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

摘要

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–Bäcklund 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→0+$krightarrow 0^{+}$, 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.