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

The authors study the equation

u x +(a(u)) x -u xxt =b ' (u)u x 2 2+b(u)u xx x (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+p+2 2u p+1 , the authors establish the local well-posedness of the Cauchy problem for (1) in the Sobolev space H s with any s>3 2. The stability and instability problem of a solitary wave solution of (1) are considered.

35Q35PDEs in connection with fluid mechanics
37K45Stability problems (infinite-dimensional systems)
35G25Initial value problems for nonlinear higher-order PDE
76B25Solitary waves (inviscid fluids)
35B35Stability of solutions of PDE