We present a review of the normal form theory for weakly dispersive nonlinear wave equations where the leading order phenomena can be described by the KdV equation. This is an infinite dimensional extension of the well-known Poincaré-Dulac normal form theory for ordinary differential equations. We also provide a detailed analysis of the interaction problem of solitary waves as an important appl...

We study the modulational instability of a shallow water model, with and without surface tension, which generalizes the Whitham equation to include bi-directional propagation. Without surface tension, the small amplitude periodic traveling waves are modulationally unstable if their wave number is greater than a critical wave number predicting a Benjamin-Feir type instability and the result qual...

A shallow water spectral wave prediction model based on a numerical solution of the radiative transfer equation is presented. The model is second generation and uses a simple yet effective representation for the nonlinear source term. In addition, the model pays particular attention to the shallow water processes of refraction, shoaling, bottom friction, and wave breaking. The flexibility of th...

Topological concepts have been introduced into electronic, photonic, and phononic systems, but have not been studied in surface-water-wave systems. Here we study a one-dimensional periodic resonant surface-water-wave system and demonstrate its topological transition. By selecting three different water depths, we can construct different types of water waves - shallow, intermediate and deep water...

The parabolic approximation is developed to study the combined refraction/diffraction of weakly nonlinear shallow-water waves. Two methods of approach are used. In the first method Boussinesq equations are used to derive evolution equations for spectral-wave components in a slowly varying two-dimensional domain. The second method modifies the K-P equation (Kadomtsev & Petviashvili 1970) to incl...

We study the numerical recovery of Manning’s roughness coefficient for the diffusive wave approximation of the shallow water equation. We describe a conjugate gradient method for the numerical inversion. Numerical results for one-dimensional model are presented to illustrate the feasibility of the approach. Also we provide a proof of the differentiability of the weak form with respect to the co...

In this paper, a quasi-rogue wave solution of the Camassa-Holm equation is obtained by discrete method and Hirota’s bilinear transform method. The wave’s amplitude suddenly changes high and then suddenly disappears. The whole process experiences a very short time. Waveform’s characteristics are similar to rogue waves in shallow water. This obtained solution can be used to describe the possible ...