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 (...

In this paper, the Exp-function method is used to seek new generalized solitonary solutions of the Riccati equation. Based on the Riccati equation and one of its generalized solitonary solutions, new exact solutions with three arbitrary functions of the (2+1)-dimensional dispersive long wave equations are obtained. Compared with the tanh-function method and its extensions, the proposed method i...

In this paper we implement the unified rational expansion methods, which leads to find exact rational formal polynomial solutions of nonlinear partial differential equations (NLPDEs), to the (1+1)dimensional dispersive long wave and Clannish Random Walker’s parabolic (CRWP) equations. By using this scheme, we get some solutions of the (1+1)-dimensional dispersive long wave and CRWP equations in...

In this paper, a new generalized F-expansion method is proposed to seek exact solutions of nonlinear evolution equations. With the aid of symbolic computation, we choose the (2+1)-dimensional dispersive long wave equations to illustrate the validity and advantages of the proposed method. As a result, many new and more general exact non-traveling wave and coefficient function solutions are obtai...

It is important to seek for more explicit exact solutions of nonlinear partial differential equations (NLPDEs) in mathematical physics. With the help of symbolic computation software like Maple or Mathematica, much work has been focused on the various extensions and applications of the known methods to construct exact solutions of NLPDEs. Mathematical modelling of physical systems often leads t...

A family of effective equations that capture the long time dispersive effects of wave propagation in heterogeneous media in an arbitrary large periodic spatial domain Ω ⊂ Rd over long time is proposed and analyzed. For a wave equation with highly oscillatory periodic spatial tensors of characteristic length ε, we prove that the solution of any member of our family of effective equations are ε-c...

Abstract With the help of bifurcation theory dynamical differential system and maple software, we shall devote ourselves to research travelling wave solutions bifurcations (2 + 1)-dimensional dissipative long equation. The study for equation derives a planar Hamiltonian system. Based on phase portraits, obtain exact explicit expressions some bounded traveling important singular solutions, under...

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 ...