Modulational instability in some shallow water wave models

Pandey, Ashish Kumar

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https://hdl.handle.net/2142/101023 Copy

Description

Title

Modulational instability in some shallow water wave models

Author(s)

Pandey, Ashish Kumar

Issue Date

2018-04-19

Director of Research (if dissertation) or Advisor (if thesis)

Hur, Vera M

Doctoral Committee Chair(s)

Bronski, Jared C.

Committee Member(s)

• Laugesen, Richard
• Kirr, Eduard W

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

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

Abstract

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.

Graduation Semester

2018-05

Type of Resource

text

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

Copyright and License Information

Copyright 2018 Ashish Pandey

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