Oscillatory Solutions for Sine-Gordon Equation

Year: 2010 Authors: Ion Bica

Core claim

A limit of the 2-soliton formula yields a singular oscillatory solution of the Sine-Gordon equation.

Topics

Sine-Gordon equation, 2-soliton solution, singular oscillatory behavior, limiting process

Domains

nonlinear partial differential equations, soliton theory, inverse scattering method

Methods

2-soliton limiting process, Taylor series expansion, analytic formula derivation

Media

equation formulas, 3D solution figure, personal FotoShopped picture

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2010: Mathematics, Music, Art, Architecture, Culture

Oscillatory Solutions for Sine-Gordon Equation

Ion Bica Grant MacEwan University Department of Mathematics & Statistics 10700-104 Avenue, Edmonton, Alberta, Canada T5J 4S2 Phone: 1-780-633-3910 E-mail: bicai@macewan.ca

Abstract

In this paper I show how from the 2-soliton solution of the Sine-Gordon equation I create a new solution of this equation. The new solution is oscillatory, but singular.

Motivation

The Sine-Gordon equation, $u_{it}-u_{xx}+\sin u=0$ [1,2,5], has a $N$ -soliton formula [5], which describes the interaction of an arbitrary number $N$ of solitons. These types of oscillatory solutions derived from the N-soliton formula can possibly give a better understanding of singular phenomena that can happen in a system, like rogue (freak) waves for example, where the massive wave front can be understood as a singularity created by an unusual phenomenon (like an earthquake). The Sine-Gordon equation is a nontrivial model of the Field Theory as well. These types of oscillatory solutions could bring a better understanding of unusual phenomena in this field. There is a lot of study ahead, but in this paper I want to bring to attention these types of solutions, which usually bring controversy because of the singularity.

Constructing the soliton-like solutions for the Sine-Gordon equation

The Sine-Gordon equation has the analogue representation:

I construct the oscillatory solutions for the Sine-Gordon equation (1) by applying a limiting process to the 2-soliton solution of the Sine-Gordon equation (1) [5]:

where $I$ is the $2\times 2$ identity matrix and $V$ is the $2\times 2$ matrix with the following entries:

The parameters ${\lambda }_1,{\lambda }_2$ and $c_1,c_2$ are complex parameters (they can be assigned to be real as well). In the formula (2) we consider: ${\lambda }_1=1/{\mu }_1-{\mu }_1\epsilon i,{\lambda }_2=-{\overline{\lambda }}_1,c_1=2\epsilon {\mu }_1\exp (1/((1/p_1)+\epsilon )+{\alpha }_1\epsilon /2)$ ,and $c_2=-2\epsilon {\mu }_1\exp (-1/((1/p_1)+\epsilon )+{\alpha }_1\epsilon /2)$ . Taking Taylor series expansion about $\epsilon =0$ , and taking a limiting process as $\epsilon \to 0$ , the formula (2) becomes:

where:

Bica

The solution (3) is an oscillatory solution of (1), and it is a singular solution as well. The singularity is a result of the decaying behavior of the oscillations in space and time. The formula (3) is governed by the real parameters ${\mu }_{\mathrm{t}}\ne 0$ , $p_{\mathrm{t}}\ne 0$ , and ${\alpha }_{\mathrm{t}}$ .

Figure: Oscillatory solution in 3D. The first picture shows the regular part of the solution. The second picture shows the singularity. Parameters used: ${\mu }_{\mathrm{t}}=-1$ , $p_{\mathrm{t}}=\ln (2)$ , ${\alpha }_{\mathrm{t}}=50$ .

Here is a beautiful picture showing oscillations near singularity and a personal picture with a twist (using the described solutions):

http://www.universaltheory.org/Singularity.

Personal FotoShopped picture

References

[1] J. M. Ablowitz, A. P. Clarkson, Solitons, Non-Linear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Notes, Cambridge University Press, 1991. [2] G. P. Drazin, S. R. Johnson, Solitons: an introduction, Cambridge University Press, 1989. [3] B. B. Kadomtsev, I. V. Petviashvili, On the stability of solitary waves in weakly dispersive media, Soviet Physics - Doklady, Fluid Mechanics, 1970, Vol. 15, No. 6, pp.539-541. [4] M. Kovalyov, I. Bica, Some properties of slowly decaying oscillatory solutions of $KP$ , Chaos, Solitons and Fractals, 2005, Vol. 25, pp.979-989. [5] V. Novikov, S. V. Manakov, P. L. Pitaevskii, E. V. Zakharov, Theory of Solitons, The Inverse Scattering Method, Contemporary Soviet Mathematics, Consultants Bureau, Plenum Publishing Corporation, 1984.