Complex Solitary Waves and Soliton Trains in KdV and mKdV Equations

TL;DR

This paper demonstrates the existence of complex solitary wave and periodic solutions in KdV and mKdV equations, revealing novel $ ext{PT}$-symmetric properties and connections to supersymmetry, with potential hydrodynamic analogs.

Contribution

It introduces complex solutions of KdV and mKdV equations that are $ ext{PT}$-symmetric and connects them to supersymmetric quantum mechanics, a novel approach in soliton theory.

Findings

Complex solutions appear in conjugate pairs with specific symmetry properties.

$ ext{PT}$-odd solitons are connected to classical solutions via supersymmetry.

Solutions resemble hydrodynamic analogs of Bloch solitons.

Abstract

We demonstrate the existence of complex solitary wave and periodic solutions of the Kortweg de-vries (KdV) and modified Kortweg de-Vries (mKdV) equations. The solutions of the KdV (mKdV) equation appear in complex-conjugate pairs and are even (odd) under the simultaneous actions of parity ($\cal{P}$) and time-reversal ($\cal{T}$) operations. The corresponding localized solitons are hydrodynamic analogs of Bloch soliton in magnetic system, with asymptotically vanishing intensity. The $\cal{PT}$-odd complex soliton solution is shown to be iso-spectrally connected to the fundamental $sech^2$ solution through supersymmetry.