Extended shallow water wave equations are derived, using the method of asymptotic expansions, from Euler (or wave) equations. These extended models valid one order beyond usual weakly nonlinear, long approximation, incorporating all appropriate dispersive and nonlinear terms. Specifically, first we derive Korteweg–de Vries (KdV) equation, then proceed with Benjamin–Bona–Mahony Camassa–Holm in (...

3 Distance defined by optimal transportation problem 33 3.1 A distance functional in the spatially periodic case. Introduction This thesis deals with two strongly nonlinear evolution Partial Differential Equation (in the following named P.D.E.) arising from mathematical physics. The first one was introduced first by Fokas and Fuchssteiner [28] as a bi-Hamiltonian equation, and then was rediscov...

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing three translation equation, fourth-order ordinary differential is obtained solved in terms an incomplete elliptic integral. Moreover, with aid Kudryashov’s approach, more closed-form solutions cons...

It is known that the governing equations for the evolution of the envelope of weakly nonlinear capillary-gravity wave packets, the so-called Davey-Stewartson equation, admit dromion solutions in the limit of shallow water and strong surface tension, i.e., when kh≪1 and the Bond number (Bo) ≡T/ρgh^{2}>1/3 (where T is the surface tension coefficient and h the water depth). Here we show that capil...

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...

Third-order dispersive evolution equations are widely adopted to model one-dimensional long waves and have extensive applications in fluid mechanics, plasma physics nonlinear optics. The typical representatives the KdV equation, Camassa–Holm equation Degasperis–Procesi equation. They share many common features such as complete integrability, Lax pairs bi-Hamiltonian structure. In this paper we ...

in this work g'/g-expansion method has been employed to solve (2+1)-dimensional dispersive long wave equation. it is shown that g'/g-expansion method, with the help of symbolic computation, provides a very effective and powerful mathematical tool, for solving this equation.