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Infinitely Many Solitary Waves Due to the Second-Harmonic Generation in Quadratic Media

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Abstract

In this paper, we consider the following coupled Schrödinger system with χ(2) nonlinearities

$\left\{ \begin{array}{l} - \Delta {u_1} + {V_1}\left( x \right){u_1} = \alpha {u_1}{u_2},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x \in {^N}, \\ - \Delta {u_2} + {V_2}\left( x \right){u_2} = \frac{\alpha }{2}u_1^2 + \beta u_2^2,\,\,\,\,\,\,\,\,\,\,\,\,\,x \in {^N}, \\ \end{array} \right.$

which arises from second-harmonic generation in quadratic media. Here V1(x) and V2(x) are radially positive functions, 2 ≤ N < 6, α > 0 and α > β. Assume that the potential functions V1(x) and V2(x) satisfy some algebraic decay at infinity. Applying the finite dimensional reduction method, we construct an unbounded sequence of non-radial vector solutions of synchronized type.

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Acknowledgments

The authors sincerely thank Prof. S. Peng for his helpful discussions and suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics and Hubei Key Laboratory Mathematical Sciences, Central China Normal University, Wuhan, 430079, China

    Chunhua Wang  (王春花) & Jing Zhou  (周静)

  2. School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan, 430000, China

    Jing Zhou  (周静)

Authors
  1. Chunhua Wang  (王春花)
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  2. Jing Zhou  (周静)
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Corresponding author

Correspondence to Chunhua Wang  (王春花).

Additional information

This paper was partially supported by NSFC (11671162; 11601194), CCNU18CXTD04 and CZQ13017

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Wang, C., Zhou, J. Infinitely Many Solitary Waves Due to the Second-Harmonic Generation in Quadratic Media. Acta Math Sci 40, 16–34 (2020). https://doi.org/10.1007/s10473-020-0102-3

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  • DOI: https://doi.org/10.1007/s10473-020-0102-3

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