Lyapunov stability of ground states of nonlinear dispersive evolution equations.(English)Zbl 0594.35005

The orbital stability of the ground state solutions of the nonlinear Schrödinger equation $(1)\quad i\phi_ t(x,t)+\Delta \phi (x,t)+f(| \phi (x,t)|^ 2)\phi (x,t)=0,\quad x\in {\mathbb{R}}^ N$ and the generalized Korteweg-de Vries equation $(2)\quad w_ t(x,t)+a(w(x,t))w_ x(x,t)+w_{xxx}(x,t)=0,\quad x\in {\mathbb{R}}$ are studied using a Lyapunov function which is a conserved energy integral. In the case of (1) the ground state is orbitally stable, in the particular case where $$f(| \phi |^ 2)=| \phi |^{2\sigma}$$, if $$\sigma <2/N$$ and $$N=1$$ or $$N=3$$. A more general class of functions f is also considered. In the case of (2), if R(E) is a ground state with energy E, then a solitary wave $$\psi(x-ct)$$ is orbitally stable if $$\phi(c)=(d/dc)\| R(c)\|^ 2>0.$$
Reviewer: S.P.Banks

MSC:

 35B35 Stability in context of PDEs 35Q99 Partial differential equations of mathematical physics and other areas of application 35J10 Schrödinger operator, Schrödinger equation 58J47 Propagation of singularities; initial value problems on manifolds
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References:

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