# Professor Thomas J Bridges

## About

### Biography

Further details can be found on my personal web page.

View a full list of my publications.

## Research

### Research interests

### Research projects

Department of Mathematics, University of Surrey and Offshore Wave Energy Ltd.

## Teaching

In Summer 2023, I supervised two MSc projects in Astrodynamics for the MSc in Space Engineering. In the 2023-2024 academic year I am lecturing on

- MAT3048 Lagrangian Fluid Dynamics of Planet Earth (Autumn 2023)
- MATM032 Geometric Mechanics (Autumn 2023).
- MAT1031 Seminars in Algebra (Autumn 2023)
- MSc projects (TBC) in Spring/Summer 2024

## Publications

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 newcharacterization 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.

Many models for physical phenomena in oceanography, atmospheric dynamics, optical fibre transmission, nerve conduction, acoustical and gas dynamic flows are conservative translation-invariant evolution equations with a Hamiltonian structure. Solitary waves and fronts form an important class of solutions of such equations and the calculus of variations, critical point theory and symplectic structure have played a major role in the analysis of their stability and instability. For example, the characterisation of solitary waves as critical points of the Hamiltonian (energy) constrained to level sets of the momentum (or momentum and other constants of motion) leads to a powerful framework for proving nonlinear Lyapunov stability – when the second variation, evaluated at the constrained critical point, has a finite number of negative eigenvalues (e.g. BENJAMIN2, BONA3, HOLM ET AL14, GRILLAKIS ET AL12, 13, MADDOCKS & SACHS16 and references therein)…

For linear systems with a multi-symplectic structure, arising from the linearization of Hamiltonian partial differential equations about a solitary wave, the Evans function can be characterized as the determinant of a matrix, and each entry of this matrix is a restricted symplectic form. This variant of the Evans function is useful for a geometric analysis of the linear stability problem. But, in general, this matrix of two-forms may have branch points at isolated points, shrinking the natural region of analyticity. In this paper, a new construction of the symplectic Evans matrix is presented, which is based on individual vectors but is analytic at the branch points—indeed, maximally analytic. In fact, this result has greater generality than just the symplectic case; it solves the following open problem in the literature: can the Evans function be constructed in a maximally analytic way when individual vectors are used? Although the non-symplectic case will be discussed in passing, the paper will concentrate on the symplectic case, where there are geometric reasons for evaluating the Evans function on individual vectors. This result simplifies and generalizes the multi-symplectic framework for the stability analysis of solitary waves, and some of the implications are discussed.

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.

Two generalisations of the Evans function, for the analysis of the linearisation about solitary waves, are shown to be equivalent. The generalisation introduced by Alexander, Gardner and Jones [J. reine Angew. Math. 410 (1990) 167] is based on exterior algebra and the generalisation introduced by Swinton [Phys. Lett. A 163 (1992) 57] is based on a matrix formulation and adjoint systems. In regions of the complex plane where both formulations are defined, the equivalence is geometric: we show that the formulations are dual and the duality can be made explicit using Hodge duality and the Hodge star operator. Swinton's formulation excludes potential branch points at which the Alexander, Gardner and Jones formulation is well-defined. Therefore we consider the implications of equivalence on the analytic continuation of the two formulations.

The spectral problem associated with the linearization about solitary waves of the generalized fifth-order KdV equation is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. A numerical framework, based on a fast robust shooting algorithm on exterior algebra spaces is introduced. The complete algorithm has several new features, including a rigorous numerical algorithm for choosing starting values, a new method for numerical analytic continuation of starting vectors, the role of the Grassmannian G2(C5) in choosing the numerical integrator, and the role of the Hodge star operator for relating Ʌ2(C5) and Ʌ3(C5) and deducing a range of numerically computable forms for the Evans function. The algorithm is illustrated by computing the stability and instability of solitary waves of the fifth-order KdV equation with polynomial nonlinearity.

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.

