# 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

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 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.

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.

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.

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.

*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.

*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 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 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.

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.

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.

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

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

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 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.

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.

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.

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] *u*_{0}, 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 (*u*_{0} - *u*)^{1/2} with *u* > *u*_{0}. 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.

*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.

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.

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.

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.

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 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.

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

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