In this short note we discuss a new formula for solving the nonlocal $\overline{\partial}$-problem, and application to Manakov--Zakharov dressing method. We then explicitly apply complex (2+1)D Kadomtsev--Petviashvili equation completely integrable generalization of Kaup--Broer (or Kaup--Boussinesq) system. will also how real (1+1)D solutions are expressed using formalism. It is simple express ...

A class of model equations that describe the bi-directional propagation of small amplitude long waves on the surface of shallow water is derived from two-dimensional potential flow equations at various orders of approximation in two small parameters, namely the amplitude parameter a 1⁄4 a=h0 and wavelength parameter b 1⁄4 ðh0=lÞ2, where a and l are the actual amplitude and wavelength of the sur...

We apply a multiple–time version of the reductive perturbation method to study long waves as governed by the Boussinesq model equation. By requiring the absence of secular producing terms in each order of the perturbative scheme, we show that the solitary–wave of the Boussinesq equation can be written as a solitary–wave satisfying simultaneously all equations of the KdV hierarchy, each one in a...

The accurate numerical simulation of wave disturbance within harbours requires consideration of both nonlinear and dispersive wave processes in order to capture such physical effects as wave refraction and diffraction, and nonlinear wave interactions such as the generation of harmonic waves. The Boussinesq equations are the simplest class of mathematical model that contain all these effects in ...