We prove that line solitons of the two-dimensional hyperbolic nonlinear Schrodinger equation are unstable with respect to transverse perturbations of arbitrarily small periods, {em i.e.}, short waves. The analysis is based on the construction of Jost functions for the continuous spectrum of Schr{o}dinger operators, the Sommerfeld radiation conditions, and the Lyapunov--Schmidt decomposition. Precise asymptotic expressions for the instability growth rate are derived in the limit of short periods.
Nonlinear wave run-up on the beach caused by harmonic wave maker located at some distance from the shore line is studied experimentally. It is revealed that under certain wave excitation frequencies a significant increase in run-up amplification is observed. It is found that this amplification is due to the excitation of resonant mode in the region between the shoreline and wave maker. Frequency and magnitude of the maximum amplification are in good correlation with the numerical calculation results represented in the paper (T.S. Stefanakis et al. PRL (2011)). These effects are very important for understanding the nature of rougue waves in the coastle zone.
The problem of tsunami wave run-up on a beach is discussed in the framework of the rigorous solutions of the nonlinear shallow-water theory. We present an analysis of the run-up characteristics for various shapes of the incoming symmetrical solitary tsunami waves. It will be demonstrated that the extreme (maximal) wave characteristics on a beach (run-up and draw-down heights, run-up and draw-down velocities and breaking parameter) are weakly dependent on the shape of incident wave if the definition of the significant wave length determined on the 2/3 level of the maximum height is used. The universal analytical expressions for the extreme wave characteristics are derived for the run-up of the solitary pulses. They can be directly applicable for tsunami warning because in many case the shape of the incident tsunami wave is unknown.
Long linear wave transformation in the basin of varying depth is studied for a case of a convex bottom profile in the framework of one-dimensional shallow water equation. The existence of travelling wave solutions in this geometry and the uniqueness of this wave class is established through construction of a 1:1 transformation of the general 1D wave equation to the analogous wave equation with constant coefficients. The general solution of the Cauchy problem consists of two travelling waves propagating in opposite directions. It is found that generally a zone of a weak current is formed between these two waves. Waves are reflected from the coastline so that their profile is inverted with respect to the calm water surface. Long wave runup on a beach with this profile is studied for sine pulse, KdV soliton and N-wave. Shown is that in certain cases the runup height along the convex profile is considerably larger than for beaches with a linear slope. The analysis of wave reflection from the bottom containing a shallow coastal area of constant depth and a section with the convex profile shows that a transmitted wave always has a sign-variable shape.
