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The Lagrangian Formulation for Wave Motion with a Shear Current and Surface Tension

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Abstract

The Lagrangian formulation for the irrotational wave motion is straightforward and follows from a Lagrangian functional which is the difference between the kinetic and the potential energy of the system. In the case of fluid with constant vorticity, which arises for example when a shear current is present, the separation of the energy into kinetic and potential is not at all obvious and neither is the Lagrangian formulation of the problem. Nevertheless, we use the known Hamiltonian formulation of the problem in this case to obtain the Lagrangian density function, and utilising the Euler–Lagrange equations we proceed to derive some model equations for different propagation regimes. While the long-wave regime reproduces the well known KdV equation, the short- and intermediate long wave regimes lead to highly nonlinear and nonlocal evolution equations.

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Acknowledgements

The authors are thankful to two anonymous referees for their corrections and constructive suggestions, which have improved the quality of the manuscript. This publication has emanated from research conducted with the financial support of Science Foundation Ireland under Grant No. 21/FFP-A/9150.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Technological University Dublin, City Campus, Grangegorman Lower, Dublin, D07 ADY7, Ireland

    Conor Curtin & Rossen Ivanov

  2. Institute for Advanced Physical Studies, 111 Tsarigradsko shose Blvd., 1784, Sofia, Bulgaria

    Rossen Ivanov

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  1. Conor Curtin
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  2. Rossen Ivanov
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Correspondence to Rossen Ivanov.

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Curtin, C., Ivanov, R. The Lagrangian Formulation for Wave Motion with a Shear Current and Surface Tension. J. Math. Fluid Mech. 25, 87 (2023). https://doi.org/10.1007/s00021-023-00831-6

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