady doubly periodic patterns. A central observation is that mean flow can be characterized geometrically by associating it with symmetries. The theory gives precise information about the role of the ten natural parameters in the problem which govern the wave–mean flow interaction for steady patterns in finite depth. The formulation is applied to the problem of interaction of capillary–gravity short-crested waves with oblique travelling waves, leading to several new observations for this class of waves. Moreover, by including oblique travelling waves and short-crested waves in the same analysis, new bifurcations of short-crested waves are found, which give rise to mixed waves which may have complicated spatial structure.

F Chardard, F Dias, TJ Bridges (2009)Computing the Maslov index of solitary waves, Part 1: Hamiltonian systems on a four-dimensional phase space, In: PHYSICA D-NONLINEAR PHENOMENA238(18)pp. 1841-1867 ELSEVIER SCIENCE BV
T. Brooke Benjamin, Thomas J. Bridges (1997)Reappraisal of the Kelvin–Helmholtz problem. Part 1. Hamiltonian structure, In: Journal of Fluid Mechanics

This paper and Part 2 report various new insights into the classic Kelvin–Helmholtz problem which models the instability of a plane vortex sheet and the complicated motions arising therefrom. The full nonlinear version of the hydrodynamic problem is treated, with allowance for gravity and surface tension, and the account deals in precise fashion with several inherently peculiar properties of the mathematical model. The main achievement of the paper, presented in §3, is to demonstrate that the problem admits a canonical Hamiltonian formulation, which represents a novel variational definition of a functional representing perturbations in kinetic energy. The Hamiltonian structure thus revealed is then used to account systematically for relations between symmetries and conservation laws, and none of those examined appears to have been noticed before. In §4, a generalized, non-canonical Hamiltonian structure is shown to apply when the vortex sheet becomes folded, so requiring a parametric representation, as is well known to occur in the later stages of evolution from Kelvin–Helmholtz instability. Further invariant properties are demonstrated in this context. Finally, §5, the linearized version of the problem – reviewed briefly in §2.1 – is reappraised in the light of Hamiltonian structure, and it is shown how Kelvin–Helmholtz instability can be interpreted as the coincidence of wave modes characterized respectively by positive and negative values of the Hamiltonian functional representing perturbations in total energy.

TJ Bridges, G Derks (2001)The symplectic Evans matrix and the instability of solitary waves and fronts, In: Archives for Rational Mechanics and Analysis156pp. 1-87 Springer
Hamiltonian evolution equations which are equivariant with respect to the action of a Lie group are models for physical phenomena such as oceanographic flows, optical fibres and atmospheric flows, and such systems often have a wide variety of solitary wave or front solutions. In this paper, we present a new symplectic framework for analyzing the spectral problem associated with the linearization about such solitary waves and fronts. At the heart of the analysis is a multi-symplectic formulation of Hamiltonian partial differential equations where a distinct symplectic structure is assigned for the time and space directions, with a third symplectic structure -- with two-form denoted by Omega - associated with a coordinate frame moving at the speed of the wave. This leads to a geometric decomposition and symplectification of the Evans function formulation for the linearization about solitary waves and fronts. We introduce the concept of the "symplectic Evans matrix", a matrix consisting of restricted "Omega-symplectic" forms. By applying Hodge duality to the exterior algebra formulation of the Evans function, we find that the zeros of the Evans function correspond to zeros of the determinant of the symplectic Evans matrix. Based on this formulation, we prove several new properties of the Evans function. Restricting the spectral parameter lambda to the real axis, we obtain rigorous results on the derivatives of the Evans function near the origin, based solely on the abstract geometry of the equations, and results for the large $|____lambda|$ behaviour which use primarily the symplectic structure, but also extend to the non-symplectic case. The Lie group symmetry affects the Evans function by generating zero eigenvalues of large multiplicity in the so-called systems at infinity. We present a new geometric theory which describes precisely how these zero eigenvalues behave under perturbation. By combining all these results, a new rigorous sufficient condition for instability of solitary waves and fronts is obtained. The theory applies to a large class of solitary waves and fronts including waves which are biasymptotic to a nonconstant manifold of states as $|x|$ tends to infinity. To illustrate the theory, it is applied to three examples: a Boussinesq model from oceanography, a class of nonlinear Schrodinger equations from optics and a nonlinear Klein-Gordon equation from atmospheric dynamics.
TJ Bridges (2015)Dimension Breaking from Spatially-Periodic Patterns to KdV Planforms, In: JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS27(3-4)pp. 443-456 SPRINGER
TJ Bridges (2014)Emergence of dispersion in shallow water hydrodynamics via modulation of uniform flow, In: JOURNAL OF FLUID MECHANICS761ARTN R CAMBRIDGE UNIV PRESS
A new theory for the emergence of dispersion in shallow water hydrodynamics in two horizontal space dimensions is presented. Starting with the key properties of uniform flow in open-channel hydraulics, it is shown that criticality is the key mechanism for generating dispersion. Modulation of the uniform flow then leads to model equations. The coefficients in the model equations are related precisely to the derivatives of the mass flux, momentum flux and mass density. The theory gives a new perspective - from the viewpoint of hydraulics - on how and why key shallow water models like the Korteweg-de Vries equation and Kadomtsev-Petviashvili equations arise in the theory of water waves.
T. J. Bridges (2004)On the susceptibility of bright nonlinear Schrödinger solitons to long-wave transverse instability, In: Proceedings of the Royal Society of London A460pp. 2605-2615

A new theory for transverse instability of bright solitons of equations of nonlinear Schrödinger (NLS) type is presented, based on a natural deformation of the solitons into a four-parameter family. This deformation induces a set of four diagnostic functionals which encode information about transverse instability. These functionals include the deformed power, the deformed momentum and two new functionals. The main result is that a sufficient condition for long-wave transverse instability is completely determined by these functionals. Whereas longitudinal instability is determined by a single partial derivative (the Vakhitov-Kolokolov criterion), the condition for transverse instability requires 10 partial derivatives. The theory is illustrated by application to scalar NLS equations with general potential, and vector NLS equations for optical media with χ(2) nonlinearity.

