The integrable shallow-water models with cubic nonlinearity
报告人:刘跃教授,The University of Texas at Arlington 时间:2019年12月30日10:00
摘要: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.