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The large-time asymptotic solution of the mKdV equation

journal contribution
posted on 2023-05-18, 15:45 authored by Leach, JA, Andrew BassomAndrew Bassom
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.

History

Publication title

European Journal of Applied Mathematics

Volume

26

Issue

6

Pagination

931-943

ISSN

0956-7925

Department/School

School of Natural Sciences

Publisher

Cambridge Univ Press

Place of publication

40 West 20Th St, New York, USA, Ny, 10011-4211

Rights statement

?Copyright Cambridge University Press 2015

Repository Status

  • Restricted

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