×

Stability of peakons for the generalized Camassa-Holm equation. (English) Zbl 1088.35060

The author considers the existence of minimizers for the constrained variational problem: \[ \text{minimize} \;V(u)=-\int_{\mathbb R}(uu_x^2+F(u)+ku^2)dx \]
\[ \text{subject to} \;I(u)=\int_{\mathbb R}(u_x^2+u^2)dx=\lambda >0 \] in \(H^1(\mathbb R)\) in the case of \(k\geq 0\). These minimizers are stable wave solutions for the generalized Camassa-Holm equation, and their derivative may have a singularity, thereby containing peakons. It is shown that under certain conditions on \(F(u)\) and \(V(u)\) and except for translations in the space variable \(x\), any minimizing sequence of the above problem is precompact in \(H^1(\mathbb R)\), and any minimizer is a positive function decaying exponentially at infinity. The result is established by a method developed by the author in a previous work [O. Lopes, ESAIM, Control Optim. Calc. Var. 5, 501–528 (2000; Zbl 0969.35046)]. Some examples are given to explain the resulting theory.
Reviewer: Ma Wen-Xiu (Tampa)

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A15 Variational methods applied to PDEs
49J10 Existence theories for free problems in two or more independent variables

Citations:

Zbl 0969.35046
PDFBibTeX XMLCite
Full Text: EuDML EMIS