## 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

### Keywords:

nonlinear Schrödinger equation
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