講座題目：SINGULARITY FORMATION AND SOLITARY WAVES TO THE SHALLOW WATER MODELING WITH CUBIC NONLINEARITY
主 講 人：劉躍 教授
In the present study several integrable equations with cubic nonlinearity are derived as asymptotic models from the classical shallow water theory. The starting point in our derivation is the Euler equation for an incompressible fluid with the simplest bottom and surface conditions. The approximate model equations are generated by introducing suitable scalings and by truncating asymptotic expansions of the quantities to appropriate order combining with the Kodama transformation. The so obtained equations can be related to the following integrable systems: the Novikov equation, the modified Camassa-Holm equation, and the Camassa-Holm type equation with cubic nonlinearity. These model equations have a formal bi-Hamiltonian structure and possess single and muti-peaked solutions. Their solutions corresponding to physically relevant initial perturbations are more accurate on a much longer time scale. The effect of the nonlocal higher nonlinearities on wave-breaking phenomena to these quasi-linear model equations are also investigated. Our analysis is approached by applying the method of characteristics and conserved quantities to the Riccati-type differential inequality.
劉躍，美國德克薩斯大學阿靈頓分校教授，1994年博士畢業于美國布朗大學，主要從事非線性水波模型問題的研究，在偏微分方程，應用分析和流體力學，一大類淺水波模型的推導、分析、穩定性理論、奇異性形成、局部和整體適定性等方面做出了許多國際一流的工作。其研究成果發表在《Comm. Pure Appl. Math.》《Adv. Math.》《Comm. Math. Phys.》《Arch. Ration.Mech. Anal.》《J. Funct. Anal.》等國際著名刊物上。