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

✍ Scribed by J.A. Leach

• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 233 KB
• Volume: 24
• Category: Article
• ISSN: 0893-9659
• DOI: 10.1016/j.aml.2010.09.006

✦ Synopsis

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

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.