Weakly Nonlinear and Dispersive Wave Equations for Random Waves

Computational Models for Weakly Dispersive Nonlinear Water Waves

Numerical methods for the two and three dimensional Boussinesq equations governing weakly non-linear and dispersive water waves are presented and investigated. Convenient handling of grids adapted to the geometry or bottom topography is enabled by nite element discretization in space. Staggered nite diierence schemes are used for the temporal dis-cretization, resulting in only two linear system...

Slow Passage through Resonance for a Weakly Nonlinear Dispersive Wave

Nonlinear dispersive equations with random initial data

In the first part of this thesis we consider the defocusing nonlinear wave equation of power-type on R 3. We establish an almost sure global existence result with respect to a suitable randomization of the initial data. In particular, this provides examples of initial data of supercritical regularity which lead to global solutions. The proof is based upon Bourgain's high-low frequency decomposi...

Model Equations for Waves in Nonlinear Dispersive Systems

The present discussion focuses on models for unidirectional wave propagation in which nonlinear, dispersive and dissipative effects are simulated realistically and in such a way that shock formation and other singular behaviour is avoided. While comparatively narrow, the range of discussion nevertheless covers a number of interesting and challenging scientific issues, several of which still rem...

Stability of Periodic Traveling Waves for Nonlinear Dispersive Equations

We study the stability and instability of periodic traveling waves for Korteweg-de Vries type equations with fractional dispersion and related, nonlinear dispersive equations. We show that a local constrained minimizer for a suitable variational problem is nonlinearly stable to period preserving perturbations. We then discuss when the associated linearized equation admits solutions exponentiall...