### Professor Thomas Bridges

Professor
PhD, MSc, MA, BSc

### Publications

Bridges TJ, Pennant J, Zelik S (2014) Degenerate Hyperbolic Conservation Laws with Dissipation: Reduction to and Validity of a Class of Burgers-Type Equations, ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS 214 (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.
Bridges TJ, Dias F (2007) Enhancement of the Benjamin-Feir instability with dissipation, PHYSICS OF FLUIDS 19 (10) ARTN 104104 AMER INST PHYSICS
Avitabile D, Bridges TJ (2010) Numerical implementation of complex orthogonalization, parallel transport on Stiefel bundles, and analyticity, PHYSICA D-NONLINEAR PHENOMENA 239 (12) pp. 1038-1047 ELSEVIER SCIENCE BV
Chardard F, Dias F, Bridges TJ (2011) Computing the Maslov index of solitary waves, Part 2: Phase space with dimension greater than four, Physica D: Nonlinear Phenomena 240 (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.
Bridges TJ (2014) Dimension Breaking from Spatially-Periodic Patterns to KdV Planforms, Journal of Dynamics and Differential Equations
© 2014 Springer Science+Business Media New York.The problem of dimension breaking, for gradient elliptic partial differential equations in the plane, from a family of one-dimensional spatially periodic patterns (rolls) is considered. Conditions on the family of rolls are determined that lead to dimension breaking in the plane governed by a KdV equation relative to the periodic state. Since the KdV equation is time-independent, the (Formula presented.)-pulse solutions of KdV provide a sequence of multi-pulse planforms in the plane bifurcating from the rolls. The principal examples are the nonlinear Schrödinger equation, with evolution in the plane, and the steady Swift?Hohenberg equation with weak transverse variation.
Bridges TJ (2008) Conservation laws in curvilinear coordinates: A short proof of Vinokur's Theorem using differential forms, APPLIED MATHEMATICS AND COMPUTATION 202 (2) pp. 882-885 ELSEVIER SCIENCE INC
Turner MR, Bridges TJ, Alemi Ardakani H (2013) Dynamic coupling in Cooker's sloshing experiment with baffles., Phys Fluids 25 112102 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: ï : 1 resonances, as well as ?: m: ï : n for natural numbers ?, 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.
Dias F, Bridges TJ (2006) The numerical computation of freely propagating time-dependent irrotational water waves, FLUID DYNAMICS RESEARCH 38 (12) pp. 803-830 ELSEVIER SCIENCE BV
Turner MR, Bridges TJ (2013) Nonlinear energy transfer between fluid sloshing and vessel motion, Journal of Fluid Mechanics 719 pp. 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.
Bridges TJ (2013) A universal form for the emergence of the Korteweg-de Vries equation, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES 469 (2153) ARTN 20120707 ROYAL SOC
Bridges TJ (2008) Degenerate relative equilibria, curvature of the momentum map, and homoclinic bifurcation, JOURNAL OF DIFFERENTIAL EQUATIONS 244 (7) pp. 1629-1674 ACADEMIC PRESS INC ELSEVIER SCIENCE
Bridges TJ, Donaldson NM (2010) Variational principles for water waves from the viewpoint of a time-dependent moving mesh, Mathematika 57 pp. 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.
Chardard F, Dias F, Bridges TJ (2009) On the Maslov index of multi-pulse homoclinic orbits, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES 465 (2109) pp. 2897-2910 ROYAL SOC
Turner MR, Alemi Ardakani H, Bridges TJ (2015) Instability of sloshing motion in a vessel undergoing pivoted oscillations, 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.
Grudzien CJ, Bridges TJ, Jones CKRT (2016) Geometric phase in the Hopf bundle and the stability of non-linear waves, Physica D: Nonlinear Phenomena 334 pp. 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.
Bridges TJ, Donaldson NM (2006) Secondary criticality of water waves. Part 1. Definition, bifurcation and solitary waves, JOURNAL OF FLUID MECHANICS 565 pp. 381-417 CAMBRIDGE UNIV PRESS
