# 【预告】钩深讲坛：SINGULARITY FORMATION AND SOLITARY WAVES TO THE SHALLOW WATER MODELING WITH CUBIC NONLINEARITY

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.

刘跃，美国德克萨斯大学阿灵顿分校教授，1994年博士毕业于美国布朗大学，主要从事非线性水波模型问题的研究，在偏微分方程，应用分析和流体力学，一大类浅水波模型的推导、分析、稳定性理论、奇异性形成、局部和整体适定性等方面做出了许多国际一流的工作。其研究成果发表在《Comm. Pure Appl. Math.》《Adv. Math.》《Comm. Math. Phys.》《Arch. Ration.Mech. Anal.》《J. Funct. Anal.》等国际著名刊物上。 