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The Cauchy problem for the Gross–Pitaevskii equation. (English) Zbl 1122.35133
The author considers the initial-value problem for the Gross-Pitaevskii equation $$iu{_ {t}}+\Delta u=(| u| {^{2}}-1)u,$$ where $$x\in{\mathbb R}{^{d}}$$ with $$d=2,3.$$ Using some Strichartz estimates it is shown that the initial-value problem is globally well posed on the space $$\{u\in H_{\text{loc}}^{1}({\mathbb {R}}{^{d}}): \nabla u\in L{^{2}}({\mathbb R}{^{d}}),\;| u| {^{2}}-1\in L{^{2}}({\mathbb {R}}{^{d}})\}.$$

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 37L50 Noncompact semigroups, dispersive equations, perturbations of infinite-dimensional dissipative dynamical systems 37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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