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Quadratic-Argument Approach to Nonlinear Schrödinger Equation and Coupled Ones

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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. The coupled two-dimensional cubic nonlinear Schrödinger equations are used to describe the interaction of electromagnetic waves with different polarizations in nonlinear optics. Mathematically, they are fundamental nonlinear partial differential equations of elliptic type. In this paper, we solve the above equations by imposing a quadratic condition on the related argument functions and using their symmetry transformations. More complete families of exact solutions of such type are obtained. Many of them are the periodic, quasi-periodic, aperiodic and singular solutions that may have practical significance.

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Authors and Affiliations

  1. Hua Loo-Keng Mathematical Laboratory, Institute of Mathematics, Academy of Mathematics & System Sciences, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Xiaoping Xu

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  1. Xiaoping Xu
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Correspondence to Xiaoping Xu.

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Research supported by China NSF 10871193.

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Xu, X. Quadratic-Argument Approach to Nonlinear Schrödinger Equation and Coupled Ones. Acta Appl Math 110, 749–769 (2010). https://doi.org/10.1007/s10440-009-9472-y

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  • DOI: https://doi.org/10.1007/s10440-009-9472-y

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