Stability of Solitary Waves to the Shallow Water Models

主讲人：刘跃

摘要：In this talk, we will present our recent results on dynamical stability of localized smooth solitary waves to the Camassa-Holm (CH) equation and the modified Camassa-Holm (mCH) equation on the real line. Those quasilinear equations are completely integrable and arise as asymptotic models for the unidirectional propagation of shallow water waves.  We show orbital stability of the smooth solitary-wave solution of the mCH equation in the energy space $ H^1 \cap W^{1, 4}$ under small disturbances by variational methods without the assumption on the positive momentum density initially. We also demonstrate the dynamical stability of the shapes of all smooth multi-solitons under small disturbances in a suitable Sobolev space. Our approach in the spectral analysis consists in an invariant for the multi-solitons  with new operator identities motivated by the bi-Hamilton structure and integrable property of the recursion operator of the  CH equation.

主讲人简介：刘跃，美国德克萨斯⼤学阿灵顿分校教授，1994年获布朗⼤学博⼠学位。刘跃教授是⽬前国际上偏微分⽅程研究尤其是浅⽔波领域的⼀流专家。在偏微分⽅程，应⽤分析和流体⼒学，可积系统与孤⼦理论，⾮线性波⽅程的稳定性理论、奇异性形成、局部和整体适定性等领域取得国际领先的成果, 发表在《Comm. Pure Appl. Math.》, 《Ann. PDE》, 《J. Reine Angew. Math.》, 《J. Math. Pures Appl.》，《Adv. Math.》, 《Comm. Math. Phys.》, 《Arch. Ration. Mech. Anal.》, 《Math. Ann.,》，《J. Funct. Anal.》, 《Int. Math. Res. Not.,》，《Trans. AMS》等国际著名刊物上。

邀请人：李骥

时间：2023年6月18日（星期日）15:30-17:00