## Abstract

The problem of two-dimensional capillary-gravity waves on an inviscid fluid of finite depth interacting with a linear shear current is considered. The shear current breaks the symmetry of the irrotational problem and supports simultaneously counter-propagating waves of different types: Korteweg de-Vries (KdV)-type long solitary waves and wave-packet solitary waves whose envelopes are associated with the nonlinear Schrödinger equation. A simple intuition for the broken symmetry is that the current modifies the Bond number differently for left- and right-propagating waves. Weakly nonlinear theories are developed in general and for two particular resonant cases: the case of second harmonic resonance and long-wave/short-wave interaction. Traveling-wave solutions and their dynamics in the full Euler equations are computed numerically using a time-dependent conformal mapping technique, and compared to some weakly nonlinear solutions. Additional attention is paid to branches of elevation generalized solitary waves of KdV type: although true embedded solitary waves are not detected on these branches, it is found that periodic wavetrains on their tails can be arbitrarily small as the vorticity increases. Excitation of waves by moving pressure distributions and modulational instabilities of the periodic waves in the resonant cases described above are also examined by the fully nonlinear computations.

Original language | English |
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Pages (from-to) | 1036-1057 |

Number of pages | 22 |

Journal | Studies in Applied Mathematics |

Volume | 147 |

Issue number | 3 |

Early online date | 6 Jul 2021 |

DOIs | |

Publication status | Published - 31 Oct 2021 |

### Bibliographical note

Funding Information:Z.W. was supported by the National Natural Science Foundation of China (no. 11772341) and the Strategic Priority Research Program of the Chinese Academy of Sciences (no. XDB22040203). P.A.M. was supported by the EPSRC grant (no. EP/N018176/1).

## Keywords

- gravity-capillary wave
- solitary wave
- water wave

## ASJC Scopus subject areas

- Applied Mathematics