Helical solitons in vector modified Korteweg-de Vries equations

TL;DR

This paper investigates the existence, construction, and stability of helical solitons in both integrable and non-integrable vector mKdV equations, relevant to physical systems like plasma and elastic chains.

Contribution

It introduces a numerical method for constructing helical solitons in non-integrable equations and analyzes their bifurcation and stability properties.

Findings

Helical solitons exist along a curve in the parameter plane.

Numerical solutions match exact solutions in the integrable case.

Helical solitons are stable under small perturbations.

Abstract

We study existence of helical solitons in the vector modified Korteweg-de Vries (mKdV) equations, one of which is integrable, whereas another one is non-integrable. The latter one describes nonlinear waves in various physical systems, including plasma and chains of particles connected by elastic springs. By using the dynamical system methods such as the blow-up near singular points and the construction of invariant manifolds, we construct helical solitons by the efficient shooting method. The helical solitons arise as the result of co-dimension one bifurcation and exist along a curve in the velocity-frequency parameter plane. Examples of helical solitons are constructed numerically for the non-integrable equation and compared with exact solutions in the integrable vector mKdV equation. The stability of helical solitons with respect to small perturbations is confirmed by direct numerical simulations.