Stability and asymptotic stability for subcritical gKdV equations.(English)Zbl 1017.35098

Summary: We prove the stability and asymptotic stability in $$H^1$$ of a decoupled sum of $$N$$ solitons for the subcritical generalized KdV equations $$u_t+(u_{xx}+u^p)_x=0$$ ($$1<p<5$$). The proof of the stability result is based on energy arguments and monotonicity of the local $$L^2$$ norm. Note that the result is new even for $$p=2$$ (the KdV equation). The asymptotic stability result then follows directly from a rigidity theorem in [Y. Martel and F. Merle, Arch. Ration. Mech. Anal. 157, 219-254 (2001; Zbl 0981.35073)].

MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems 76B25 Solitary waves for incompressible inviscid fluids

Zbl 0981.35073
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