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Center manifold for nonintegrable nonlinear Schrödinger equations on the line. (English) Zbl 1003.37045

The author deals with the nonintegrable, nonlinear Schrödinger equation, \[ i \frac{\partial u}{\partial t}= -\frac{d^2u}{dx^2}+ V(x) u(t,x)+ f(x,|u|) \frac{u(t,x)} {|u(t,x)|}, \qquad u(0,x)= \varphi(x), \tag{1} \] where \(u\) is a complex-valued function for \(t,x\in \mathbb{R}\), \(f(x,\cdot)\in C^1(\mathbb{R},\mathbb{R})\), \(\frac{\partial} {\partial x} f(x,\cdot)\in C(\mathbb{R},\mathbb{R})\), \(f(x,0)= 0\) and, \[ \biggl|\frac{\partial} {\partial u}f(x,u) \biggr|\leq C|u|^{p-1}, \quad \biggl|\frac{\partial} {\partial x}f(x,u) \biggr|\leq C|u|^p, \quad\text{for some }p>2. \] Under some generic condition on the potential \(V\), the author proves that all small solutions approach a particular periodic orbit in the center manifold as \(t\to\pm \infty\). In general, the periodic orbits are different for \(t\to\pm \infty\).

MSC:

37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
35B35 Stability in context of PDEs
35B41 Attractors
35Q55 NLS equations (nonlinear Schrödinger equations)
81U05 \(2\)-body potential quantum scattering theory
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