Water waves generated by a moving bottom

Tsunamis are often generated by a moving sea bottom. This paper deals with the case where the tsunami source is an earthquake. The linearized water-wave equations are solved analytically for various sea bottom motions. Numerical results based on the analytical solutions are shown for the free-surface profiles, the horizontal and vertical velocities as well as the bottom pressure.Comment: 41 pages, 13 figures. Accepted for publication in a book: "Tsunami and Nonlinear Waves", Kundu, Anjan (Editor), Springer 2007, Approx. 325 p., 170 illus., Hardcover, ISBN: 978-3-540-71255-8, available: May 200

Rogue waves and solitons on a cnoidal background

Solutions of the nonlinear Schr¨odinger equation, appearing as rogue waves on a spatially-periodic background envelope, are obtained using the Darboux transformation scheme. Several particular examples are illustrated numerically. These include soliton and breather solutions on a periodic background as well as higher-order structures. The results enrich our knowledge of possible analytic solutions that describe the appearance of rogue waves in a variety of situations.The authors acknowledge the support of the Australian Research Council (Discovery Project number DP110102068). N.A. and A.A. acknowledge support from the Volkswagen Stiftung

Dispersive shock waves: from water waves to nonlinear optics

Dispersive shock waves are strongly oscillating wave trains that spontaneously form and expand thanks to the action of weak dispersion, which contrasts the tendency, driven by the nonlinearity, to develop a gradient catastrophe. Here we review the basic concepts and recent progresses made in the description of such nonlinear waves, both in terms of experimental results and modelling. In particular, we discuss the formation of dispersive shocks in shallow water, which can be described in terms of Korteweg-de Vries or Whitham nonlocal equations. We contrast such results with those obtained in the field of nonlinear optics, described in terms of local or nonlocal nonlinear Schr¨odinger equations. Finally we show that a dispersive shock propagating under the action of small perturbations can radiate. A perturbative approach allows for the accurate prediction of the radiated frequencies