Modulational instability in some shallow water wave models

shallow water models;modulational instability;BBM equation;Camassa-Holm equation;full-dispersion;surface tension

Pandey, Ashish Kumar

关键词: shallow water models;    modulational instability;    BBM equation;    Camassa-Holm equation;    full-dispersion;    surface tension;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/101023/PANDEY-DISSERTATION-2018.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

Modulational or Benjamin-Feir instability is a well known phenomenon of Stokes' periodic waves on the water surface. In this dissertation, we study this phenomenon for periodic traveling wave solutions of various shallow water wave models. We study the spectral stability or instability with respect to long wave length perturbations of small amplitude periodic traveling waves of shallow water wave models like Benjamin-Bona-Mahony and Camassa-Holm equations. We propose a bi-directional shallow water model which generalizes Whitham equation to contain the nonlinearities of nonlinear shallow water equations. The analysis yields a modulational instability index for each model which is solely determined by the wavenumber of underlying periodic traveling wave. For a fixed wavenumber, the sign of the index determines modulational instability. We also includes the effects of surface tension in full-dispersion shallow water models and study its effects on modulational instability.