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The semiclassical focusing nonlinear Schrödinger equation. (English) Zbl 1142.35083

Bonilla, L. L. (ed.) et al., Recent advances in nonlinear partial differential equations and applications. Conference in honor of Peter D. Lax and Louis Nirenberg on their 80th birthdays, Toledo, Spain, June 7–10, 2006. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4211-9/hbk). Proceedings of Symposia in Applied Mathematics 65, 47-80 (2007).
This paper studies the leading asymptotic behavior of the solution to the semi-classical focusing nonlinear Schrödinger (NLS) equation as the differential terms become small. This equation is important since it models the nonlinear transmission in optical fibers. There are many questions which remain unsolved resulted from the modulational instabilities of this equation. The paper gives a wonderful history and review of the literature and results to date describing the approaches and open problems that are addressed in the paper.
For the entire collection see [Zbl 1124.35002].

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
78M35 Asymptotic analysis in optics and electromagnetic theory
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
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