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The large-time asymptotic solution of the mKdV equation

Citation

Leach, JA and Bassom, AP, The large-time asymptotic solution of the mKdV equation, European Journal of Applied Mathematics, 26, (6) pp. 931-943. ISSN 0956-7925 (2015) [Refereed Article]

Copyright Statement

Copyright Cambridge University Press 2015

DOI: doi:10.1017/S095679251500025X

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

ut − 6u2ux + uxxx = 0,     −∞ < x < ∞,      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),     −∞ < x < ∞,

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.

Item Details

Item Type:Refereed Article
Keywords:asymptotic methods, large time solution, solitons and kinks
Research Division:Mathematical Sciences
Research Group:Applied Mathematics
Research Field:Theoretical and Applied Mechanics
Objective Division:Expanding Knowledge
Objective Group:Expanding Knowledge
Objective Field:Expanding Knowledge in the Mathematical Sciences
Author:Bassom, AP (Professor Andrew Bassom)
ID Code:105711
Year Published:2015
Deposited By:Mathematics and Physics
Deposited On:2016-01-13
Last Modified:2016-11-25
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