Initial-boundary value problem for the Camassa-Holm equation with linearizable boundary condition
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)].
scientific article; zbMATH DE number 5900552
scientific article