Variational analysis of a nonlinear Klein-Gordon equation

Abstract

Many nonlinear Klein-Gordon equations have been studied numerically, and in a few cases, analytical solutions have been found. We used the variational method to study three different equations in this family. The first one to be studied here was the linear equation, Utt - Uzz + U = 0, where U is a real Klein-Gordon field. Attempts to find non-stationary radiative-type solutions of this equation were not successful. Next we studied the nonlinear equation Utt - U:= ± IUl 2U = O, with U complex, which represents a nonlinear massless scalar field. Here we searched for possible stationary solutions using the variational approximation, however to no avail. Next, we added a linear term to this second equation, which then became Utt - Uzll: ± IUl2U + µU = 01 whereµ can always be scaled to ±1. Here we found that we can find approximate variational solutions of the form A(t)e^i{k(x-z0(t))+a)e / 2w2(z) . This third equation is a generalization of the tf,4 equation, which has many physical applications. However, the variational solution found required different signs on the coefficients of this equation than are found in the O4 equation. Properties and features of this variational solution will be discussed.

Notes

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Thesis Completion

2008

Semester

Spring

Advisor

Kaup, David J.

Degree

Bachelor of Science (B.S.)

College

College of Sciences

Degree Program

Mathematics

Subjects

Dissertations, Academic -- Sciences;Sciences -- Dissertations, Academic

Format

Print

Identifier

DP0022321

Language

English

Access Status

Open Access

Length of Campus-only Access

None

Document Type

Honors in the Major Thesis

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