Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



The large-time development of the solution to an initial-value problem for the Korteweg-de Vries equation: IV. Time dependent coefficients

Author: J. A. Leach
Journal: Quart. Appl. Math.
MSC (2010): Primary 35Q53
DOI: https://doi.org/10.1090/qam/1481
Published electronically: September 20, 2017
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Abstract: In this paper, we consider an initial-value problem for the Korteweg-de Vries equation with time dependent coefficients. The normalized variable coefficient Korteweg-de Vries equation considered is given by

$\displaystyle u_{t}+ \Phi (t) u u_{x}+ \Psi (t) u_{xxx}=0, \quad -\infty <x<\infty , \quad t>0,$    

where $ x$ and $ t$ represent dimensionless distance and time respectively, whilst $ \Phi (t)$, $ \Psi (t)$ are given functions of $ t (>0)$. In particular, we consider the case when the initial data has a discontinuous expansive step, where $ u(x,0)=u_{+}$ for $ x \ge 0$ and $ u(x,0)=u_{-}$ for $ x<0$. We focus attention on the case when $ \Phi (t)=t^{\delta }$ (with $ \delta >-\frac {2}{3}$) and $ \Psi (t)=1$. The constant states $ u_{+}$, $ u_{-}$ ($ <u_{+}$) and $ \delta $ are problem parameters. The method of matched asymptotic coordinate expansions is used to obtain the large-$ t$ asymptotic structure of the solution to this problem, which exhibits the formation of an expansion wave in $ x \ge \frac {u_{-} }{(\delta +1)}t^{(\delta +1)}$ as $ t \to \infty $, while the solution is oscillatory in $ x<\frac {u_{-}}{(\delta +1)}t^{(\delta +1)}$ as $ t \to \infty $. We conclude with a brief discussion of the structure of the large-$ t$ solution of the initial-value problem when the initial data is step-like being continuous with algebraic decay as $ \vert x\vert \to \infty $, with $ u(x,t) \to u_{+}$ as $ x \to \infty $ and $ u(x,t) \to u_{-} (<u_{+})$ as $ x \to -\infty $.

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Additional Information

J. A. Leach
Affiliation: School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, U.K.

DOI: https://doi.org/10.1090/qam/1481
Received by editor(s): March 27, 2017
Received by editor(s) in revised form: August 2, 2017
Published electronically: September 20, 2017
Article copyright: © Copyright 2017 Brown University

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