A note on the large-time development of the solution to an initial-value problem for a variable coefficient Korteweg-de Vries equation

作者：Leach J A^*

来源：IMA Journal of Applied Mathematics, 2017, 82(4): 717-725.

DOI：10.1093/imamat/hxx009

• Korteweg-de Vries equation
• non-linear waves
• asymptotic methods

In this article, we consider an initial-value problem for a variable coefficient Korteweg-de Vries equation. The normalized variable coefficient Korteweg-de Vries equation considered is given by
u(t) + uu(x) + e(alpha t)u(xxx) = 0, -infinity < x < infinity, t > 0,
where x and t represent dimensionless distance and time respectively, and alpha(> 0) is a constant. In particular, we consider the case when the initial data has a discontinuous expansive step, where u(x, 0) = u(+) for x >= 0 and u(x, 0) = u_ for x < 0. The method of matched asymptotic coordinate expansions is used to obtain the large-t asymptotic structure of the solution to this problem. We find that the large-t attractor for the solution u(x, t) of the initial-value problem is based on the integral of the standard Airy function, where
u(ze(alpha/3t), t) -> [(u(-) 2u(+))/3 + (u(+) - u(-)) integral((alpha/3)1/3z)(0) Ai(s) ds]
as t -> infinity with z = xe(-alpha/3t) = O(1). Further, this large-t attractor forms in a stretching frame of reference of thickness x = O (e(alpha t/3)) as t -> infinity.

• 出版日期2017-8