## Abstract

Using a nonlocal version of the center-manifold theorem and a normal form reduction, we prove the existence of small-amplitude generalized solitary-wave solutions and modulated solitary-wave solutions to the steady gravity-capillary Whitham equation with weak surface tension. Through the application of the center-manifold theorem, the nonlocal equation for the solitary wave profiles is reduced to a four-dimensional system of ODEs inheriting reversibility. Along particular parameter curves, relating directly to the classical gravity-capillary water wave problem, the associated linear operator is seen to undergo either a reversible (Formula presented.) bifurcation or a reversible (Formula presented.) bifurcation. Through a normal form transformation, the reduced system of ODEs along each relevant parameter curve is seen to be well approximated by a truncated system retaining only second-order or third-order terms. These truncated systems relate directly to systems obtained in the study of the full gravity-capillary water wave equation and, as such, the existence of generalized and modulated solitary waves for the truncated systems is guaranteed by classical works, and they are readily seen to persist as solutions of the gravity-capillary Whitham equation due to reversibility. Consequently, this work illuminates further connections between the gravity-capillary Whitham equation and the full two-dimensional gravity-capillary water wave problem.

Original language | English |
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Pages (from-to) | 773-812 |

Journal | Studies in Applied Mathematics |

Volume | 148 |

Issue number | 2 |

Early online date | 19 Oct 2021 |

DOIs | |

Publication status | Published - 28 Feb 2022 |

### Bibliographical note

Funding Information:The work of Johnson was partially funded by the Simons Foundation Collaboration Grant number 714021. Truong gratefully acknowledges the support of the Swedish Research Council, Grant number 2016‐04999. Also, Truong appreciates the discussions and support she has received from her supervisor Erik Wahlén. Finally, the authors thank the reviewers for their careful reading, insightful comments, and suggestions.

## Keywords

- Whitham equation
- center-manifold reduction
- surface tension

## ASJC Scopus subject areas

- Applied Mathematics