Modulational instability in some shallow water wave models

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2018

Author dc:creator

• Pandey, Ashish Kumar

Contributors dc:contributor

• Hur, Vera M
• Bronski, Jared C.
• Laugesen, Richard
• Kirr, Eduard W

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

Statement dc:rights

• Copyright 2018 Ashish Pandey

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/101023

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/101023