Solving bi-directional soliton equations in the KP hierarchy by gauge transformation

TL;DR

This paper develops a systematic method to construct solutions for bi-directional soliton equations within the KP hierarchy, classifying various soliton types and their propagation directions, including multi-peak and stationary solutions.

Contribution

It introduces a new systematic approach to generate and classify solutions of bi-directional soliton equations in the KP hierarchy, including multi-peak and stationary solitons, based on symmetry and root distribution.

Findings

Constructed explicit one- and two-soliton solutions.

Classified solitons by propagation directions and symmetries.

Identified existence of multi-peak and stationary solitons.

Abstract

We present a systematic way to construct solutions of the (n=5)-reduction of the BKP and CKP hierarchies from the general tau function of the KP hierarchy. We obtain the one-soliton, two-soliton, and periodic solution for the bi-directional Sawada-Kotera (bSK), the bi-directional Kaup-Kupershmidt (bKK) and also the bi-directional Satsuma-Hirota (bSH) equation. Different solutions such as left- and right-going solitons are classified according to the symmetries of the 5th roots of exp(i epsilon). Furthermore, we show that the soliton solutions of the n-reduction of the BKP and CKP hierarchies with n= 2 j +1, j=1, 2, 3, ..., can propagate along j directions in the 1+1 space-time domain. Each such direction corresponds to one symmetric distribution of the nth roots of exp(i epsilon). Based on this classification, we detail the existence of two-peak solitons of the n-reduction from the Grammian tau function of the sub-hierarchies BKP and CKP. If n is even, we again find two-peak solitons. Last, we obtain the "stationary" soliton for the higher-order KP hierarchy.