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The large-time asymptotic solution of the mKdV equation
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.
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 MathematicsVolume
26Issue
6Pagination
931-943ISSN
0956-7925Department/School
School of Natural SciencesPublisher
Cambridge Univ PressPlace of publication
40 West 20Th St, New York, USA, Ny, 10011-4211Rights statement
?Copyright Cambridge University Press 2015Repository Status
- Restricted