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Stability of solitary waves for a nonlinearly dispersive equation. (English) Zbl 1059.35012

Summary: Solitary-wave solutions of a nonlinearly dispersive equation of the type

u t -u xxt +ωu x +3uu x =γ(2u x u xx +uu xxx )

are considered. It is found that solitary waves are peaked or smooth waves, depending on the wave speed. The stability of the smooth solitary waves also depends on the wave speed. Orbital stability is proved for some wave speeds, while instability is proved for others.

MSC:
35B35Stability of solutions of PDE
35G25Initial value problems for nonlinear higher-order PDE
37L50Noncompact semigroups; dispersive equations; perturbations of Hamiltonian systems
35Q51Soliton-like equations