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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

Citations:

Zbl 0853.35112
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