The large-time development of the solution to an initial-value problem for the generalized Korteweg–de Vries equation

Abstract

In this work we address an initial-value problem for the generalized Korteweg–de Vries equation. The normalized generalized Korteweg–de Vries (gKdV) equation considered is given by u τ + u k u x + u x x x = 0 , − ∞ < x < ∞ , τ > 0 , where x and τ represent dimensionless distance and time respectively and k ( > 1 ) is an odd positive integer. We consider the case with the initial data having a discontinuous expansive step, where u ( x , 0 ) = u 0 for x ≥ 0 and u ( x , 0 ) = 0 for x < 0 . In particular, we present the large- τ asymptotic structure of the solution to this problem, which exhibits the formation of an expansion wave in x ≥ 0 , while the solution is oscillatory in x < 0 , with the oscillatory envelope being of O ( τ − 1 2 ) as τ → ∞ . This work extends the asymptotic theory developed by Leach and Needham [J.A. Leach, D.J. Needham, The large-time development of the solution to an initial-value problem for the Korteweg–de Vries equation. I. Initial data has a discontinuous expansive step, Nonlinearity 21 (2008) 2391–2408] for this problem when k = 1 .