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Dynamics near the solitary waves of the supercritical gKDV equations. (English) Zbl 1423.35066

Summary: This work is devoted to study the dynamics of the supercritical gKDV equations near solitary waves in the energy space \(H^1\). We construct smooth local center-stable, center-unstable and center manifolds near the manifold of solitary waves and give a detailed description of the local dynamics near solitary waves. In particular, the instability is characterized as follows: any forward flow not starting from the center-stable manifold will leave a neighborhood of the manifold of solitary waves exponentially fast. Moreover, orbital stability is proved on the center manifold, which implies the uniqueness of the center manifold and the solutions on it exist globally and asymptotically approach the solitary waves.

MSC:

35C08 Soliton solutions
35Q53 KdV equations (Korteweg-de Vries equations)
35B40 Asymptotic behavior of solutions to PDEs
37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
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