Action, symplecticity, signature and complex instability are fundamental concepts in Hamiltonian dynamics which can be characterized in terms of the symplectic structure. In this paper, Hamiltonian PDEs on unbounded domains are characterized in terms of a multisymplectic structure where a distinct differential two-form is assigned to each space direction and time. This leads to a new geometric formulation of the conservation of wave action for linear and nonlinear Hamiltonian PDEs, and, via Stokes's theorem, a conservation law for symplecticity. Each symplectic structure is used to define a signature invariant on the eigenspace of a normal mode. The first invariant in this family is classical Krein signature (or energy sign, when the energy is time independent) and the other (spatial) signatures are energy flux signs, leading to a classification of instabilities that includes information about directional spatial spreading of an instability. The theory is applied to several examples: the Boussinesq equation, the water-wave equations linearized about an arbitrary Stokes's wave, rotating shallow water flow and flow past a compliant surface. Some implications for non-conservative systems are also discussed.

TJ Bridges, F Dias (2007)Enhancement of the Benjamin-Feir instability with dissipation, In: PHYSICS OF FLUIDS19(10)ARTN 1pp. ?-? AMER INST PHYSICS
TJ Bridges (2012)Emergence of unsteady dark solitary waves from coalescing spatially periodic patterns, In: PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES468(2148)pp. 3784-3803 ROYAL SOC
TJ Bridges, JE Furter (1993)Singularity theory and equivariant symplectic maps1558 Springer
The monograph will appeal to researchers and graduate students in the areas of symplectic maps, Hamiltonian systems, singularity theory and equivariant ...
TJ Bridges (2015)Breakdown of the Whitham Modulation Theory and the Emergence of Dispersion, In: Studies in Applied Mathematics
A Wiley Company. The Whitham modulation theory for periodic traveling waves of PDEs generated by a Lagrangian produces first-order dispersionless PDEs that are, generically, either hyperbolic or elliptic. In this paper, degeneracy of the Whitham equations is considered where one of the characteristic speeds is zero. In this case, the Whitham equations are no longer valid. Reformulation and rescaling show that conservation of wave action morphs into the Korteweg-de Vries (KdV) equation on a longer time scale thereby generating dispersion in the Whitham modulation equations even for finite amplitude waves.
T. J. Bridges, F. E. Laine-Pearson (2005)The Long-Wave Instability of Short-Crested Waves, Via Embedding in the Oblique Two-Wave Interaction, In: Journal of Fluid Mechanics543(-1)pp. 147-182

The motivation for this work is the stability problem for short-crested Stokes waves. A new point of view is proposed, based on the observation that an understanding of the linear stability of short-crested waves (SCWs) is closely associated with an understanding of the stability of the oblique non-resonant interaction between two waves. The proposed approach is to embed the SCWs in a six-parameter family of oblique non-resonant interactions. A variational framework is developed for the existence and stability of this general two-wave interaction. It is argued that the resonant SCW limit makes sense a posteriori, and leads to a new stability theory for both weakly nonlinear and finite-amplitude SCWs. Even in the weakly nonlinear case the results are new: transverse weakly nonlinear long-wave instability is independent of the nonlinear frequency correction for SCWs whereas longitudinal instability is influenced by the SCW frequency correction, and, in parameter regions of physical interest there may be more than one unstable mode. With explicit results, a critique of existing results in the literature can be given, and several errors and misconceptions in previous work are pointed out. The theory is developed in some generality for Hamiltonian PDEs. Water waves and a nonlinear wave equation in two space dimensions are used for illustration of the theory.

TJ Bridges, MD Groves, DP Nicholls (2016)Lectures on the Theory of Water Waves Cambridge University Press
This book is an ideal introduction for PhD students and researchers looking for a research project. It may also be used as a supplementary text for advanced courses in mathematics or fluid dynamics.
TJ Bridges (2014)Bifurcation from rolls to multi-pulse planforms via reduction to a parabolic Boussinesq model, In: Physica D: Nonlinear Phenomena275pp. 8-18
A mechanism is presented for the bifurcation from one-dimensional spatially periodic patterns (rolls) into two-dimensional planar states (planforms). The novelty is twofold: the planforms are solutions of a Boussinesq partial differential equation (PDE) on a periodic background and secondly explicit formulas for the coefficients in the Boussinesq equation are derived, based on a form of planar conservation of wave action flux. The Boussinesq equation is integrable with a vast array of solutions, and an example of a new planform bifurcating from rolls, which appears to be generic, is presented. Adding in time leads to a new time-dependent PDE, which models the nonlinear behaviour emerging from a generalization of Eckhaus instability. The class of PDEs to which the theory applies is evolution equations whose steady part is a gradient elliptic PDE. Examples are the 2+1 Ginzburg-Landau equation with real coefficients, and the 2+1 planar Swift-Hohenberg equation. © 2014 Elsevier B.V. All rights reserved.
H. Alemi Ardakani, T.J. Bridges, F. Gay-Balmaz, Y.H. Huang, C. Tronci (2019)A variational principle for fluid sloshing with vorticity, dynamically coupled to vessel motion, In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences Royal Society
A variational principle is derived for two-dimensional incompressible rotational fluid flow with a free surface in a moving vessel when both the vessel and fluid motion are to be determined. The fluid is represented by a stream function and the vessel motion is represented by a path in the planar Euclidean group. Novelties in the formulation include how the pressure boundary condition is treated, the introduction of a stream function into the Euler- Poincaré variations, the derivation of free surface variations, and how the equations for the vessel path in the Euclidean group, coupled to the fluid motion, are generated automatically.
TJ Bridges (2006)Canonical multi-symplectic structure on the total exterior algebra bundle, In: Proceedings Royal Society London A462(2069)pp. 1531-1551 ROYAL SOCIETY
The aim of this paper is to construct multi-symplectic structures starting with the geometry of an oriented Riemannian manifold, independent of a Lagrangian or a particular partial differential equation (PDE). The principal observation is that on an n-dimensional orientable manifold All there is a canonical quadratic form Theta associated with the total exterior algebra bundle on M. On the fibre, which has dimension 2(n), the form Theta can be locally decomposed into n classical symplectic structures. When concatenated, these n-symplectic structures define a partial differential operator, J(partial derivative), which turns out to be a Dirac operator with multi-symplectic structure. The operator J(partial derivative) generalizes the product operator J(d/dt) in classical symplectic geometry, and M is a generalization of the base manifold (i.e. time) in classical Hamiltonian dynamics. The, structure generated by 19 provides a natural setting for analysing a class of covariant nonlinear gradient, elliptic operators. The operator J(partial derivative) is elliptic, and the generalization of Hamiltonian systems, J(partial derivative)Z=del S(Z), for a section Z of the total exterior algebra bundle, is also an elliptic PDE. The inverse problem-find S(Z) for a given elliptic PDE-is shown to be related to a variant of the Legendre transform on k-forms. The theory is developed for flat base manifolds, but the constructions are coordinate free and generalize to Riemannian manifolds with non-trivial curvature. Some applications and implications of the theory are also discussed.
TJ Bridges, NM Donaldson (2010)Variational principles for water waves from the viewpoint of a time-dependent moving mesh, In: Mathematika57pp. 147-173 University College London
The time-dependent motion of water waves with a parametrically defined free surface is mapped to a fixed time-independent rectangle by an arbitrary transformation. The emphasis is on the general properties of transformations. Special cases are algebraic transformations based on transfinite interpolation, conformal mappings, and transformations generated by nonlinear elliptic partial differential equations. The aim is to study the effect of transformation on variational principles for water waves such as Luke’s Lagrangian formulation, Zakharov’s Hamiltonian formulation, and the Benjamin–Olver Hamiltonian formulation. Several novel features are exposed using this approach: a conservation law for the Jacobian, an explicit form for surface re-parameterization, inner versus outer variations and their role in the generation of hidden conservation laws of the Laplacian. Also some of the differential geometry of water waves becomes explicit. The paper is restricted to the case of planar motion, with a preliminary discussion of the extension to three-dimensional water waves.
T J Bridges, N M Donaldson (2005)Degenerate Periodic Orbits and Homoclinic Torus Bifurcation, In: Physical Review Letters95(10)

