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Nonlinear Schrödinger and Davey–Stewartson Equations

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Algebraic Approaches to Partial Differential Equations

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Abstract

The two-dimensional cubic nonlinear Schrödinger equation is used to describe the propagation of an intense laser beam through a medium with Kerr nonlinearity. Coupled two-dimensional cubic nonlinear Schrödinger equations are used to describe the interaction of electromagnetic waves with different polarizations in nonlinear optics. In this chapter, we solve these equations by imposing a quadratic condition on the related argument functions and using their symmetry transformations. More complete families of exact solutions of this type are obtained, many of which are periodic, quasi-periodic, aperiodic, and singular solutions that may have practical significance. The Davey–Stewartson equations are used to describe the long time evolution of three-dimensional packets of surface waves. Assuming that the argument functions are quadratic in spatial variables, we find various exact solutions for the Davey–Stewartson equations.

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Authors and Affiliations

  1. Institute of Mathematics, Academy of Mathematics and System Science, Beijing, People’s Republic of China

    Xiaoping Xu

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  1. Xiaoping Xu
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Xu, X. (2013). Nonlinear Schrödinger and Davey–Stewartson Equations. In: Algebraic Approaches to Partial Differential Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36874-5_6

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