# Professor Thomas Bridges

### Biography

### Research interests

Details can be found on my personal web page.

A full list of publications can be found here.

### Teaching

- MAT3048 Lagrangian Fluid Dynamics of Planet Earth (Autumn 2018)
- MATM032 Geometric Mechanics (Spring 2019)

### My publications

### Publications

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

The 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.

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 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 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.

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.

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

,Journal of Fluid Mechanics 565 pp. 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.

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.

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.

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.

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.

rectilinear vessel motion in a system of

connected vessels,

of a one-dimensional array of vessels, partially filled with fluid, being connected together

by nonlinear springs. The fluid motion induces a hydrodynamic force on the side walls of

the vessel, which induces the vessel to move. The vessel movement is also controlled via a

restoring force of the attached springs, which in turn causes the fluid to alter its motion.

The simplest coupled liquid sloshing system consists of one vessel connected to a side wall via

a spring. We review this example and investigate the techniques documented in the literature

to study the linear problem of such a system. Then we extend this linear theory to multivessel systems, in particular focussing on the 2-vessel system which introduces the notion of

modes being in-phase or out-of-phase with each other. In the general N-vessel system we

also identify the existence in parameter space of internal resonances, where different modes

oscillates at the same frequency. Such a resonance provides a mechanism for energy exchange

between modes in the nonlinear system.

The nonlinear dynamics of the coupled liquid sloshing system are studied by employing a

symplectic integration scheme based on a variational principle, to the shallow-water form of

the governing equations. We present results for the shallow-water scheme in the linear, weakly

nonlinear and fully nonlinear regimes. The shallow-water numerical scheme has trouble

dealing with breaking waves, so we perform a feasibility study where we attempt to use a

similar approach for a non-shallow fluid system. The governing equations are derived and

results presented in the linear amplitude regime for the 1-vessel system.

Finally, we develop variational principles for the coupled liquid sloshing system in the Eulerian

framework based on the principle of constrained variations to derive the governing fluid

equations and free-surface boundary conditions, from a natural Lagrangian functional. We

use this constrained variational approach to derive the fully rotational 2D Euler equations

and its stream function formulation.