Alice-Bob systems, $P_s$-$T_d$-$C$ principles and multi-soliton solutions

TL;DR

This paper introduces new models called Alice-Bob systems to describe two-place physical problems, utilizing various symmetry principles and constructing explicit multi-soliton solutions for different integrable systems.

Contribution

It proposes a framework using $P_s$, $T_d$, $C$ symmetries to analyze Alice-Bob systems and explicitly constructs multi-soliton solutions for several types of integrable models.

Findings

Explicit multi-soliton solutions for KdV-KP-Toda, mKdV-sG, NLS, and discrete $H_1$ AB systems.

Use of symmetry principles to classify and solve two-place physical models.

New group-invariant solutions demonstrating the applicability of $P_s$-$T_d$-$C$ symmetries.

Abstract

To describe two-place physical problems, many possible models named Alice-Bob (AB) systems are proposed. To find and to solve these systems, the Parity (P), time reversal (T), charge conjugation (C), shifted-parity ($P_s$, parity with a shift), delayed time reversal ($T_d$, time reversal with a delay) and their possible combinations such as PT, PC, $P_sC$, $P_sT_d$ and $P_sT_dC$ etc. can be successively used. Especially, some special types of $P_s$-$T_d$-$C$ group invariant multi-soliton solutions for the KdV-KP-Toda type, mKdV-sG type, NLS type and discrete $H_1$ type AB systems are explicitly constructed.