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I. Exact solution of some nonlinear evolution equations. II. The similarity solution for the Korteweg-De Vries equation and the related Painleve Transcendent

Citation

Rosales, Rodolfo Ruben (1977) I. Exact solution of some nonlinear evolution equations. II. The similarity solution for the Korteweg-De Vries equation and the related Painleve Transcendent. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:03262013-081944588

Abstract

In Part I, a method for finding solutions of certain diffusive dispersive nonlinear evolution equations is introduced. The method consists of a straightforward iteration procedure, applied to the equation as it stands (in most cases), which can be carried out to all terms, followed by a summation of the resulting infinite series, sometimes directly and other times in terms of traces of inverses of operators in an appropriate space.

We first illustrate our method with Burgers' and Thomas' equations, and show how it quickly leads to the Cole-Hopft transformation, which is known to linearize these equations.

We also apply this method to the Korteweg and de Vries, nonlinear (cubic) Schrödinger, Sine-Gordon, modified KdV and Boussinesq equations. In all these cases the multisoliton solutions are easily obtained and new expressions for some of them follow. More generally we show that the Marcenko integral equations, together with the inverse problem that originates them, follow naturally from our expressions.

Only solutions that are small in some sense (i.e., they tend to zero as the independent variable goes to ∞) are covered by our methods. However, by the study of the effect of writing the initial iterate u_1 = u_(1)(x,t) as a sum u_1 = ^∼/u_1 + ^≈/u_1 when we know the solution which results if u_1 = ^∼/u_1, we are led to expressions that describe the interaction of two arbitrary solutions, only one of which is small. This should not be confused with Backlund transformations and is more in the direction of performing the inverse scattering over an arbitrary “base” solution. Thus we are able to write expressions for the interaction of a cnoidal wave with a multisoliton in the case of the KdV equation; these expressions are somewhat different from the ones obtained by Wahlquist (1976). Similarly, we find multi-dark-pulse solutions and solutions describing the interaction of envelope-solitons with a uniform wave train in the case of the Schrodinger equation.

Other equations tractable by our method are presented. These include the following equations: Self-induced transparency, reduced Maxwell-Bloch, and a two-dimensional nonlinear Schrodinger. Higher order and matrix-valued equations with nonscalar dispersion functions are also presented.

In Part II, the second Painleve transcendent is treated in conjunction with the similarity solutions of the Korteweg-de Vries equat ion and the modified Korteweg-de Vries equation.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Applied Mathematics
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied Mathematics
Thesis Availability:Restricted to Caltech community only
Research Advisor(s):
  • Whitham, Gerald Beresford
Thesis Committee:
  • Unknown, Unknown
Defense Date:24 May 1977
Record Number:CaltechTHESIS:03262013-081944588
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:03262013-081944588
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7554
Collection:CaltechTHESIS
Deposited By: Dan Anguka
Deposited On:09 Oct 2013 17:08
Last Modified:09 Oct 2013 17:08

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