arXiv subjects

Thomas J. Bridges

Publications and source records attributed to Thomas J. Bridges.

8 recordsLinked to original sources

arXiv2026

Heteroclinic connections between finite-amplitude periodic orbits emerging from a codimension two singularity

Heteroclinic connections between two distinct hyperbolic periodic orbits in conservative systems are important in a wide range of applications. On the other hand, it is theoretically challenging to find large amplitude connections from scratch and compute them numerically. In this paper, we use a codimension two singularity, in a family of periodic orbits, as an organizing center for the emergence of heteroclinic connections. A normal form is derived whose unfolding produces two distinct finite amplitude periodic orbits with an explicit heteroclinic connection. We also construct heteroclinic connections far from the singularity by numerical continuation, using two numerical strategies: shooting and the core-farfield decomposition. A key geometric tool in the numerics is cylindrical foliations for the stable and unstable manifolds and their intersection. We introduce a new property of heteroclinic connections - the action - and show it is an invariant along foliations, it has a jump at a surface of section, and it appears in a central way in the normal form theory. We find that the difference in asymptotic phase between minus and plus infinity is also a key property. The theory is applied to the Swift-Hohenberg equation, the nonlinear Schrodinger with fourth order dispersion, and coupled Boussinesq equations from water waves, all of which have an energy and action conservation law.

math.DS↗

arXiv2024

Modulation leading to frequency downshifting of water waves in the vicinity of the Benjamin-Feir transition

For Stokes waves in finite depth within the neighbourhood of the Benjamin-Feir stability transition, there are two families of periodic waves, one modulationally unstable and the other stable. In this paper we show that these two families can be joined by a heteroclinic connection, which manifests in the fluid as a travelling front. By shifting the analysis to the setting of Whitham modulation theory, this front is in wavenumber and frequency space. An implication of this jump is that a permanent frequency downshift of the Stokes wave can occur in the absence of viscous effects. This argument, which is built on a sequence of asymptotic expansions of the phase dynamics, is confirmed via energetic arguments, with additional corroboration obtained by numerical simulations of a reduced model based on the Benney-Roskes equation.

physics.flu-dyn↗

arXiv2022

Reappraisal of Whitham's 1967 theory for wave-mean flow interaction in shallow water

The modulation equations for Stokes waves in shallow water coupled to wave generated mean flow, derived in Whitham (1967), based on an averaged Lagrangian are revisited. Firstly, it is shown that they can be recast into two coupled classical shallow water equations, with modified gravity having the sign of the Whitham index: sign($ω_0'' ω_2$). Secondly, it is shown that the amplitude of the mean flow and amplitude of the wave are, in general, independent. Thirdly, the implications of the coalescing characteristics, whose unfolding is associated with the Benjamin-Feir instability, are studied.

physics.flu-dyn↗

arXiv2022

The Kawahara Equation: Traveling Wave Solutions Joining Periodic Waves

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

nlin.PS↗

arXiv2020

Nonlinear theory for coalescing characteristics in multiphase Whitham modulation theory

The multiphase Whitham modulation equations with $N$ phases have $2N$ characteristics which may be of hyperbolic or elliptic type. In this paper a nonlinear theory is developed for coalescence, where two characteristics change from hyperbolic to elliptic via collision. Firstly, a linear theory develops the structure of colliding characteristics involving the topological sign of characteristics and multiple Jordan chains, and secondly a nonlinear modulation theory is developed for transitions. The nonlinear theory shows that coalescing characteristics morph the Whitham equations into an asymptotically valid geometric form of the two-way Boussinesq equation. That is, coalescing characteristics generate dispersion, nonlinearity and complex wave fields. For illustration, the theory is applied to coalescing characteristics associated with the modulation of two-phase travelling-wave solutions of coupled nonlinear Schrödinger equations, highlighting how collisions can be identified and the relevant dispersive dynamics constructed.

math.AP↗

arXiv2020

Validity of the hyperbolic Whitham modulation equations in Sobolev spaces

It is proved that modulation in time and space of periodic wave trains, of the defocussing nonlinear Schrödinger equation, can be approximated by solutions of the Whitham modulation equations, in the hyperbolic case, on a natural time scale. The error estimates are based on existence, uniqueness, and energy arguments, in Sobolev spaces on the real line. An essential part of the proof is the inclusion of higher-order corrections to Whitham theory, and concomitant higher-order energy estimates.

math.AP↗

arXiv2009

Computational aspects of the Maslov index of solitary waves

When solitary waves are characterized as homoclinic orbits of a finite-dimensional Hamiltonian system, they have an integer-valued topological invariant, the Maslov index. We are interested in developing a robust numerical algorithm to compute the Maslov index, to understand its properties, and to study the implications for the stability of solitary waves. The algorithms reported here are developed in the exterior algebra representation, which leads to a robust and fast algorithm with some novel properties. We use two different representations for the Maslov index, one based on an intersection index and one based on approximating the homoclinic orbit by a sequence of periodic orbits. New results on the Maslov index for solitary wave solutions of reaction-diffusion equations, the fifth-order Korteweg-De Vries equation, and the longwave-shortwave resonance equations are presented. Part 1 considers the case of four-dimensional phase space, and Part 2 considers the case of $2n-$dimensional phase space with $n>2$.

math.AP↗

arXiv2007

Enhancement of the Benjamin-Feir instability with dissipation

It is shown that there is an overlooked mechanism whereby some kinds of dissipation can enhance the Benjamin-Feir instability of water waves. This observation is new, and although it is counterintuitive, it is due to the fact that the Benjamin-Feir instability involves the collision of modes with opposite energy sign (relative to the carrier wave), and it is the negative energy perturbations which are enhanced.

physics.class-ph↗