## Nonlinear Water Waves: Cetraro, Italy 2013This volume brings together four lecture courses on modern aspects of water waves. The intention, through the lectures, is to present quite a range of mathematical ideas, primarily to show what is possible and what, currently, is of particular interest. Water waves of large amplitude can only be fully understood in terms of nonlinear effects, linear theory being not adequate for their description. Taking advantage of insights from physical observation, experimental evidence and numerical simulations, classical and modern mathematical approaches can be used to gain insight into their dynamics. The book presents several avenues and offers a wide range of material of current interest. The lectures provide a useful source for those who want to begin to investigate how mathematics can be used to improve our understanding of water wave phenomena. In addition, some of the material can be used by those who are already familiar with one branch of the study of water waves, to learn more about other areas. |

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### Contents

1 | |

Breaking Water Waves | 83 |

Asymptotic Methods for Weakly Nonlinear and Other Water Waves | 120 |

Some Basic Models and Problems | 197 |

### Common terms and phrases

ˆ ˆ ˆ 0-isocline analysis analytic approach asymptotic expansion boundary conditions Boussinesq equation Camassa–Holm equation cKdV classical completely integrable conformal mapping constant vorticity Constantin corresponding crest curve defined derivation dimensionless discussion Dq.R edge waves existence flat bed flat free surface function geodesic geodesic flow given global governing equations horizontal initial data intersects irrotational KdV equation Korteweg–de Vries equation leading order Liénard equation limit cycle linear Math mathematical moving frame non-dimensional nonlinear obtain operator parameter periodic travelling phase plane physical real-analytic relation in 56 relevant result scale Sect semi-trajectory shallow water equation small amplitude solitary wave soliton speed Stokes waves surface wave Theorem theory transformation travelling waves trough uniform flow valid variable velocity vorticity water waves wave profile wave propagation wavelength waves of small x-variable