On Quasi-integrable Deformation Scheme of The KdV System

TL;DR

The paper develops a general framework to quasi-deform the KdV equation by deforming the Hamiltonian and leverages an $sl(2)$ loop-algebra Abelianization to generate anomalous charges. It shows that with parity-even deformations and localized solutions, a subset of charges can be quasi-conserved, yielding asymptotic integrability and connections to RLW/mRLW and to quasi-NLS under weak coupling. It further constructs integrable-scale deformations and higher-derivative hierarchies, and discusses perturbative and non-holonomic perspectives, with a clear mapping to NLS for validation. The work suggests rich avenues for numerical exploration of soliton-like structures and extensions to other KdV-type hierarchies and nonlocal systems.

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 $u \to u^{1+ε}$ 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.

Figures (4)

• Figure 1: The effect of quasi-deformation on the weak-coupling map from NLS to KdV solutions. The NLS single soliton (a) maps to a KdV soliton train (b), a property retained over the quasi-deformation though the effect of quasi-deformation is clear (c and d). Herein, $\epsilon=1.5$, $V=1=\rho$, $X_0=0=T_0$$\varepsilon=0.1$ and $\omega_0=1=k_0$.
• Figure 2: Numerical solutions $u_d$ corresponding to the deformation $u^3\rightarrow u^{3++3\epsilon}$ in the Hamiltonian. The very localized and parity-even single-soliton structure for $\epsilon=0$ (\ref{['F2a']}) gets significantly distorted (\ref{['F2b']}, \ref{['F2c']}, \ref{['F2d']}) even for small values $\epsilon$. They still remain fairly localized suggesting asymptotic conservation of the corresponding charges. These plots are evaluated at time $t=1$.
• Figure 3: Approximate soliton train-like solution for power-scaling $u\rightarrow u^{1+\epsilon}$ of the nonlinear term in the Hamiltonian with $\epsilon=0.7$ and $c_0=1$.
• Figure 4: ${\cal O}(\epsilon)$ correction to the deformed KdV solution. It clearly deviates from even-parity structure eventually leading to non-conserved charges.