Based on the traveling wave reduction method with a perturbed initial solution and F-expansion method, class of explicit exact solutions (2+1)-dimensional CDGKS equation are obtained through symbolic computation. Moreover, both interaction behavior between parameters perturbation degree periodic Gauss to rational pulse wave, correlation superposition energy solitary discussed. Finally, numerica...

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...

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 ...

Spatially homogeneous models in quantum supergravity with a nonvanishing cosmological constant are studied. A class of exact nontrivial solutions of the supersymmetry and Lorentz constraints is obtained in terms of the Chern-Simons action on the spatially homogeneous 3-manifold, both in Ashketar variables where the solution is explicit up to reality conditions, and, more concretely, in the tetr...

The semilinear normal parabolic equations corresponding to 3D Navier-Stokes system have been derived. The explicit formula for solution of normal parabolic equations with periodic boundary conditions has been obtained. It was shown that phase space of corresponding dynamical system consists of the set of stability (where solutions tends to zero as time t → ∞), the set of explosions (where solut...

We find the general behaviour of homogeneous and isotropic cosmological models in some fourth-order theories of gravity. Explicit, exact, general solutions are given for both empty universes and those filled with a perfect fluid. For the vacuum case, solutions are found with closed, open and flat geometries, whilst the perfect fluid solutions are all spatially flat. Both early and late-time lim...

A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order group foliation system whose independent and dependent variables respectively consist of the invariants and differential invariants of a given one-dimensional gr...

An approach is proposed to obtain some exact explicit solutions in terms of the Weierstrass’ elliptic function ℘ to a generalized Benjamin-Bona-Mahony (BBM) equation. Conditions for periodic and solitary wave like solutions can be expressed compactly in terms of the invariants of ℘. The approach unifies recently established ad-hoc methods to a certain extent. Evaluation of a balancing principle...

When one solves differential equations, modeling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. Based on general theory for positivity, with an explicit third-order Runge-Kutta method (we will refer to it as RK3 method) pos...