A one-parameter family of periodic orbits with frequency omega and energy E of an autonomous Hamiltonian system is degenerate when E-'(omega)=0. In this paper, new features of the nonlinear bifurcation near this degeneracy are identified. A new normal form is found where the coefficient of the nonlinear term is determined by the curvature of the energy-frequency map. An important property of the bifurcating "homoclinic torus" is the homoclinic angle and a new asymptotic formula for it is derived. The theory is constructive, and so is useful for physical applications and in numerics.

TJ Bridges, PE Hydon, JK Lawson (2010)Multisymplectic structures and the variational bicomplex, In: MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY148pp. 159-178 CAMBRIDGE UNIV PRESS
Andrei L. Afendikov, Thomas J. Bridges (2001)Instability of the Hocking-Stewartson pulse and its implications for three-dimensional Poiseuille flow, In: Proceedings of the Royal Society of London A457pp. 257-272

The linear stability problem for the Hocking-Stewartson pulse, obtained by linearizing the complex Ginzburg-Landau (cGL) equation, is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to stability exponents. A numerical algorithm based on the compound matrix method is developed for computing the Evans function. Using values in the cGL equation associated with spanwise modulation of plane Poiseuille flow, we show that the Hocking-Stewartson pulse associated with points along the neutral curve is always linearly unstable due to a real positive eigenvalue. Implications for the spanwise structure of nonlinear Poiseuille problem between parallel plates are also discussed.

TJ Bridges, S Reich (2006)Numerical methods for Hamiltonian PDEs, In: JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL39(19)pp. 5287-5320 IOP PUBLISHING LTD
Thomas Bridges, Daniel Ratliff (2016)Double criticality and the two-way Boussinesq equation in stratifed shallow water hydrodynamics, In: Physics of Fluids28(6)062103 American Institute of Physics
Double criticality and its nonlinear implications are considered for stratified N–layer shallow water flows with N = 1,  2,  3. Double criticality arises when the linearization of the steady problem about a uniform flow has a double zero eigenvalue. We find that there are two types of double criticality: non-semisimple (one eigenvector and one generalized eigenvector) and semi-simple (two independent eigenvectors). Using a multiple scales argument, dictated by the type of singularity, it is shown that the weakly nonlinear problem near double criticality is governed by a two-way Boussinesq equation (non-semisimple case) and a coupled Korteweg-de Vries equation (semisimple case). Parameter values and reduced equations are constructed for the examples of two-layer and three-layer stratified shallow water hydrodynamics.
L. Brevdo, Thomas J. Bridges (1997)Local and global instabilities of spatially developing flows: cautionary examples, In: Proceedings of the Royal Society of London A453pp. 1345-1364

In the analysis of the linear stability of basic states in fluid mechanics that are slowly varying in space, the quasi-homogeneous hypothesis is often invoked, where the stability exponents are defined locally and treated as slowly varying functions of a spatial coordinate. The set of local stability exponents is then used to predict the global perturbation dynamics and an implicit hypothesis is that the local analysis provides at least a conservative estimate of the global stability properties of the flow. In this paper cautionary examples are presented that demonstrate a contradiction between the results of the local and global analyses. For example, a local analysis may predict stability everywhere even when the exact PDE with non-constant coefficients is ill-posed, demonstrating that global stability exponents are not, in general, bounded by the maximal local stability exponents. A key observation in this paper is the importance of distinguishing between the discrete spectrum and the continuous spectrum when comparing global and local stability exponents. This distinction is particularly significant for spatially periodic flows where, for the global flow, only the continuous spectrum is present and, hence, instability arises always in the absence of discrete spectra. New exact definitions for global absolute and convective instabilities are also given for a class of spatially periodic basic states and applied to an example based on the complex Ginzburg–Landau equation. The consequences of this example, and of the argument involved for basic states that are slowly varying in space but non-periodic, and for some problems in fluid mechanics are also presented.

