Numerical evaluations of quasi-periodic wave solutions to two supersymmetric integrable equations

Abstract

A new algorithm named global Levenberg–Marquardt (LM) algorithm is proposed to explicitly construct quasi-periodic wave solutions of the supersymmetric integrable systems, which combines the super Riemann theta functions and Hirota’s bilinear method. In particular, the numerical quasi-periodic wave solutions for the supersymmetric KdV–Sawada–Kotera–Ramani equation and supersymmetric Ito’s equation are investigated. The quasi-periodic wave solvability problem is successfully transformed into a system of over-determined equations, which can be formulated into a least square problem and then solved by using the global LM algorithm. The quasi-periodic wave solutions can be classified into two categories including quasi-periodic parallel waves and quasi-periodic cross waves. A distinct analysis of the factors that influence the emergence of “influencing band” and the bandwidth of “influencing band” is presented. Furthermore, by using an analytical method related to the characteristic lines for the quasi-periodic waves, the dynamic characteristics including the periods, wave propagation direction, peak points, trough points, velocities and the distance between two peaks can be intuitively displayed. In addition, based on the small amplitude limit method, the asymptotic properties of $N$-periodic waves to the supersymmetric integrable systems are presented.