标题： 椭圆函数解对于修正KdV方程的聚焦形式的稳定性 显示英文标题

标题： Stability of elliptic function solutions for the focusing modified KdV equation

摘要： 我们研究了聚焦修正Korteweg-de Vries (mKdV) 方程的椭圆函数解的谱稳定性和轨道稳定性，并构造相应的呼吸子解来展示稳定的或不稳定的动态行为。 mKdV方程的椭圆函数解和相关的基本Lax对解可以精确地表示为θ函数。 基于‘修正平方波函数’(MSW) 方法，我们构造了线性化mKdV方程的所有线性独立解，然后提供了关于次谐波扰动的椭圆函数解谱稳定性的充分必要条件。 在谱稳定的情况下，在适当的Hilbert空间中建立了椭圆函数解的轨道稳定性。 利用Darboux-Bäcklund变换，我们构造呼吸子解来展示不稳定或稳定的动态行为。 通过分析渐近行为，我们发现，在cn型解背景下，呼吸子解等价于加上一个小扰动后的椭圆函数解$t\to\pm\infty$。 显示英文摘要

摘要： We study the spectral and orbital stability of elliptic function solutions for the focusing modified Korteweg-de Vries (mKdV) equation and construct the corresponding breather solutions to exhibit the stable or unstable dynamic behavior. The elliptic function solutions of the mKdV equation and related fundamental solutions of the Lax pair are exactly represented by theta functions. Based on the `modified squared wavefunction' (MSW) method, we construct all linear independent solutions of the linearized mKdV equation and then provide a necessary and sufficient condition of the spectral stability for elliptic function solutions with respect to subharmonic perturbations. In the case of spectrum stability, the orbital stability of elliptic function solutions is established in a suitable Hilbert space. Using Darboux-B\"acklund transformation, we construct breather solutions to exhibit unstable or stable dynamic behavior. Through analyzing the asymptotic behavior, we find that the breather solution under the cn-type solution background is equivalent to the elliptic function solution adding a small perturbation as $t\to\pm\infty$.