The Cauchy problem for the damped Boussinesq equation with small initial data is considered in two space dimensions. Existence and uniqueness of its classical solution is proved and the solution is constructed in the form of a series. The major term of its long-time asymptotics is calculated explicitly and a uniform in space estimate of the residual term is given.

A. general theory for determining Hamiltonian model equations from noncanonical perturbation expansions of Hamiltonian systems is applied to the Boussinesq expan sion fcr long, small amplitude waves in shallow water, leading to the Korteweg-deVries equation. New Hamiltonian model equations, including a natural "Hamiltonian ver-cn» of the KdV equation, are proposed. The method also provides a di...

We construct the traveling wave solutions of the 1 1 -dimensional modified Benjamin-BonaMahony equation, the 2 1 -dimensional typical breaking soliton equation, the 1 1 -dimensional classical Boussinesq equations, and the 2 1 -dimensional Broer-Kaup-Kuperschmidt equations by using an extended G′/G -expansion method, where G satisfies the second-order linear ordinary differential equation. By us...

We develop a new accurate and robust numerical method for the Boussinesq paradigm equation (BPE). To design the method we first introduce a change of variables, for which the BPE takes the form of a nonlinear wave equation with the global pressure, and rewrite the wave equation as a system of conservation laws with a global flux. We then apply a Godunov-type central-upwind scheme together with ...