The FORQ equation, which is bi-Hamiltonian, integrable, and hosts of a range of soliton solutions, is viewed afresh from the viewpoint of multi-symplectic structures. A multi-symplectic formulation is derived and used as a guide in the construction of numerical methods for simulation; in particular, a multi-symplectic Preissmann box scheme is constructed for the FORQ equation, which is second-order accurate in both space and time. Simulations show that the cuspons and the W/M-shape-peaks solitons for the FORQ equation are reproduced accurately. To verify the precision and the validity of the numerical results, the relative errors between the numerical results and the theoretical solutions, and the maximum values of the errors of the local conservation laws are presented.

The Kawahara equation is a weakly nonlinear longwave model of dispersive waves that emerges when leading, third order, dispersive effects are in balance with the next, fifth-order correction. Traveling wave solutions of the Kawahara equation satisfy a fourth-order ordinary differential equation in which the traveling wave speed is a parameter. The fourth-order equation has Hamiltonian structure and admits a two-parameter family of single-phase periodic solutions with varying speed and Hamiltonian. A set of jump conditions is derived for pairs of distinct periodic solutions with equal speed and Hamiltonian. These are necessary conditions for the existence of traveling waves that asymptote to the periodic orbits at ± ∞ . Bifurcation theory and parameter continuation are used to construct multiple solution branches of the jump conditions. For example pairs of compatible periodic solutions, the heteroclinic orbit representing the traveling wave is constructed from the intersection of stable and unstable manifolds of the periodic orbits. Each branch terminates at an equilibrium-to-periodic solution in which the equilibrium is the background for a solitary wave that connects to the associated periodic solution.

The modulation equations for Stokes waves in shallow water coupled to wave-generated meanflow, derived in Whitham (1967), based on an averaged Lagrangian are re-visited. Firstly, it is shown that they can be recast into two coupled classical shallow water equations, with modified gravity having the sign of the Whitham index: sign(omega 0 ''omega 2). Secondly, it is shown that the amplitude of the meanflow and amplitude of the wave are, in general, independent. Thirdly, the implications of the coalescing characteristics, whose unfolding is associated with the Benjamin-Feir instability, are studied.(c) 2022 Published by Elsevier B.V.

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.

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.

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.

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.

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.

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.

Unsteadiness and the Benjamin-Feir Instability From the Viewpoint of Hydraulics

, In: Journal of Fluid Mechanics565pp. 419-439

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

This paper studies the linear stability problem for solitary wave solutions of Hamiltonian PDEs. The linear stability problem is formulated in terms of the Evans function, a complex analytic function denoted by D(λ), where λ is the spectral parameter. The main result is the introduction of a new factor, denoted Π, in the Pego & Weinstein (1992) derivative formula D''(0) = χΠ dI/dc , where I is the momentum of the solitary wave and c is the speed. Moreover this factor turns out to be related to transversality of the solitary wave, modelled as a homoclinic orbit: the homoclinic orbit is transversely constructed if and only if Π is nonzero. The sign of Π is a symplectic invariant, an intrinsic property of the solitary wave, and is a key new factor affecting the linear stability. The factor χ was already introduced by Bridges & Derks (1999) and is based on the asymptotics of the solitary wave. A supporting result is the introduction of a new abstract class of Hamiltonian PDEs built on a nonlinear Dirac-type equation, which model a wide range of PDEs in applications. Examples where the theory applies, other than Dirac operators, are the coupled mode equation in fluid mechanics and optics, the massive Thirring model, and coupled nonlinear wave equations. A calculation of D''(0) for solitary wave solutions of the latter class is included to illustrate the theory.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

*SIAM Journal on Mathematical Analysis*33pp. 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.

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.

The monograph will appeal to researchers and graduate students in the areas of symplectic maps, Hamiltonian systems, singularity theory and equivariant ...

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.

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.

*Mathematical Proceedings of the Cambridge Philosophical Society*121(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.

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 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.

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.

*SIAM Journal on Applied Mathematics*64pp. 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.