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Existence, regularity and symmetry of periodic traveling waves for Gardner–Ostrovsky type equations

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Abstract

We study the existence, regularity and symmetry of periodic traveling solutions to a class of Gardner–Ostrovsky type equations, including the classical Gardner–Ostrovsky equation, the (modified) Ostrovsky and the reduced (modified) Ostrovsky equation. The modified Ostrovsky equation is also known as the short pulse equation. The Gardner-Ostrovsky equation is a model for internal ocean waves of large amplitude. We prove the existence of nontrivial, periodic traveling wave solutions using local bifurcation theory, where the wave speed serves as the bifurcation parameter. Moreover, we give a regularity analysis for periodic traveling solutions in the presence as well as absence of the Boussinesq dispersion. We see that the presence of Boussinesq dispersion implies the smoothness of periodic traveling wave solutions, while its absence may lead to singularities in the form of peaks or cusps. Eventually, we study the symmetry of periodic traveling solutions by the method of moving planes. A novel feature of the symmetry results in the absence of Boussinesq dispersion is that we do not need to impose a traditional monotonicity condition or a recently developed reflection criterion on the wave profiles to prove the statement on the symmetry of periodic traveling waves.

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Notes

  1. Note that the results still hold for other \(\gamma >0\) after proper adjustment in the formulation.

  2. Without loss of generality, one can shift the wave profile such that the condition in the table is satisfied by \(\phi \) at \(x=0\) due to the translation invariance of equation (6.2).

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Acknowledgements

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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Authors and Affiliations

  1. Centre for Mathematical Sciences, Lund University, 22100, Lund, Sweden

    Gabriele Bruell

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

    Long Pei

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  1. Gabriele Bruell
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Bruell, G., Pei, L. Existence, regularity and symmetry of periodic traveling waves for Gardner–Ostrovsky type equations. Monatsh Math 202, 685–711 (2023). https://doi.org/10.1007/s00605-023-01891-6

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