Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation. (English) Zbl 1076.35098

Hakkaev, Sevdzhan; Kirchev, Kiril

Commun. Partial Differ. Equations 30, No. 5-6, 761-781 (2005).

The authors study the equation \[ u_x + (a(u))_x -u_{xxt} =\left(b'(u)\frac{u^2_x}{2} + b(u)u_{xx}\right)_x \tag{1} \] which is a generalization of the famous Camassa-Holm equation. Applying the method of the pseudoparabolic regularization for \(b(u) = u^p\) and \(a(u) =2ku +\frac{p+2}{2}u^{p+1}\), the authors establish the local well-posedness of the Cauchy problem for (1) in the Sobolev space \(H^s\) with any \(s >\frac{3}{2}\). The stability and instability problem of a solitary wave solution of (1) are considered.

Reviewer: Vasily A. Chernecky (Odessa)

MSC:

35Q35	PDEs in connection with fluid mechanics
37K45	Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
35G25	Initial value problems for nonlinear higher-order PDEs
76B25	Solitary waves for incompressible inviscid fluids
35B35	Stability in context of PDEs

Keywords:

Camassa-Holm equation; initial value problem; existence; pseudoparabolic regularization; stability; stability and instability

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