A short review of the theory of weakly nonlinear water waves, prepared for the forthcoming Encyclopedia of Nonlinear Science

We consider the modulational instability of nonlinearly interacting two-dimensional waves in deep water, which are described by a pair of two-dimensional coupled nonlinear Schrödinger equations. We derive a nonlinear dispersion relation. The latter is numerically analyzed to obtain the regions and the associated growth rates of the modulational instability. Furthermore, we follow the long term ...

Solitons are a unique class of pulse-shaped waves that propagate in nonlinear dispersive media. In the course of propagation, they maintain spatial confinement of wave energy within the pulse shape without dispersion and exhibit remarkable nonlinear dynamics. Solitons are found throughout nature where fine balance between nonlinearity and dispersion is achieved. Dispersion alone would flatten o...

Nonlinear partial differential equations are more suitable to model many physical phenomena in science and engineering. In this paper, we consider three nonlinear partial differential equations such as Novikov equation, an equation for surface water waves and the Geng-Xue coupled equation which serves as a model for the unidirectional propagation of the shallow water waves over a flat bottom. T...

When modeling water waves, one often assume that water is incompressible and inviscid and the flow is irrotational. The evolution of nonlinear water waves is governed by the Euler equations. There are two important parameters associated with long waves. One is the ratio of amplitude to depth, represented by a, and the other one is the ratio of depth to wavelength, represented by e. Under the as...

Unidirectional deep-water waves are studied theoretically and experimentally. Theoretically we apply the theory of the nonlinear Schroedinger equation (NLS) using the inverse scattering transform, a kind of generalized, nonlinear Fourier analysis. We discover from the theoretical study that there are essentially four kinds of physical effects that can lead to extreme waves: (1) the superpositio...

is well known as the universal model for the description of the unstable envelope of strongly dispersive and weakly nonlinear waves. The other interpretation of ( 1 ) is connected with the nonlinear optics of the focusing media (see, e.9., refs. [ 1-3] ). The field of the important physical applications of the NLS equation is extromely wide. To mention just a few: nonlinear deep water waves [4 ...

This paper discusses the applications of linear and nonlinear shallow water wave equations in practical tsunami simulations. We verify which hydrodynamic theory would be most appropriate for different ocean depths. The linear and nonlinear shallow water wave equations in describing tsunami wave propagation are compared for the China Sea. There is a critical zone between 400 and 500 m depth for ...

Large amplitude water waves on deep water has long been known in the sea faring community, and the cause of great concern for, e.g., oil platform constructions. The concept of such freak waves is nowadays, thanks to satellite and radar measurements, well established within the scientific community. There are a number of important models and approaches for the theoretical description of such wav...