Symplectic transversality and the Pego-Weinstein theory

TL;DR

This paper introduces a new symplectic invariant factor in the stability analysis of solitary waves in Hamiltonian PDEs, linking transversality to spectral stability through the Evans function.

Contribution

It presents a novel factor $\Pi$ in the Pego-Weinstein derivative formula, connecting transversality of homoclinic orbits with spectral stability, and introduces a new class of Hamiltonian PDEs modeling various physical systems.

Findings

The factor $\Pi$ is nonzero if and only if the homoclinic orbit is transversely constructed.

The sign of $\Pi$ is a symplectic invariant and influences stability.

The theory applies to models like the coupled mode equations, the massive Thirring model, and nonlinear wave equations.

Abstract

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