On Quasi-integrable Deformation Scheme of The KdV System

TL;DR

This paper introduces a method for quasi-deforming the KdV equation through Hamiltonian deformation, leading to a hierarchy of (quasi-)integrable systems with potential applications to soliton solutions and connections to known quasi-NLS models.

Contribution

It presents a novel Hamiltonian deformation scheme for the KdV system, establishing conditions for quasi-integrability and linking to known quasi-NLS solitons.

Findings

Infinite anomalous conservation laws derived

Deformed KdV solutions can be parity-even for quasi-conservation

Numerical soliton solutions obtained for specific deformations

Abstract

We put forward a general approach to quasi-deform the KdV equation by deforming the corresponding Hamiltonian. Following the standard Abelianization process based on the inherent $sl(2)$ loop algebra, an infinite number of anomalous conservation laws are obtained, which yield conserved charges if the deformed solution has definite space-time parity. Judicious choice of the deformed Hamiltonian leads to an integrable system with scaled parameters as well as to a hierarchy of deformed systems, some of which possibly being quasi-integrable. As a particular case, one such deformed KdV system maps to the known quasi-NLS soliton in the already known weak-coupling limit, whereas a generic scaling of the KdV amplitude $u \to u^{1+\epsilon}$ also goes to possible quasi-integrability under an order-by-order expansion. Following a generic parity analysis of the deformed system, these deformed KdV solutions need to be parity-even for quasi-conservation which may be the case here following our analytical approach. From the established quasi-integrability of RLW and mRLW systems [Nucl. Phys. B 939 (2019) 49-94], which are particular cases of the present approach, exact solitons of the quasi-KdV system could be obtained numerically.