MR Turner, H Alemi Ardakani, TJ Bridges (2015)Instability of sloshing motion in a vessel undergoing pivoted oscillations, In: Journal of Fluids and Structures
Suspending a rectangular vessel partially filled with an inviscid fluid from a single rigid pivoting rod produces an interesting physical model for investigating the dynamic coupling between the fluid and vessel motion. The fluid motion is governed by the Euler equations relative to the moving frame of the vessel, and the vessel motion is given by a modified forced pendulum equation. The fully nonlinear, two-dimensional, equations of motion are derived and linearised for small-amplitude vessel and free-surface motions, and the natural frequencies of the system analysed. It is found that the linear problem exhibits an unstable solution if the rod length is shorter than a critical length which depends on the length of the vessel, the fluid height and the ratio of the fluid and vessel masses. In addition, we identify the existence of 1:1 resonances in the system where the symmetric sloshing modes oscillate with the same frequency as the coupled fluid/vessel motion. The implications of instability and resonance on the nonlinear problem are also briefly discussed.
MR Turner, TJ Bridges, H Alemi Ardakani (2015)The pendulum-slosh problem: simulation using a time-dependent conformal mapping, In: Journal of Fluids and Structures59pp. 202-223 Elsevier
Suspending a rectangular vessel which is partially filled with fluid from a single rigid pivoting pole produces an interesting theoretical model with which to investigate the dynamic coupling between fluid motion and vessel rotation. The exact equations for this coupled system are derived with the fluid motion governed by the Euler equations relative to the moving frame of the vessel, and the vessel motion governed by a modified forced pendulum equation. The nonlinear equations of motion for the fluid are solved numerically via a time-dependent conformal mapping, which maps the physical domain to a rectangle in the computational domain with a time dependent conformal modulus. The numerical scheme expresses the implicit free-surface boundary conditions as two explicit partial differential equations which are then solved via a pseudo-spectral method in space. The coupled system is integrated in time with a fourth-order Runge-Kutta method. The starting point for the simulations is the linear neutral stability contour discovered by Turner, Alemi Ardakani ____& Bridges (2014, {____it J. Fluid Struct.} {____bf 52}, 166-180). Near the contour the nonlinear results confirm the instability boundary, and far from the neutral curve (parameterised by longer pole lengths) nonlinearity is found to significantly alter the vessel response. Results are also presented for an initial condition given by a superposition of two sloshing modes with approximately the same frequency from the linear characteristic equation. In this case the fluid initial conditions generate large nonlinear vessel motions, which may have implications for systems designed to oscillate in a confined space or on the slosh-induced-rolling of a ship.
TJ Bridges (2000)Universal Geometric Condition for the Transverse Instability of Solitary Waves, In: Physical Review Letters8412pp. 2614-2617 American Physical Society
Transverse instabilities correspond to a class of perturbations traveling in a direction transverse to the direction of the basic solitary wave. Solitary waves traveling in one space direction generally come in one-parameter families. We embed them in a two-parameter family and deduce a new geometric condition for transverse instability of solitary waves. This condition is universal in the sense that it does not require explicit properties of the solitary wave—or the governing equation. In this paper the basic idea is presented and applied to the Zakharov-Kuznetsov equation for illustration. An indication of how the theory applies to a large class of equations in physics and oceanography is also discussed.
TJ Bridges, DJ Needham (2011)Breakdown of the shallow water equations due to growth of the horizontal vorticity, In: JOURNAL OF FLUID MECHANICS679pp. 655-666 CAMBRIDGE UNIV PRESS
Thomas Bridges, Daniel Ratliff (2018)Nonlinear modulation near the Lighthill instability threshold in 2+1 Whitham theory, In: Philosophical Transactions A376(2117) The Royal Society
The dispersionless Whitham modulation equations in 2+1 (two space dimensions and time) are reviewed and the instabilities identified. The modulation theory is then reformulated, near the Lighthill instability threshold, with a slow phase, moving frame, and different scalings. The resulting nonlinear phase modulation equation near the Lighthill surfaces is a geometric form of the 2+1 two-way Boussinesq equation. This equation is universal in the same sense as Whitham theory. Moreover, it is dispersive, and it has a wide range of interesting multiperiodic, quasiperiodic and multi-pulse localized solutions. For illustration the theory is applied to a complex nonlinear 2+1 Klein-Gordon equation which has two Lighthill surfaces in the manifold of periodic travelling waves.
Matthew Turner, Thomas Bridges, H Alemi Ardakani (2017)Lagrangian particle path formulation of multilayer shallow-water flows dynamically coupled to vessel motion, In: Journal of Engineering Mathematics106(1)pp. 75-106 Springer Verlag
The coupled motion, between multiple inviscid, incompressible, immiscible fluid layers in a rectangular vessel with a rigid lid and the vessel dynamics, is considered. The fluid layers are assumed to be thin and the shallow-water assumption is applied. The governing form of the Lagrangian functional in the Lagrangian Particle Path (LPP) framework is derived for an arbitrary number of layers, while the corresponding Hamiltonian is explicitly derived in the case of two- and three-layer fluids. The Hamiltonian formulation has nice properties for numerical simulations, and a fast, effective and symplectic numerical scheme is presented in the two- and three-layer cases, based upon the implicit-midpoint rule. Results of the simulations are compared with linear solutions and with the existing results of Alemi Ardakani, Bridges & Turner [1] (J. Fluid Struct. 59 432-460) which were obtained using a finite volume approach in the Eulerian representation. The latter results are extended to non-Boussinesq regimes. The advantages and limitations of the LPP formulation and variational discretization are highlighted.
Thomas J. Bridges (2019)The Pressure Boundary Condition and the Pressure as Lagrangian for Water Waves, In: Water Waves: An interdisciplinary journal Springer Nature
The pressure boundary condition for the full Euler equations with a free surface and general vorticity field is formulated in terms of a generalized Bernoulli equation deduced from the Gavrilyuk–Kalisch–Khorsand conservation law. The use of pressure as a Lagrangian density, as in Luke’s variational principle, is reviewed and extension to a full vortical flow is attempted with limited success. However, a new variational principle for time-dependent water waves in terms of the stream function is found. The variational principle generates vortical boundary conditions but with a harmonic stream function. Other aspects of vorticity in variational principles are also discussed.

Several new results on the bifurcation and instability of nonlinear periodic travelling waves, at the interface between two fluids in relative motion, in a parametric neighbourhood of a Kelvin–Helmholtz unstable equilibrium are presented. The organizing centre for the analysis is a canonical Hamiltonian formulation of the Kelvin–Helmholtz problem presented in Part 1. When the density ratio of the upper and lower fluid layers exceeds a critical value, and surface tension is present, a pervasive superharmonic instability is found, and as u [rightward arrow] u0, where u is the velocity difference between the two layers and u0 is the Kelvin–Helmholtz threshold, the amplitude at which the superharmonic instability occurs scales like (u0 - u)1/2 with u > u0. Other results presented herein include (a) new results on the structure of the superharmonic instability, (b) the discovery of isolated branches and intersecting branches of travelling waves near a critical density ratio, (c) the appearance of Benjamin–Feir instability along branches of waves near the Kelvin–Helmholtz instability threshold and (d) the interaction between the Kelvin–Helmholtz, superharmonic and Benjamin–Feir instability at low amplitude.

