Numerical calculation ofN-periodic wave solutions to coupled KdV–Toda-type equations

Abstract

In this paper, we study the N -periodic wave solutions of coupled Korteweg–de Vries (KdV)–Toda-type equations. We present a numerical process to calculate the N -periodic waves based on the direct method of calculating periodic wave solutions proposed by Akira Nakamura. Particularly, in the case of N = 3, we give some detailed examples to show the N -periodic wave solutions to the coupled Ramani equation, the Hirota–Satsuma coupled KdV equation, the coupled Ito equation, the Blaszak–Marciniak lattice, the semi-discrete KdV equation, the Leznov lattice and a relativistic Toda lattice.

Numerical calculation of N-periodic wave solutions of the negative-order Korteweg-de Vries equations

“…Matlab [46][47][48][49][50]. N -periodic wave solutions can be seen as a periodic extension of the N -soliton solutions or multiple collisions of N -soliton.…”

Introduction to special feature dedicated to Prof. Allan Fordy on the occasion of his 70th birthday

“…An important classical problem (the Schottky problem) is to characterize the period matrices that can appear in these theta functions: precisely, to determine which such period matrices correspond to periods of algebraic curves (which is the same as characterizing Jacobians among abelian varieties). Rather than starting from a given spectral curve and calculating period integrals, an alternative approach to finding N-phase solutions of coupled KdV equations and Toda-type systems is presented in [27], where Zhang and co-authors obtain numerical solutions of the overdetermined systems for the entries of the period matrices and other parameters appearing in the theta functions. The periods of the spectral curve depend in a very nontrivial way on its moduli, and the question of how best to describe this in terms of invariant differential operators acting on first and second kind differentials is addressed in Athorne’s paper [15].…”