Chardard F, Dias F, Bridges TJ (2006) Fast computation of the Maslov index for hyperbolic linear systems with periodic coefficients, JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL 39 (47) pp. 14545-14557 IOP PUBLISHING LTD
Bridges T, Pennant J, Zelik S (2014) Degenerate Hyperbolic Conservation Laws with Dissipation: Reduction to and Validity of a Class of Burgers-Type Equations, Archive for Rational Mechanics and Analysis 214 (2) pp. 671-716
© 2014, Springer-Verlag Berlin Heidelberg.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.
Bridges TJ, Laine-Pearson FE (2005) The long-wave instability of short-crested waves, via embedding in the oblique two-wave interaction, Journal of Fluid Mechanics 543 pp. 147-182 Cambridge University Press
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.
Bridges TJ, Furter JE (1993) Singularity theory and equivariant symplectic maps, 1558 Springer
The monograph will appeal to researchers and graduate students in the areas of symplectic maps, Hamiltonian systems, singularity theory and equivariant ...
Chardard F, Dias F, Bridges TJ (2009) Computing the Maslov index of solitary waves, Part 1: Hamiltonian systems on a four-dimensional phase space, PHYSICA D-NONLINEAR PHENOMENA 238 (18) pp. 1841-1867 ELSEVIER SCIENCE BV
Ardakani HA, Bridges TJ, Turner MR (2015) Dynamic coupling between horizontal vessel motion and two-layer shallow-water sloshing, JOURNAL OF FLUIDS AND STRUCTURES 59 pp. 432-460 ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD
Bridges TJ (2015) Breakdown of the Whitham Modulation Theory and the Emergence of Dispersion, Studies in Applied Mathematics 135 (3) pp. 277-294
© 2015 Wiley Periodicals, Inc.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.
Chardard F, Bridges TJ (2015) Transversality of homoclinic orbits, the Maslov index and the symplectic Evans function, Nonlinearity 28 (1) pp. 77-102
© 2015 IOP Publishing Ltd & London Mathematical Society Printed.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.
Alemi Ardakani H, Bridges TJ, Turner MR (2012) Resonance in a model for Cooker's sloshing experiment, European Journal of Mechanics, B/Fluids 36 pp. 25-38
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. The numerical evaluation of the resonant and nonresonant modes is presented. The implications of the resonance for the fluid dynamics, and for the nonlinear coupled dynamics near the resonance are also briefly discussed. © 2012 Elsevier Masson SAS. All rights reserved.
Bridges TJ, Needham DJ (2011) Breakdown of the shallow water equations due to growth of the horizontal vorticity, JOURNAL OF FLUID MECHANICS 679 pp. 655-666 CAMBRIDGE UNIV PRESS
Alemi Ardakani H, Bridges TJ (2010) Dynamic coupling between shallow-water sloshing and horizontal vehicle motion, European Journal of Applied Mathematics 21 (6) pp. 479-517
Bridges TJ, Donaldson NM (2009) Criticality manifolds and their role in the generation of solitary waves for two-layer flow with a free surface, EUROPEAN JOURNAL OF MECHANICS B-FLUIDS 28 (1) pp. 117-126 GAUTHIER-VILLARS/EDITIONS ELSEVIER
Bridges TJ, Ratliff DJ (2015) Phase dynamics of periodic waves leading to the Kadomtsev-Petviashvili equation in 3+1 dimensions, Proceedings of the Royal Society of London: Mathematical, Physical and Engineering Sciences 471 (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.
Bridges TJ, Donaldson NM (2006) Secondary criticality of water waves. Part 2. Unsteadiness and the Benjamin-Feir instability from the viewpoint of hydraulics, JOURNAL OF FLUID MECHANICS 565 pp. 419-439 CAMBRIDGE UNIV PRESS
Bridges TJ (2009) Wave breaking and the surface velocity field for three-dimensional water waves, NONLINEARITY 22 (5) pp. 947-953 IOP PUBLISHING LTD
Bridges TJ, Derks G (2001) The symplectic Evans matrix and the instability of solitary waves and fronts, Archives for Rational Mechanics and Analysis 156 pp. 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.