F Chardard, F Dias, TJ Bridges (2006)Fast computation of the Maslov index for hyperbolic linear systems with periodic coefficients, In: JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL39(47)pp. 14545-14557 IOP PUBLISHING LTD
MR Turner, TJ Bridges (2013)Nonlinear energy transfer between fluid sloshing and vessel motion, In: Journal of Fluid Mechanics719pp. 606-636
This paper examines the dynamic coupling between a sloshing fluid and the motion of the vessel containing the fluid. A mechanism is identified which leads to an energy exchange between the vessel dynamics and fluid motion. It is based on a 1:1 resonance in the linearized equations, but nonlinearity is essential for the energy transfer. For definiteness, the theory is developed for Cooker's pendulous sloshing experiment. The vessel has a rectangular cross section, is partially filled with a fluid, and is suspended by two cables. A nonlinear normal form is derived close to an internal 1:1 resonance, with the energy transfer manifested by a heteroclinic connection which connects the purely symmetric sloshing modes to the purely anti-symmetric sloshing modes. Parameter values where this pure energy transfer occurs are identified. In practice, this energy transfer can lead to sloshing-induced destabilization of fluid-carrying vessels.
T. J. Bridges, Fiona E. Laine-Pearson (2004)Nonlinear Counterpropagating Waves, Multisymplectic Geometry, and the Instability of Standing Waves, In: the SIAM Journal on Applied Mathematics64pp. 2096-2120

Standing waves are a fundamental class of solutions of nonlinear wave equations with a spatial reflection symmetry, and they routinely arise in optical and oceanographic applications. At the linear level they are composed of two synchronized counterpropagating periodic traveling waves. At the nonlinear level, they can be defined abstractly by their symmetry properties. In this paper, general aspects of the modulational instability of standing waves are considered. This problem has difficulties that do not arise in the modulational instability of traveling waves. Here we propose a new geometric formulation for the linear stability problem, based on embedding the standing wave in a four-parameter family of nonlinear counterpropagating waves. Multisymplectic geometry is shown to encode the stability properties in an essential way. At the weakly nonlinear level we obtain the surprising result that standing waves are modulationally unstable only if the component traveling waves are modulation unstable. Systems of nonlinear wave equations will be used for illustration, but general aspects will be presented, applicable to a wide range of Hamiltonian PDEs, including water waves.

T.J. Bridges, F. E. Laine-Pearson (2005)The long-wave instability of short-crested waves, via embedding in the oblique two-wave interaction, In: Journal of Fluid Mechanics543pp. 147-182

The motivation for this work is the stability problem for short-crested Stokes waves. A new point of view is proposed, based on the observation that an understanding of the linear stability of short-crested waves (SCWs) is closely associated with an understanding of the stability of the oblique non-resonant interaction between two waves. The proposed approach is to embed the SCWs in a six-parameter family of oblique non-resonant interactions. A variational framework is developed for the existence and stability of this general two-wave interaction. It is argued that the resonant SCW limit makes sense a posteriori, and leads to a new stability theory for both weakly nonlinear and finite-amplitude SCWs. Even in the weakly nonlinear case the results are new: transverse weakly nonlinear long-wave instability is independent of the nonlinear frequency correction for SCWs whereas longitudinal instability is influenced by the SCW frequency correction, and, in parameter regions of physical interest there may be more than one unstable mode. With explicit results, a critique of existing results in the literature can be given, and several errors and misconceptions in previous work are pointed out. The theory is developed in some generality for Hamiltonian PDEs. Water waves and a nonlinear wave equation in two space dimensions are used for illustration of the theory.

The theory for criticality presented in Part I is extended to the unsteady problem, and a new formulation of the Benjamin-Feir instability for Stokes waves in finite depth coupled to a mean flow, which takes the criticality matrix as an organizing centre, is presented. The generation of unsteady dark solitary waves at points of stability changes and their connection with the steady dark solitary waves of Part I are also discussed.

TJ Bridges, DJ Ratliff (2015)Phase dynamics of periodic waves leading to the Kadomtsev-Petviashvili equation in 3+1 dimensions, In: Proceedings of the Royal Society of London: Mathematical, Physical and Engineering Sciences471(2178) The Royal Society
The Kadomstev-Petviashvili (KP) equation is a well-known modulation equation normally derived by starting with the trivial state and an appropriate dispersion relation. In this paper it is shown that the KP equation is also the relevant modulation equation for bifurcation from periodic travelling waves when the wave action flux has a critical point. Moreover, the emergent KP equation arises in a universal form, with the coefficients determined by the components of the conservatio n of wave action. The theory is derived for a general class of partial differential equations generated by a Lagrangian using phase modulation. The theory extends to any space dimension and time, but the emphasis in the paper is on the ca se of 3+1. Motivated by light bullets and quantum vortex dynamics, the theory is illustrated by showing how defocussing NLS in 3+1 bifurcates to KP in 3+1 at criticality. The generalization to $N>3$ is also discussed.
Thomas Bridges, J Pennant, Sergey Zelik (2014)Degenerate Hyperbolic Conservation Laws with Dissipation: Reduction to and Validity of a Class of Burgers-Type Equations, In: ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS214(2)pp. 671-716 SPRINGER
A conservation law is said to be degenerate or critical if the Jacobian of the flux vector evaluated on a constant state has a zero eigenvalue. In this paper, it is proved that a degenerate conservation law with dissipation will generate dynamics on a long time scale that resembles Burger’s dynamics. The case of k-fold degeneracy is also treated, and it is shown that it leads to a reduction to a quadratically coupled k-fold system of Burgers-type equations. Validity of the reduction and existence for the reduced system is proved in the class of uniformly local spaces, thereby capturing both finite and infinite energy solutions. The theory is applied to some examples, from stratified shallow-water hydrodynamics, that model the birth of hydraulic jumps.
T. J. Bridges (2001)Transverse instability of solitary-wave states of the water-wave problem, In: Journal of Fluid Mechanics

Transverse stability and instability of solitary waves correspond to a class of perturbations that are travelling in a direction transverse to the direction of the basic solitary wave. In this paper we consider the problem of transverse instability of solitary waves for the water-wave problem, from both the model equation point of view and the full water-wave equations. A new universal geometric condition for transverse instability forms the backbone of the analysis. The theory is first illustrated by application to model PDEs for water waves such as the KP equation, and then it is applied to the full water-wave problem. This is the first theory proposed for transverse instability of solitary waves of the full water-wave problem. The theory suggests the introduction of a new functional for water waves, whose importance is suggested by the mathematical structure. Without explicit calculation, the theory is used to argue that the basic class of solitary waves of the water-wave problem, which bifurcate at Froude number unity, are likely to be stable to transverse perturbations, even at large amplitude.

