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On the Steadiness of Symmetric Solutions to Two Dimensional Dispersive Models

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Abstract

In this paper, we consider the steadiness of symmetric solutions to two dispersive models in shallow water and hyperelastic mechanics, respectively. These models are derived previously in the two-dimensional setting and can be viewed as the generalization of the Camassa–Holm and Kadomtsev–Petviashvili equations. For these two models, we prove that the symmetry of classical solutions implies steadiness in the horizontal direction. We also confirm the connection between symmetry and steadiness for solutions in weak formulation, which covers in particular the peaked solutions.

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Notes

  1. These are modified Sobolev spaces. See Remark 1 for the definition.

  2. Although the KP-I and KP-II equations are integrable, the b-family Camassa-Holm equations are not integrable except for the Camassa-Holm (with \(b=1\)) and Degasperis-Procesi (with \(b=2\)) equation (see [25]). The hyperelastic compressible plate model can be viewed as a combination of a member of the b-family Camassa-Holm equation and some terms with y-derivatives. Its integrability is still unknown.

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Acknowledgements

The authors are very grateful to the referee for the valuable comments and suggestions, which lead to the current version of the manuscript. The authors would like to thank Dr. Lu Li for the many discussions on the ideas and proofs during the preparation of the initial version, and for her careful proofreading before the submission. The author L.P. gratefully acknowledges financial support from the National Natural Science Foundation for Young Scientists of China (Grant No. 12001553), the Fundamental Research Funds for the Central Universities (Grant No. 20lgpy151), the Science and Technology Program of Guangzhou (Grant No. 202102080474) and the Guangdong Basic, and Applied Basic Research Foundation (Grant No. 2023A1515010599).

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  1. School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, 519082, Guangdong, China

    Long Pei, Fengyang Xiao & Pan Zhang

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  1. Long Pei
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Correspondence to Long Pei.

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Pei, L., Xiao, F. & Zhang, P. On the Steadiness of Symmetric Solutions to Two Dimensional Dispersive Models. J. Math. Fluid Mech. 26, 34 (2024). https://doi.org/10.1007/s00021-024-00869-0

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