The Static Elliptic $N$-soliton Solutions of the KdV Equation

TL;DR

This paper constructs static elliptic N-soliton solutions for the KdV equation using Bäcklund transformations, extending the known single-soliton elliptic solutions to infinitely many multi-soliton solutions.

Contribution

It introduces a method to generate static elliptic N-soliton solutions for the KdV equation, a significant extension beyond the previously known single-soliton solutions.

Findings

Successfully constructed static elliptic N-soliton solutions

Extended the class of known elliptic solutions for KdV

Demonstrated the use of Bäcklund transformations for multi-soliton solutions

Abstract

Regarding $N$-soliton solutions, the trigonometric type, the hyperbolic type, and the exponential type solutions are well studied. While for the elliptic type solution, we know only the one-soliton solution so far. Using the commutative B\"{a}cklund transformation, we have succeeded in constructing the KdV static elliptic $N$-soliton solution, which means that we have constructed infinitely many solutions for the $\wp$-function type differential equation.

Full text

The Static Elliptic $N$-soliton Solutions of the KdV Equation

Masahito Hayashi

Osaka Institute of Technology, Osaka 535-8585, Japan

Kazuyasu Shigemoto

Tezukayama University, Nara 631-8501, Japan

Takuya Tsukioka

Bukkyo University, Kyoto 603-8301, Japan masahito.hayashi@oit.ac.jpshigemot@tezukayama-u.ac.jptsukioka@bukkyo-u.ac.jp