Bridges TJ (2000) Universal Geometric Condition for the Transverse Instability of Solitary Waves, Physical Review Letters 84 12 pp. 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.
Bridges TJ (2015) Breakdown of the Whitham Modulation Theory and the Emergence of Dispersion, Studies in Applied Mathematics
© 2015 Wiley Periodicals, Inc., 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.
Bridges TJ (2006) Canonical multi-symplectic structure on the total exterior algebra bundle, Proceedings Royal Society London A 462 (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.
Alemi Ardakani H, Bridges TJ (2012) Shallow-water sloshing in vessels undergoing prescribed rigid-body motion in two dimensions, European Journal of Mechanics, B/Fluids 31 (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.
Bridges TJ, Donaldson NM (2007) Reappraisal of criticality for two-layer flows and its role in the generation of internal solitary waves, PHYSICS OF FLUIDS 19 (7) ARTN 072111 AMER INST PHYSICS
Bridges TJ (2009) Degeneracy of the action-frequency map: A mechanism for homoclinic bifurcation of invariant tori, PHYSICAL REVIEW E 79 (6) ARTN 066603 AMER PHYSICAL SOC
Bridges TJ (2014) Emergence of dispersion in shallow water hydrodynamics via modulation of uniform flow, JOURNAL OF FLUID MECHANICS 761 ARTN R1 CAMBRIDGE UNIV PRESS
Ardakani HA, Bridges TJ, Turner MR (2016) Adaptation of f-wave finite volume methods to the two-layer shallow-water equations in a moving vessel with a rigid-lid, JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 296 pp. 462-479 ELSEVIER SCIENCE BV
Bridges TJ (2014) Bifurcation from rolls to multi-pulse planforms via reduction to a parabolic Boussinesq model, Physica D: Nonlinear Phenomena 275 pp. 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.
Bridges TJ, Reich S (2006) Numerical methods for Hamiltonian PDEs, JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL 39 (19) pp. 5287-5320 IOP PUBLISHING LTD
Ardakani HA, Bridges TJ (2010) Dynamic coupling between shallow-water sloshing and horizontal vehicle motion, EUROPEAN JOURNAL OF APPLIED MATHEMATICS 21 (6) pp. 479-517 CAMBRIDGE UNIV PRESS
Bridges TJ, Donaldson NM (2008) Criticality manifolds and their role in the generation of solitary waves for two-layer flow with a free surface, European Journal of Mechanics - B/Fluids 28 pp. 117-126 Elsevier Masson SAS
The role of criticality manifolds is explored both for the classification of all uniform flows and for the bifurcation of solitary waves, in the context of two fluid layers of differing density with an upper free surface. While the weakly nonlinear bifurcation of solitary waves in this context is well known, it is shown herein that the critical nonlinear behaviour of the bifurcating solitary waves and generalized solitary waves is determined by the geometry of the criticality manifolds. By parametrizing all uniform flows, new physical results are obtained on the implication of a velocity difference between the two layers on the bifurcating solitary waves.
Bridges TJ, Groves MD, Nicholls DP (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.
Turner M, Bridges T (2015) Time-dependent conformal mapping of doubly-connected regions, Advances in Computational Mathematics 42 (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.
Bridges TJ (2012) Geometric lift of paths of hamiltonian equilibria and homoclinic bifurcation, International Journal of Bifurcation and Chaos 22 (12)
A saddle-center transition of eigenvalues in the linearization about Hamiltonian equilibria, and the attendant planar homoclinic bifurcation, is one of the simplest and most well-known bifurcations in dynamical systems theory. It is therefore surprising that anything new can be said about this bifurcation. In this tutorial, the classical view of this bifurcation is reviewed and the lifting of the planar system to four dimensions gives a new view. The principal practical outcome is a new formula for the nonlinear coefficient in the normal form which generates the homoclinic orbit. The new formula is based on the intrinsic curvature of the lifted path of equilibria. © 2012 World Scientific Publishing Company.
Alemi Ardakani H, Bridges T (2010) Shallow-water sloshing in vessels undergoing prescribed rigid-body motion in three dimensions, Journal of Fluid Mechanics 667 in press pp. 474-519 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.
Ardakani HA, Bridges TJ, Turner MR (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, JOURNAL OF COMPUTATIONAL PHYSICS 314 pp. 590-617 ACADEMIC PRESS INC ELSEVIER SCIENCE
Bridges TJ (2012) Emergence of unsteady dark solitary waves from coalescing spatially periodic patterns, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES 468 (2148) pp. 3784-3803 ROYAL SOC
Turner MR, Bridges TJ, Ardakani HA (2015) The pendulum-slosh problem: Simulation using a time-dependent conformal mapping, JOURNAL OF FLUIDS AND STRUCTURES 59 pp. 202-223 ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD
Bridges TJ, Hydon PE, Lawson JK (2010) Multisymplectic structures and the variational bicomplex, MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY 148 pp. 159-178 CAMBRIDGE UNIV PRESS
Bridges T, Ratliff D (2017) On the elliptic-hyperbolic transition in whitham modulation theory, SIAM Journal on Applied Mathematics 77 (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.
Ratliff DJ, Bridges TJ (2016) Multiphase wavetrains, singular wave interactions and the emergence of the Korteweg-de Vries equation, Proceedings of the Royal Society A 472 (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.
Bridges TJ, Ratliff DJ (2016) Double criticality and the two-way Boussinesq equation in stratifed shallow water hydrodynamics, Physics of Fluids 28 (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.
Alemi Ardakani H, Bridges T, Turner M (2012) Resonance in a model for Cooker's sloshing experiment, European Journal of Mechanics B - Fluids 36 (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.
Alemi Ardakani H, Bridges T, Turner M (2012) Resonance in a model for Cooker's sloshing experiment, European Journal of Mechanics B - Fluids 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.
Turner Matthew, Bridges Thomas, Alemi Ardakani H (2017) Lagrangian particle path formulation of multilayer shallow-water flows dynamically coupled to vessel motion, Journal of Engineering Mathematics 106 (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.
Bridges Thomas, Ratliff Daniel (2018) Nonlinear modulation near the Lighthill instability threshold in 2+1 Whitham theory, Philosophical Transactions A 376 (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.
Ratliff Daniel, Bridges Tom (2018) Reduction to modified KdV and its KP-like generalization via phase modulation, Nonlinearity 31 (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.
Bridges T, Laine-Pearson F (2005) The Long-Wave Instability of Short-Crested Waves, Via Embedding in the Oblique Two-Wave Interaction, Journal of Fluid Mechanics 543 (-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.

