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+$k\rightarrow 0^{+}$, 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.
We study the spectral (linear) stability and orbital (nonlinear) stability of the elliptic solutions for the focusing modified Korteweg-de Vries (mKdV) equation with respect to subharmonic perturbations and construct the corresponding breather solutions to exhibit the unstable or stable dynamic behavior. The elliptic function solutions of mKdV equation and the fundamental solutions of Lax pair are exactly represented by using the theta function. Based on the 'modified squared wavefunction' (MSW) method, we construct all linear independent solutions of the linearized KdV equation, and then provide a necessary and sufficient condition of the spectral stability for the elliptic function solutions with respect to subharmonic perturbations. In the case of spectrum stable, the orbital stability of the elliptic function solutions with respect to subharmonic perturbations is established under a suitable Hilbert space. Using Darboux-Bäcklund transformation, we construct the breather solutions to exhibit the unstable or stable dynamic behavior. Through analyzing the asymptotical behavior, we find the breather solution under the cn-background is equivalent to the elliptic function solution adding a small perturbation as t → ±∞.
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