Periodic multi-pulses and spectral stability in Hamiltonian PDEs with symmetry

TL;DR

This paper analyzes the existence and spectral stability of periodic multi-pulse solutions in Hamiltonian PDEs, revealing bifurcation structures and instability bubbles through analytical and numerical methods.

Contribution

It introduces a novel application of Lin's method to construct and analyze spectral properties of multi-pulses in Hamiltonian PDEs with symmetry, including bifurcation and instability phenomena.

Findings

Demonstrated bifurcation structure for periodic double pulses

Reduced spectral problem to determinant of a block matrix

Identified and characterized instability bubbles during eigenvalue collisions

Abstract

We consider the existence and spectral stability of periodic multi-pulse solutions in Hamiltonian systems which are translation invariant and reversible, for which the fifth-order Korteweg-de Vries equation is a prototypical example. We use Lin's method to construct multi-pulses on a periodic domain, and in particular demonstrate a pitchfork bifurcation structure for periodic double pulses. We also use Lin's method to reduce the spectral problem for periodic multi-pulses to computing the determinant of a block matrix, which encodes both eigenvalues resulting from interactions between neighboring pulses and eigenvalues associated with the essential spectrum. We then use this matrix to compute the spectrum associated with periodic single and double pulses. Most notably, we prove that brief instability bubbles form when eigenvalues collide on the imaginary axis as the periodic domain size is altered. These analytical results are all in good agreement with numerical computations, and numerical timestepping experiments demonstrate that these instability bubbles correspond to oscillatory instabilities.