The large-time development of the solution to an initial-value problem for the Korteweg–de Vries equation: III. Pure soliton solutions

In this paper, we consider an initial-value problem for the Korteweg-de Vries equation. The normalized Korteweg-de Vries equation considered is given by u(tau) + uu(x) + u(xxx) = 0, -infinity 0, where x and tau represent dimensionless distance and time respectively. In particular, we consider the case when the initial data is given by u(x, 0) = 6N(N + 1)k(2)sech(2) (kx), where k > 0 and N is a positive integer. The method of matched asymptotic coordinate expansions is used to obtain the large-tau asymptotic structure of the solution to this problem, which exhibits the formation of a N soliton solution structure in x > 0. Further, this solution is a pure soliton solution with no propagating oscillatory behavior in x <0. For N > 1 we determine the correction to the propagation speed of each of the N solitons as tau -> infinity and the rate of convergence to each of the N solitons as tau -> infinity. (C) 2010 Elsevier Ltd. All rights reserved.