Bridges T (2005) Steady Dark Solitary Waves Emerging from Wave-Generated Meanflow: The Role of Modulation Equations, Chaos: An Interdisciplinary Journal of Nonlinear Science 15 (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.

Bridges T (2004) Superharmonic instability, homoclinic torus bifurcation and water-wave breaking, Journal of Fluid Mechanics 505 pp. 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.

Bridges T (2000) Universal Geometric Condition for the Transverse Instability of Solitary Waves, Physical Review Letters 84 pp. 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.

Bridges T (2004) Superharmonic Instability, Homoclinic Torus Bifurcation and Water-Wave Breaking, Journal of Fluid Mechanics 505 pp. 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.

Bridges T, Laine-Pearson F (2005) The long-wave instability of short-crested waves, via embedding in the oblique two-wave interaction, Journal of Fluid Mechanics 543 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.

Bridges T, Donaldson N (2006) Secondary criticality of water waves. Part 1. Definition, bifurcation and solitary waves, Journal of Fluid Mechanics 565 pp. 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 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.

Bridges T (2005) Steady dark solitary waves emerging from wave-generated meanflow: The role of modulation equations, Chaos: An Interdisciplinary Journal of Nonlinear Science 037113 (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.

Bridges T, Donaldson N (2006) Secondary Criticality of Water Waves. Part 1.
Definition, Bifurcation and Solitary Waves,
Journal of Fluid Mechanics 565 pp. 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.

Bridges Thomas J., Ratliff Daniel J. (2019) Krein signature and Whitham modulation theory: the sign of characteristics and the ?sign characteristic?, Studies in Applied Mathematics 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.