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Existence of a finite-dimensional global attractor for a damped parametric nonlinear Schrödinger equation. (English) Zbl 1260.37052

In this work, the authors consider a parametric nonlinear Schrödinger equation of the form \(u_t+au-iu_{xx}+i\lambda u+i|u|^2u+i\gamma\overline{u}=0\), where \(a,\lambda>0\) and \(\gamma\) is a real function, defined on the real line. Under the assumption that \(\lim_{|x|\to +\infty}\gamma(x)=0\), they prove the existence of a global attractor in the Sobolev space \(H^1(\mathbb{R})\) for the dynamical system generated by this equation, which is also compact in \(H^3(\mathbb{R})\). Assuming also that \(\int_{\mathbb{R}}(1+x^2)\gamma(x)^2dx<+\infty\), they prove that the global attractor has finite Hausdorff dimension.

MSC:

37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q41 Time-dependent Schrödinger equations and Dirac equations
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