Nonlinear dynamics of wave packets in parity-time-symmetric optical lattices near the phase-transition point is analytically studied. A nonlinear Klein-Gordon equation is derived for the envelope of these wave packets. A variety of phenomena known to exist in this envelope equation are shown to also exist in the full equation, including wave blowup, periodic bound states, and solitary wave solu...

Submitted: Nov 12, 2013; Accepted: Dec 18, 2013; Published: Dec 22, 2013 Abstract: In this article, we have employed an enhanced (G′/G)-expansion method to find the exact solutions first and then the solitary wave solutions of the nonlinear generalized shallow water wave equation. Here we have derived solitons, singular solitons and periodic wave solutions through the enhanced (G′/G)-expansion ...

A model describing non-equilibrium processes in relaxing media is considered. Restrictions arising from the symmetry principles and the second law of thermodynamics are stated. System of ODE describing a set of travelling wave solutions is obtained via group theory reduction. Bifurcation analysis of this system reveals the existence of periodic invariant solutions as well as limiting to them so...

The generalized Camassa-Holm equation ut + 2kux − uxxt + auux = 2uxuxx + uuxxx + γuxxx is considered in this paper. Under traveling wave variable substitution, the equation is related to a planar singular system. By making a transformation this singular system becomes a regular system. Through discussing the dynamical behavior of the regular system, the explicit periodic blow-up solutions and s...

Explicit solitary waves are known to exist for the Kadomtsev-Petviashvili-I (KP-I) equation in dimension 2. We first address numerically the question of their Morse index. The results confirm that the lump solitary wave has Morse index one and that the other explicit solutions correspond to excited states. We then turn to the 2D Gross-Pitaevskii (GP) equation which in some long wave regime conv...

In this paper, employed exp-function method and F-expansion method, we study the BBM(m,n) equation with generalized evolution, four families of exact solutions of exp-function type are obtained. Under different parametric conditions, every family of solution can be reduced to some solitary wave solutions and periodic wave solutions. The results presented in this paper improve the previous resul...

In this paper, a generalized variable coefficient Gardner equation is investigated. Exp-function method combined with F-expansion method are used to determine six families of exact solutions of exp-function type for this equation. Under different parametric conditions, every family of solution can be reduced to some solitary wave solutions and periodic wave solutions. The results presented in t...

We consider the periodic initial-value problem for the family of a-b-c-d Boussinesq systems, [8], [9], and their completely symmetric analogs, [10]. We approximate their solutions by the standard Galerkin-finite element method using smooth periodic splines for discretizing in space. We prove optimal-order L error estimates for the resulting semidiscretizations. The numerical schemes are usual i...

A new method named rational expansion method of exponent function is presented to find exact traveling wave solutions of differential-difference equations. This method generalizes the so-called tanh-method and other similar methods. Some examples are dealt with and their abundant exact solutions which include solitary solutions and periodic solutions are obtained. Among them, many solutions are...