Thomas J. Bridges (2000)Universal Geometric Condition for the Transverse Instability of Solitary Waves, In: Physical Review Letters84pp. 2614-2617

Transverse instabilities correspond to a class of perturbations traveling in a direction transverse to the direction of the basic solitary wave. Solitary waves traveling in one space direction generally come in one-parameter families. We embed them in a two-parameter family and deduce a new geometric condition for transverse instability of solitary waves. This condition is universal in the sense that it does not require explicit properties of the solitary wave—or the governing equation. In this paper the basic idea is presented and applied to the Zakharov-Kuznetsov equation for illustration. An indication of how the theory applies to a large class of equations in physics and oceanography is also discussed.

Thomas J. Bridges, Sebastian Reich (2001)Computing Lyapunov exponents on a Stiefel manifold, In: Physica D156pp. 219-238

The problem of numerical computation of a few Lyapunov exponents (LEs) of finite-dimensional dynamical systems is considered from the viewpoint of the differential geometry of Stiefel manifolds. Whether one computes one, many or all LEs of a continuous dynamical system by time integration, discrete or continuous orthonormalization is essential for stable numerical integration. A differential-geometric view of continuous orthogonalization suggests that one restricts the linearized vectorfield to a Stiefel manifold. However, the Stiefel manifold is not in general an attracting submanifold of the ambient Euclidean space: it is a constraint manifold with a weak numerical invariant. New numerical algorithms for this problem are then designed which use the fiber-bundle characterization of these manifolds. This framework leads to a new class of systems for continuous orthogonalization which have strong numerical invariance properties and the strong skew-symmetry property. Numerical integration of these new systems with geometric integrators leads to a new class of numerical methods for computing a few LEs which preserve orthonormality to machine accuracy. This idea is also taken a step further by making the Stiefel manifold an attracting invariant manifold in which case standard explicit Runge–Kutta algorithms can be used. This leads to an algorithm which requires only marginally more computation than a standard explicit integration without orthogonalization. These class of methods should be particularly effective for computing a few LEs for large-dimension dynamical systems. The new schemes are straightforward to implement. A test case is presented for illustration, and an example from dynamical systems is presented where a few LEs are computed for an array of coupled oscillators.

