Stability of periodic solutions to nonlinear Klein-Gordon equations

Venkatasubbu, Nanjundamurthy

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https://hdl.handle.net/2142/42221 Copy

Description

Title

Stability of periodic solutions to nonlinear Klein-Gordon equations

Author(s)

Venkatasubbu, Nanjundamurthy

Issue Date

2012-12

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

Bronski, Jared C.

Department of Study

Aerospace Engineering

Discipline

Aerospace Engineering

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

M.S.

Degree Level

Thesis

Keyword(s)

• nonlinear klein gordon equation
• modulational instability
• periodic solutions
• evans function
• stability index

Abstract

We study the stability of periodic travelling wave solutions to nonlinear Klein-Gordon equations, subject to small and localized perturbations. Using the periodic Evans function technique, we analyze the spectrum of a linearized quadratic eigenvalue pencil associated with the linearized equation, in a neighbourhood of the origin on the spectral plane. The stability criterion is expressed in terms of signature of an index involving physical parameters of the wave, thereby holding the result valid for a general nonlinearity F(u). The result is then verified for the following cases: • cubic nonlinearity; F(u) = u3 − u • sine Gordon equation; F(u) = − sin(u)

Graduation Semester

2012-12

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http://hdl.handle.net/2142/42221

Copyright and License Information

Copyright 2012 by Nanjundamurthy Venkatasubbu. All rights reserved.

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