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 u(t) - 6u(2)u(x) + u(xxx) = 0, -infinity < x < infinity, 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), -infinity < x < infinity, 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.