### Mathematics Colloquium

# Mark Groves

### (Universität des Saarlandes)

## "Existence and stability of fully localised three-dimensional gravity-capillary solitary water waves"

** Date: ** |
Mon, November 29, 2010 |

** Time: ** |
17:15 |

** Place: ** |
Research II Lecture Hall |

**Abstract:** In this talk I will give an introduction to gravity-capillary water
waves. Such waves arise as critical points of a Hamiltonian *H*
subject to an impulse constraint.

In particular, I will show that there exists a minimiser of
*H* subject to the constraint *I*=2μ where μ is a
small positive number. The existence of a solitary wave is thus
assured, and since *H* and *I* are both conserved quantities
its stability follows by a standard argument. `Stability' must
however be understood in a qualified sense due to the lack of a global
well-posedness theory for three-dimensional water waves.