Initial-boundary value problem for the Camassa-Holm equation with linearizable boundary condition
type: Item
instance of: scholarly article
title: Initial-boundary value problem for the Camassa-Holm equation with linearizable boundary condition
zbMATH Open document ID: 1230.37089
published in: Letters in Mathematical Physics
publication date: 2011-05-27T00:00:00Z
review text: Consider the initial boundary value problem for the Camassa-Holm (CH) equation on the half-line \[ u_t - u_{txx} + 2\omega u_x + 3 u u_x = 2 u_x u_{xx} +u u_{xxx}, \quad x > 0, \;t>0, \] with homogeneous boundary conditions \[ u(x,0)=u_0(x), \quad x> 0, \qquad u(0,t)=0, \quad t>0, \] where \(\omega > 0\) is a parameter and \(u_0(x)\) is assumed to be sufficiently smooth, fast decaying as \(x \rightarrow + \infty\) and satisfies \(u_0(0)=0\). The authors adapt the inverse scattering method for the initial value problem of the CH equation to IBV problems and address the subtle issue of ``full'' boundary conditions. They show that the above IBV problem is ``linearizable'' which means that its solution can be expressed in terms of the solution of a Riemann-Hilbert problem whose jump matrix involves spectral functions associated with the initial data \(u_0(x)\) only. This allows to apply the non-linear steepest descent method. The asymptotics of \(u(x,t)\) as \(t\rightarrow \infty\) are then derived in terms of these spectral functions for the soliton region (\(\frac{x}{t} > 2 - \varepsilon\), \(\varepsilon >0\)), the Zakharov-Manakov regime (\(\varepsilon < \frac{x}{t}< 2-\varepsilon\)), where the solution is described by decaying modulated oscillations, and for the Painlevé region (\(|\frac{x}{t}-2|t^{2/3}< C\)). These results were first announced in [the authors, C. R., Math., Acad.\ Sci.\ Paris 348, No. 13--14, 775--780 (2010; Zbl 1200.35205)].
reviewed by: Johanna Michor
Mathematics Subject Classification ID: 37K15
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35Q15
Mathematics Subject Classification ID: 65M12
zbMATH DE Number: 5900552
zbMATH Keywords: initial boundary value problem
zbMATH Keywords: integrable system
zbMATH Keywords: Camassa-Holm
zbMATH Keywords: long time asymptotics
zbMATH Keywords: Riemann-Hilbert problem
zbMATH Keywords: linearizable boundary condition
author: Anne Boutet De Monvel
author: Dmitry Shepelsky
MaRDI profile type: Publication
full work available at URL: s11005-010-0457-6
OpenAlex ID: W1996577409
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DOI: 10.1007/S11005-010-0457-6
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description: scientific article; zbMATH DE number 5900552