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The large-time asymptotic solution of the mKdV equation

Published online by Cambridge University Press:  04 June 2015

J. A. LEACH  and
ANDREW P. BASSOM
J. A. LEACH
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK email: [email protected]
ANDREW P. BASSOM
Affiliation:
School of Mathematics & Statistics, The University of Western Australia, Crawley WA 6009, Australia email: [email protected]
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Abstract

In this paper, an initial-value problem for the modified Korteweg-de Vries (mKdV) equation is addressed. Previous numerical simulations of the solution of

\[ u_{t} - 6u^{2} u_{x}+u_{xxx}=0, \quad -\infty<x<\infty, \quad t>0, \]
where x and t represent dimensionless distance and time respectively, have considered the evolution when the initial data is given by
\[ u(x,0)=\tanh ( Cx ), \quad -\infty<x<\infty, \]
for C constant. These computations suggest that kink and soliton structures develop from this initial profile and here the method of matched asymptotic coordinate expansions is used to obtain the complete large-time structure of the solution in the particular case C = 1/3. The technique is able to confirm some of the numerical predictions, but also forms a basis that could be easily extended to account for other initial conditions and other physically significant equations. Not only can the details of the relevant long-time structure be determined but rates of convergence of the solution of the initial-value problem be predicted.

Keywords

Asymptotic methodslarge time solutionsolitons and kinks
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 6 , December 2015 , pp. 931 - 943
Copyright
Copyright © Cambridge University Press 2015 

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