Tom Bridges, Anna Kostianko, Guido Schneider (2020)A proof of validity for multiphase Whitham modulation theory, In: Proceedings of the Royal Society of London. Series A, Containing papers of a mathematical and physical character.476(2243) The Royal Society Publishing
Proc. Roy. Soc. Lond. A (2020) It is proved that approximations which are obtained as solutions of the multiphase Whitham modulation equations stay close to solutions of the original equation on a natural time scale. The class of nonlinear wave equations chosen for the starting point is coupled nonlinear Schr\"odinger equations. These equations are not in general integrable, but they have an explicit family of multiphase wavetrains that generate multiphase Whitham equations which may be elliptic,hyperbolic, or of mixed type. Due to the change of type, the function space setup is based on Gevrey spaces with initial data analytic in a strip in the complex plane. In these spaces a Cauchy- Kowalevskaya-like existence and uniqueness theorem is proved. Building on this theorem and higher-order approximations to Whitham theory, a rigorous comparison of solutions, of the coupled nonlinear Schr\"odinger equations and the multiphase Whitham modulation equations, is obtained.
D Avitabile, TJ Bridges (2010)Numerical implementation of complex orthogonalization, parallel transport on Stiefel bundles, and analyticity, In: PHYSICA D-NONLINEAR PHENOMENA239(12)pp. 1038-1047 ELSEVIER SCIENCE BV
HA Ardakani, TJ Bridges, MR Turner (2015)Dynamic coupling between horizontal vessel motion and two-layer shallow-water sloshing, In: JOURNAL OF FLUIDS AND STRUCTURES59pp. 432-460 ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD
TJ Bridges, NM Donaldson (2009)Criticality manifolds and their role in the generation of solitary waves for two-layer flow with a free surface, In: EUROPEAN JOURNAL OF MECHANICS B-FLUIDS28(1)pp. 117-126 GAUTHIER-VILLARS/EDITIONS ELSEVIER
TJ Bridges (2008)Conservation laws in curvilinear coordinates: A short proof of Vinokur's Theorem using differential forms, In: APPLIED MATHEMATICS AND COMPUTATION202(2)pp. 882-885 ELSEVIER SCIENCE INC
TJ Bridges, NM Donaldson (2007)Reappraisal of criticality for two-layer flows and its role in the generation of internal solitary waves, In: PHYSICS OF FLUIDS19(7)ARTN 0pp. ?-? AMER INST PHYSICS
MR Turner, TJ Bridges, H Alemi Ardakani (2013)Dynamic coupling in Cooker's sloshing experiment with baffles., In: Phys Fluids25112102 American Institute of Physics
This paper investigates the dynamic coupling between fluid sloshing and the motion of the vessel containing the fluid, for the case when the vessel is partitioned using non-porous baffles. The vessel is modelled using Cooker's sloshing configuration [M. J. Cooker, “Water waves in a suspended container,” Wave Motion20, 385–395 (1994)]. Cooker's configuration is extended to include n − 1 non-porous baffles which divide the vessel into n separate fluid compartments each with a characteristic length scale. The problem is analysed for arbitrary fill depth in each compartment, and it is found that a multitude of resonance situations can occur in the system, from 1 : 1 resonances to (n + 1)−fold 1 : 1: ⋯ : 1 resonances, as well as ℓ: m: ⋯ : n for natural numbers ℓ, m, n, depending upon the system parameter values. The conventional wisdom is that the principle role of baffles is to damp the fluid motion. Our results show that in fact without special consideration, the baffles can lead to enhancement of the fluid motion through resonance.
H Alemi Ardakani, TJ Bridges (2010)Dynamic coupling between shallow-water sloshing and horizontal vehicle motion, In: European Journal of Applied Mathematics21(6)pp. 479-517
Matthew Turner, Tom Bridges (2019)Wetting and drying of two-layer shallow-water flow in a moving vessel via semi-Lagrangian simulation, In: European Journal of Mechanics B - Fluids76pp. 153-165 Elsevier
The sloshing motion of two inviscid, incompressible, immiscible, shallowwater fluid layers in a rectangular vessel with a rigid lid is considered. The vessel is forced to oscillate in a horizontal rectilinear motion, at the lowest resonant frequency of the system, such that the lower fluid hits the rigid lid causing a wetting-drying scenario. The two-layer shallow-water equations are solved using a conservative quasi-monotone semi-Lagrangian scheme which conserves the mass of the system. Two examples are considered: a low-fill example where the lower fluid does not interact with the upper-rigid lid, in order to validate the code against an existing Lagrangian particle path scheme, and a high-fill example where there is wetting and drying of the upper rigid lid. In both examples, results of the semi-Lagrangian simulations are compared to existing experimental data and good agreement is obtained.
TJ Bridges (2008)Degenerate relative equilibria, curvature of the momentum map, and homoclinic bifurcation, In: JOURNAL OF DIFFERENTIAL EQUATIONS244(7)pp. 1629-1674 ACADEMIC PRESS INC ELSEVIER SCIENCE
H Alemi Ardakani, TJ Bridges (2010)Shallow-water sloshing in vessels undergoing prescribed rigid-body motion in three dimensions, In: Journal of Fluid Mechanicsin press Cambridge University Press
New shallow-water equations, for sloshing in three dimensions (two horizontal and one vertical) in a vessel which is undergoing rigid-body motion in 3-space, are derived. The rigid-body motion of the vessel (roll-pitch-yaw and/or surge-sway-heave) is modelled exactly and the only approximations are in the fluid motion. The flow is assumed to be inviscid but vortical, with approximations on the vertical velocity and acceleration at the surface. These equations improve previous shallow-water models. The model also extends to three dimensions the essence of the Penney-Price-Taylor theory for the highest standing wave. The surface shallow-water equations are simulated using a split-step alternating direction implicit finite-difference scheme. Numerical experiments are reported, including comparisons with existing results in the literature, and simulations with vessels undergoing full three-dimensional rotations.
F Dias, TJ Bridges (2006)The numerical computation of freely propagating time-dependent irrotational water waves, In: FLUID DYNAMICS RESEARCH38(12)pp. 803-830 ELSEVIER SCIENCE BV
H Alemi Ardakani, TJ Bridges (2012)Shallow-water sloshing in vessels undergoing prescribed rigid-body motion in two dimensions, In: European Journal of Mechanics, B/Fluids31(1)pp. 30-43 Elsevier
New shallow-water equations, for sloshing in two dimensions (one horizontal and one vertical) in a vessel which is undergoing rigid-body motion in the plane, are derived. The planar motion of the vessel (pitch-surge-heave or roll-sway-heave) is exactly modelled and the only approximations are in the fluid motion. The flow is assumed to be inviscid but vortical, with approximations on the vertical velocity and acceleration at the surface. These equations improve previous shallow water models for sloshing. The model also contains the essence of the Penney-Price-Taylor theory for the highest standing wave. The surface shallow water equations are simulated using a robust implicit finite-difference scheme. Numerical experiments are reported, including simulations of coupled translation-rotation forcing, and sloshing on a ferris wheel. Asymptotic results confirm that rotations should be of order h/L where h is the mean depth and L the vessel length, but translations can be of order unity, in the shallow water limit.
H Alemi Ardakani, Thomas Bridges, Matthew Turner (2012)Resonance in a model for Cooker's sloshing experiment, In: European Journal of Mechanics B - Fluids36(Nov)pp. 25-38 Elsevier
Cooker's sloshing experiment is a prototype for studying the dynamic coupling between fluid sloshing and vessel motion. It involves a container, partially filled with fluid, suspended by two cables and constrained to remain horizontal while undergoing a pendulum-like motion. In this paper the fully-nonlinear equations are taken as a starting point, including a new derivation of the coupled equation for vessel motion, which is a forced nonlinear pendulum equation. The equations are then linearized and the natural frequencies studied. The coupling leads to a highly nonlinear transcendental characteristic equation for the frequencies. Two derivations of the characteristic equation are given, one based on a cosine expansion and the other based on a class of vertical eigenfunctions. These two characteristic equations are compared with previous results in the literature. Although the two derivations lead to dramatically different forms for the characteristic equation, we prove that they are equivalent. The most important observation is the discovery of an internal $1:1$ resonance in the fully two-dimensional finite depth model, where symmetric fluid modes are coupled to the vessel motion. Numerical evaluation of the resonant and nonresonant modes are presented. The implications of the resonance for the fluid dynamics, and for the nonlinear coupled dynamics near the resonance are also briefly discussed.
Thomas Bridges, Daniel Ratliff (2017)On the elliptic-hyperbolic transition in whitham modulation theory, In: SIAM Journal on Applied Mathematics77(6)pp. 1989-2011 Society for Industrial and Applied Mathematics
The dispersionless Whitham modulation equations in one space dimension and time are generically hyperbolic or elliptic, and breakdown at the transition, which is a curve in the frequency-wavenumber plane. In this paper, the modulation theory is reformulated with a slow phase and different scalings resulting in a phase modulation equation near the singular curves which is a geometric form of the two-way Boussinesq equation. This equation is universal in the same sense as Whitham theory. Moreover, it is dispersive, and it has a wide range of interesting multiperiodic, quasiperiodic and multi-pulse localized solutions. This theory shows that the elliptic-hyperbolic transition is a rich source of complex behaviour in nonlinear wave fields. There are several examples of these transition curves in the literature to which the theory applies. For illustration the theory is applied to the complex nonlinear Klein-Gordon equation which has two singular curves in the manifold of periodic travelling waves.
T J Bridges (2005)Steady Dark Solitary Waves Emerging from Wave-Generated Meanflow: The Role of Modulation Equations, In: Chaos: An Interdisciplinary Journal of Nonlinear Science15(3)

Various classes of steady and unsteady dark solitary waves (DSWs) are known to exist in modulation equations for water waves in finite depth. However, there is a class of steady DSWS of the full water-wave problem which are missed by the classical modulation equations such as the Hasimoto-Ono, Benney-Roskes, and Davey-Stewartson. These steady DSWs, recently discovered by Bridges and Donaldson, are pervasive in finite depth, arise through secondary criticality of Stokes gravity waves, and are synchronized with the Stokes wave. In this paper, the role of DSWs in modulation equations for water waves is reappraised. The intrinsic unsteady nature of existing modulation equations filters out some interesting solutions. On the other hand, the geometry of DSWs in modulation equations is very similar to the full water wave problem and these geometrical properties are developed. A model equation is proposed which illustrates the general nature of the emergence of steady DSWs due to wave-generated mean flow coupled to a periodic wave. Although the existing modulation equations are intrinsically unsteady, it is shown that there are also important shortcomings when one wants to use them for stability analysis of DSWs.

