## Stabilization of solutions to nonlinear Schrödinger equations.(English)Zbl 1031.35129

Commun. Pure Appl. Math. 54, No. 9, 1110-1145 (2001); erratum ibid. 58, No. 1, 147 (2004).
The nonlinear Schrödinger equation $iu_t+\Delta u+ \beta(|u|^2) u=0,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^n,\quad n\geq 3.\tag{1}$ is considered.
The author shows that starting at time $$t=0$$ near the manifold of ground states, under some restrictive conditions, as $$t\to 0$$, solutions to an equation of the form (1) look like a solitary wave plus radiation, with the latter dispersing at the same rate as the solutions to the linear Schrödinger equation with constant coefficients.
The result is an extension to $$n\geq 3$$ of the result stated for $$n= 1$$ in [V. S. Buslaev and G. S. Perel’man, Algebra Anal. 4, 63-102 (1992; Zbl 0853.35112)].

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35B35 Stability in context of PDEs 93D15 Stabilization of systems by feedback

Zbl 0853.35112
Full Text:

### References:

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