Prohibitions caused by nonlocality for Alice-Bob Boussinesq-KdV type systems

TL;DR

This paper explores how nonlocal symmetries derived from shifted parity and delayed time reversal impose prohibitions on certain multi-soliton solutions in Alice-Bob KdV and Boussinesq systems, revealing new solution structures.

Contribution

It introduces a unified model linking KdV and Boussinesq equations via nonlocalities and derives multi-soliton solutions, highlighting solution prohibitions due to nonlocal symmetry constraints.

Findings

Even multi-soliton solutions with head-on interactions are obtained.

Odd multi-soliton solutions and even solutions with pursuit interactions are prohibited.

ABKdV solutions exhibit complex structures due to arbitrary background waves.

Abstract

It is found that two different celebrate models, the Korteweg de-Vrise (KdV) equation and the Boussinesq equation, are linked to a same model equation but with different nonlocalities. The model equation is called the Alice-Bob KdV (ABKdV) equation which was derived from the usual KdV equation via the so-called consistent correlated bang (CCB) companied by the shifted parity (SP) and delayed time reversal (DTR). The same model can be called as the Alice-Bob Boussinesq (ABB) system if the nonlocality is changed as only one of SP and DTR. For the ABB systems, with help of the bilinear approach and recasting the multi-soliton solutions of the usual Boussinesq equation to an equivalent novel form, the multi-soliton solutions with even numbers and the head on interactions are obtained. However, the multi-soliton solutions with odd numbers and the multi-soliton solutions with even numbers but with pursuant interactions are prohibited. For the ABKdV equation, the multi-soliton solutions exhibit many more structures because an arbitrary odd function of $x+t$ can be introduced as background waves of the usual KdV equation.