Travelling waves and conservation laws for complex mKdV-type equations

TL;DR

This paper investigates travelling waves and conservation laws in a broad class of complex mKdV equations, introducing new solutions and explicitly deriving conserved quantities related to phase invariance and angular twist.

Contribution

It presents new complex solitary wave and kink solutions, and explicitly finds all first-order conserved densities for a class of U(1)-invariant complex mKdV equations.

Findings

New complex solitary wave solutions

New complex kink solutions

Explicit conserved densities including phase-invariant and angular twist

Abstract

Travelling waves and conservation laws are studied for a wide class of U(1)-invariant complex mKdV equations containing the two known integrable generalizations of the ordinary (real) mKdV equation. The main results on travelling waves include deriving new complex solitary waves and kinks that generalize the well-known mKdV $\sech$ and $\tanh$ solutions. The main results on conservation laws consist of explicitly finding all 1st order conserved densities that yield phase-invariant counterparts of the well-known mKdV conserved densities for momentum, energy, and Galilean energy, and a new conserved density describing the angular twist of complex kink solutions