Hamiltonian systems with an infinite number of localized travelling waves

TL;DR

This paper demonstrates that certain Hamiltonian systems with specific complex singularity structures can support infinitely many localized travelling waves, despite typical resonance restrictions.

Contribution

It establishes a general condition for the existence of infinite travelling solitons in Hamiltonian systems with resonances, supported by multiple illustrative examples.

Findings

Infinite number of travelling solitons can exist under specified singularity conditions.

Resonance does not necessarily prevent the propagation of localized waves.

Examples include fifth-order KdV-type, discrete Klein-Gordon, and nonlocal sine-Gordon equations.

Abstract

In many Hamiltonian systems, propagation of steadily travelling solitons or kinks is prohibited because of resonances with linear excitations. We show that Hamiltonian systems with resonances may admit an infinite number of travelling solitons or kinks if the closest to the real axis singularities in the complex upper half-plane of limiting asymptotic solution are of the form $z_\pm=\pm\alpha+i\beta$, $\alpha\ne 0$. This quite a general statement is illustrated by examples of the fifth-order Korteweg--de Vries-type equation, the discrete cubic-quintic Klein--Gordon equation, and the nonlocal double sine--Gordon equations.