Multi-cnoidal Solutions of Korteweg-de Vries Evolution Equation

TL;DR

This paper derives multi-cnoidal wave solutions for the KdV equation using prolongation structure theory, revealing nonlinear interactions, superpositions, and effects on wave patterns relevant for understanding complex wave phenomena.

Contribution

It introduces a method to construct multi-cnoidal solutions of KdV, extending superposition techniques and analyzing nonlinear wave interactions.

Findings

Nonlinear beating occurs in superposed cnoidal waves.

Superposition with solitons alters wave amplitudes and patterns.

Extra cnoidal waves can disrupt beating patterns.

Abstract

The N-cnoidal solution of the Korteweg-de Vries (KdV) evolution equation is presented based on the prolongation structure theory of Wahlquist and Estabrook [J. Math. Phys. \textbf{16}, 1 (1975)]. The generalized KdV cnoidal wave solutions satisfying both the evolution as well as the potential equations is obtained and the multi-cnoidal components are extracted from the regular and singular potential components. Current technique for construction of superposed cnoidal waves is the immidiate generalization of the N-soliton solution for KdV using the B\"acklund transformation proceedure. Quite analogous to the linear effect, the nonlinear beating is observed to exist also for nonlinear superposition of two cnoidal waves. It is further found that the nonlinear superposition of cnoidal wave with a soliton alters the whole periodic wave pattern declining the amplitude of the soliton significantly. In a three-wave nonlinear interaction of cnoidal waves it is remarked that introduction of extra cnoidal wave tends to destroy the beating pattern formed by the two of them. Furthermore, the superposition of two solitons with a cnoidal wave results in only one soliton hump traveling with the smaller soliton speed in the periodic background and with the other soliton disappeared. Current findings can help to better understand the nonlinear periodic wave interactions and nonlinear decomposition of the realistic experimental data to its components.