Introduction:
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In the last twenty years much interest has been devoted to the study of systems of coupled nonlinear Schröodinger equations. This interest is motivated by many physical experiments, e.g. in the theory of Bose-Einstein condensates and in nonlinear optics.
In a ultracold dilute Bose gas, condensation in different hyperfine spin states is commonly modeled by the Gross-Pitaevskii system. Different qualitative properties arise, depending on the interactions, which can be attractive or repulsive.
The passage of rays along materials induces nonlinear effects. Whereas the single Schrödinger equation well represents the auto-interaction of the beam, coupled Schrödinger equations take into account also interactions with the material.
Other models analyze different interactions, such as Liouville and Toda systems, or Schrödinger-Poisson-Slater problems, as well as relativistic versions of the Schrödinger equation, involving non local diffusion.
In these contexts, spatial solitary waves have attracted much attention due to their good properties, for instance they are usually least energy solutions, but also the study of solutions at higher energy level is an interesting issue.
Different related topics include concentration phenomena in semiclassical limits, singularly perturbed problems for strong competition, classification of entire solutions of elliptic systems. |
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