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arXiv · 2204.07395

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

Liming Ling·Xuan Sun
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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.

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Liming Ling, Xuan Sun. 2022-04-15. The multi elliptic-localized solutions and their asymptotic behaviors for the mKdV equation. https://doi.org/10.1111/sapm.12536

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