The multi elliptic-localized solutions and their asymptotic behaviors for the mKdV equation

TL;DR

This paper constructs and analyzes multi elliptic-localized solutions for the focusing mKdV equation, revealing their asymptotic behaviors, elastic collisions, and degenerations into solitons or breathers, extending solutions from constant backgrounds to periodic ones.

Contribution

It introduces a uniform Jacobi theta function expression for multi elliptic-localized solutions and analyzes their asymptotic behaviors and collision properties, extending the solution framework to periodic backgrounds.

Findings

Collisions between elliptic-breathers and solitons are elastic.

Solutions degenerate into solitons or breathers as the elliptic modulus approaches zero.

The paper visualizes the solutions to support the analytical results.

Abstract

We mainly construct and analyze the multi elliptic-localized solutions under the background of elliptic function solutions for the focusing modified Korteweg-de Vries (mKdV) equation. Based on the Darboux-B\"{a}cklund transformation, we provide a uniform expression for these solutions by the Jacobi theta functions. The asymptotic behaviors of multi elliptic-localized solutions are provided directly in two categories. By the consistent asymptotic expression of those solutions, we obtain that the collisions between the elliptic-breathers/solitons are elastic. Moreover, a sufficient condition of the strictly elastic collision between the solitons and breathers has been given by the symmetric analysis. In addition, as $k\rightarrow0^{+}$, the multi elliptic-localized solutions degenerate into solitons, breathers or soliton-breather solutions, which implies that we extend the solutions from the constant and vanishing backgrounds to the periodic solutions backgrounds. Moreover, we illustrate figures of the multi elliptic-localized solutions to visualize the above analysis.