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Higher dimensional solitary waves generated by second-harmonic generation in quadratic media

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Abstract

Schrödinger type soliton waves generated by second-harmonic generation in higher dimensional quadratic optical media are considered. The existence of ground state solutions for spatial dimension from two to five is proved, and the continuous dependence on the parameter and asymptotic behavior of ground state solutions are established. Multi-pulse solutions with certain symmetry are also obtained. In a bounded domain setting, global bifurcation diagram of multi-pulse solutions are shown by using new technique of double saddle-node bifurcation.

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Acknowledgments

The authors would like to thank an anonymous reviewer for helpful suggestions and comments. This work was done when Zhao and Zhao visited Department of Mathematics, College of William and Mary in 2013–2014, and they would like to thank the College for hospitality.

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Authors and Affiliations

  1. Department of Mathematics, Beijing University of Chemical Technology, Beijing, 100029, People’s Republic of China

    Leiga Zhao

  2. Department of Mathematics, Yunnan Normal University, Kunming, 650092, People’s Republic of China

    Fukun Zhao

  3. Department of Mathematics, College of William and Mary, Williamsburg, VA, 23187-8795, USA

    Junping Shi

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  1. Leiga Zhao
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  2. Fukun Zhao
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  3. Junping Shi
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Corresponding author

Correspondence to Junping Shi.

Additional information

Communicated by P. Rabinowitz.

This research is partially supported by the National Natural Science Foundation of China (11271264,11361078), National Science Foundation of US (DMS-1313243), Beijing Higher Education Young Elite Teacher Project (YETP0515) and China Scholarship Council.

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Zhao, L., Zhao, F. & Shi, J. Higher dimensional solitary waves generated by second-harmonic generation in quadratic media. Calc. Var. 54, 2657–2691 (2015). https://doi.org/10.1007/s00526-015-0879-1

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