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

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 uτ+ukux+uxxx=0,−∞0, where xx and ττ 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)=u0u(x,0)=u0 for x≥0x≥0 and u(x,0)=0u(x,0)=0 for x<0x<0. In particular, we present the large-ττ asymptotic structure of the solution to this problem, which exhibits the formation of an expansion wave in x≥0x≥0, while the solution is oscillatory in x<0x<0, with the oscillatory envelope being of O(τ−12) as τ→∞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=1k=1.