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Existence and bifurcation of nontrivial solutions for the coupled nonlinear Schrödinger–Korteweg–de Vries system

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Abstract

In the present paper, we study the existence and bifurcation of nontrivial solutions of the nonlinear Schrödinger–Korteweg–de Vries (NLS–KdV) and Schrödinger–Korteweg–de Vries–Korteweg–de Vries (NLS–KdV–KdV) systems which arise from fluid mechanics. On the one hand, for both the three-wave system and the two-wave system, the existence/nonexistence, continuous dependence and asymptotic behavior of positive ground state solutions are established. On the other hand, multiple positive solutions are found via a combination of Nehari manifold and bifurcation methods for the attractive interaction case, which has not been found for the conventional nonlinear Schrödinger systems with cubic nonlinearity.

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Acknowledgements

The authors thank the referee’s thoughtful reading of details of the paper and nice suggestions to improve the results. This work was supported by NNSFC (Grants 11971202, 11571140, 11671077), the Fellowship of Outstanding Young Scholars of Jiangsu Province (BK20160063), the Six big talent peaks project in Jiangsu Province (XYDXX-015) and the NSF of Jiangsu Province (BK20150478).

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Authors and Affiliations

  1. Faculty of Science, Jiangsu University, Zhenjiang, 212013, Jiangsu, People’s Republic of China

    Qiuping Geng & Jun Wang

  2. Bradley Department of Electric and Computer Engineering, Virginia Polytechnic Institute and State University, Blackburg, VA, 24060, USA

    Mian Liao

  3. School of Management, Jiangsu University, Zhenjiang, 212013, Jiangsu, People’s Republic of China

    Lu Xiao

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  1. Qiuping Geng
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  2. Mian Liao
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  3. Jun Wang
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  4. Lu Xiao
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Correspondence to Jun Wang.

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Geng, Q., Liao, M., Wang, J. et al. Existence and bifurcation of nontrivial solutions for the coupled nonlinear Schrödinger–Korteweg–de Vries system. Z. Angew. Math. Phys. 71, 33 (2020). https://doi.org/10.1007/s00033-020-1256-2

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