Riccati-type pseudo-potential approach to quasi-integrability of deformed soliton theories

TL;DR

This work surveys a unified Riccati-type pseudo-potential framework to study quasi-integrability in deformations of integrable soliton models, notably sine-Gordon, NLS, and KdV. By embedding deformations into a deformed AKNS (MAKNS) hierarchy using a deformation field $X$ and a spectral parameter $\lambda$, it derives infinite towers of anomalous conservation laws and exact non-local charges across DSG, MNLS, and deformed KdV, including dual formulations. The sine-Gordon, NLS, and KdV sectors appear as reductions of the MAKNS framework, and associated linear systems enable non-local conserved currents in addition to local charges. The results illuminate the structure and dynamics of quasi-integrable solitons and point to broad potential applications and future directions, such as non-Hermitian deformations and connections to gravity and condensed-matter physics.

Abstract

This review paper explores the Riccati-type pseudo-potential formulation applied to the quasi-integrable sine-Gordon, KdV, and NLS models. The proposed framework provides a unified methodology for analyzing quasi-integrability properties across various integrable systems, including deformations of the sine-Gordon, Bullough-Dodd, Toda, KdV, pKdV, NLS and SUSY sine-Gordon models. Key findings include the emergence of infinite towers of anomalous conservation laws within the Riccati-type approach and the identification of exact non-local conservation laws in the linear formulations of deformed models. As modified integrable models play a crucial role in diverse fields of nonlinear physics-such as Bose-Einstein condensation, superconductivity, gravity models, optics and soliton turbulence-these results may have far-reaching applications.

Figures (2)

• Figure 1: (color online) Numerical simulation of 2-soliton collision of the model (\ref{['mrkdv']}) for three successive times, $t_i$, before collision (green); $t_c$, collision (blue) and $t_f$, after collision (red). The parameter values are $\epsilon_1 = 1.2, \epsilon_2 = 0.9, \alpha = 4$.
• Figure 2: Top Fig. shows the anomaly density $(w_x v_t V_x)\,$ plotted in $x-$coordinate for three successive times, $t_i =$ before collision (green), $t_c=$ collision (blue) and $t_f=$ after collision (red), for the 2-soliton of Fig. 1. Middle Fig. shows the plot of $\int_{0}^{L} dx (w_x v_t V_x) \, \hbox{vs} \,\, t$ and the bottom Fig. shows the $t-$integrated anomaly $\int_{t_o}^{t} dt \int_{0}^{L} dx (w_x v_t V_x)\, \hbox{vs}\, \, t$. The last integral vanishes to within $10^{-9}$ precision.