The stability properties of line solitary wave solutions of the (2+1)-dimensional Boussinesq equation with respect to transverse perturbations and their consequences are considered. A geometric condition arising from a multisymplectic formulation of this equation gives an explicit relation between the parameters for transverse instability when the transverse wave number is small. The Evans function is then computed explicitly, giving the eigenvalues for the transverse instability for all transverse wave numbers. To determine the nonlinear and long-time implications of the transverse instability, numerical simulations are performed using pseudospectral discretization. The numerics confirm the analytic results, and in all cases studied, the transverse instability leads to collapse.

The symmetry properties of quantum variational principles are considered. Euler-Poincaré reduction theory is applied to the Dirac-Frenkel variational principle for Schrödinger's dynamics producing new variational principles for the different pictures of quantum mechanics (Schrödinger's, Heisenberg's and Dirac's) as well as for the Wigner-Moyal formulation. In addition, new variational principles for mixed states dynamics have been formulated. The already known geometric characterization of quantum mechanics on the complex projective space is shown to emerge naturally from the Euler-Poincaré variational principle. Semidirect-product structures are seen to produce new variational principles for Dirac's picture. On the other hand, the variational and Hamiltonian approach to Ehrenfest expectation values dynamics is proposed. In the Lagrangian framework, the quantum variational principle for Schrödinger's dynamics is extended to account for both classical and quantum degrees of freedom. First, it is shown that the mean field model of any quantum mechanical system can be derived from a classical-quantum Euler-Poincaré Lagrangian on the direct sum Lie algebra of the Heisenberg and unitary groups. Then, the semidirect-product structure (named Ehrenfest group), is constructed using the displacement operator from the theory of coherent quantum states ( the unitary action of the Heisenberg group on the space of wavefunctions). New classical-quantum equations for Ehrenfest's expectation values dynamics are derived redefining the mean-field model Euler-Poincaré Lagrangian on the Lie algebra of the Ehrenfest group. In the Hamiltonian framework, first expectation values of the canonical observables are shown to be equivariant momentum maps for the unitary action of the Heisenberg group on quantum states. Then, the Hamiltonian structure for Ehrenfest's dynamics is shown to be Lie-Poisson for the Ehrenfest group. The variational formulation is then given a corresponding Hamiltonian structure. The classical-quantum Ehrenfest dynamics equations produce classical and quantum dynamics as special limit cases. In the particular case of Gaussian states, expectation values couple to second order moments, so that GS are completely characterized by first and second moments. When the total energy is computed with respect to a Gaussian state, higher moments can be expressed in terms of the first two, so that the moment hierarchy closes for Gaussian states. Second moments are shown to be equivariant momentum maps for the action of the symplectic group on the space of Wigner functions. Eventually, Gaussian states are shown to possess a Lie-Poisson structure on the Jacobi group. This structure produces an energy-conserving variant of a class of Gaussian moment models that have appeared in the chemical physics literature.
Phase modulation has been a tool used for many years to describe system behaviour about periodic wavetrain solutions. The method has been shown to lead to well known partial differential equations (PDEs) in many contexts, describing how slowly varying perturbations to the phase of the solution behave. Typically these calculations can become involved and analyses are generally only valid for the one system consider and results cannot be transferred. This thesis discusses a format in which phase modulation leads to PDEs that emerge in a universal format, that is with coefficients determined by the wavetrain itself and the geometric properties of the system. The transition from simpler models emerging from the phase dynamics to more complex models is marked through the criticality of derivatives of the conservation laws associated with the system evaluated along the wavetrain considered. The analysis is also very general, so that the asymptotics need only be done once to apply to a large family of physical problems. We begin by discussing this method for single phase wavetrains, or more generally relative equilibrium, in the context of systems generated by a Lagrangian. By considering the modulation of single phased relative equilibrium for suitable scales, the Whitham equations are shown to emerge with a form explicitly related to the system's conservation laws. Furthermore, when the linearisation of the Whitham system develops zero characteristics (referred to as criticality), it can be shown that the modulation leads to the universal emergence of the Kadomtsev- Petviashvili (KP) and Boussinesq models. These models have coefficients which take the form of derivatives of the conservation law components evaluated on the relative equilibrium solution, as well as those that arise from a Jordan chain analysis. Additional degeneracies are shown to lead to fifth order KP and modified KP models of a similar universal form. The discussion is then shifted to the case of relative equilibria generated by multiple symmetries in 1 + 1 dimensions. It will be demonstrated how the results of the single phase case generalise naturally to additional phases, with criticality moving from scalar conditions to those of tensors. By undertaking similar analyses to the single phase, one obtains the multiphase Whitham equations, and depending on the number of zero characteristics its linearisation possesses the scalar Korteweg-de Vries (KdV), two-way Boussinesq and a new equation emerge instead. These analyses are supplemented with examples in stratified hydrodynamics and a coupled Nonlinear Schrodinger model. The final part of this thesis focusses on criticality conditions in multiphase modulations outside of the zero characteristic conditions. In particular, the aim is complete the derivation of all possible codimension 2 (equations requiring two conditions to be met simultaneously) and codimension 3 equations from the modulation approach. This recovers several well known equations, such as the modified KdV, fifth order KdV and higher order KdV equations, but also results in some novel equations such as the modified two-way Boussinesq and sixth order two-way Boussinesq equations. Examples of how these equations emerge are then given, revisiting the stratified shallow water and coupled Nonlinear Schrodinger equations to do so, demonstrating how the conditions for each reduction may be met and how the coefficients may be obtained.