Existence of periodic waves in some perturbed shallow water model


主讲人:孙宪波 广西财经学院副教授




内容介绍:We consider a generalized BBM equation with weak backward diffusion, dissipation  and Marangoni effects, and study the existence of periodic and solitary waves.  Main attention is focused on periodic and solitary waves on a manifold via  studying the number of zeros of some linear combination of Abelian integrals.  The uniqueness of the periodic waves is established when the equation contains  one coefficient in backward diffusion and dissipation terms, by showing that the  Abelian integrals form a Chebyshev set. The monotonicity of the wave speed is  proved, and moreover the upper and lower bounds of the limiting wave speeds are  obtained. Especially, when the equation involves Marangoni effect due to imposed  weak thermal gradients, it is shown that at most two periodic waves can exist.  The exact conditions are obtained for the existence of one and two periodic  waves as well as for the co-existence of one solitary and one periodic waves.  The analysis is mainly based on Chebyshev criteria and asymptotic expansions of  Abelian integrals near the solitary and singularity.