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

We mainly construct and analyze the multi elliptic-localized solutions under background of elliptic function for focusing modified Korteweg-de Vries (mKdV) equation. Based on Darboux–Bäcklund transformation, we provide a uniform expression these by Jacobi theta functions. The asymptotic behaviors are provided directly in two categories. By consistent those solutions, obtain that collisions between elliptic-breathers/solitons elastic. Moreover, sufficient condition strictly elastic collision solitons breathers has been given symmetric analysis. In addition, as k → 0 + $k\rightarrow 0^{+}$ , degenerate into solitons, breathers, or soliton-breather which implies extend from constant vanishing backgrounds to periodic backgrounds. illustrate figures visualize above