Explicit Exact Solitary Wave Solutions Research Articles

Bifurcation Behaviour of the Travelling Wave Solutions of the Perturbed Nonlinear Schrödinger Equation with Kerr Law Nonlinearity

In this paper, we study the bifurcations and dynamic behaviour of the travelling wave solutions of the perturbed nonlinear Schrödinger equation (NLSE) with Kerr law nonlinearity by using the theory of bifurcations of dynamic systems. Under the given parametric conditions, all possible representations of explicit exact solitary wave solutions and periodic wave solutions are obtained

New abundant exact solutions for the (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov system

In this paper, the extended hyperbolic function method is used for analytic treatment of the (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov (GNNV) system. We can obtained some new explicit exact solitary wave solutions, the multiple nontrivial exact periodic travelling wave solutions, the soliton solutions and complex solutions. Some known results in the literatures can be regarded as special cases. The methods employed here can also be used to solve a large class of nonlinear evolution equations.

New exact solutions to the Zakharov–Kuznetsov equation and its generalized form

In this paper, the extended hyperbolic function method is used for analytic treatment of the (2+1)-dimensional Zakharov–Kuznetsov (ZK) equation and its generalized form. We can obtained some new explicit exact solitary wave solutions, the multiple nontrivial exact periodic travelling wave solutions, the solitons solutions and complex solutions. Some known results in the literatures can be regarded as special cases. The methods employed here can also be used to solve a large class of nonlinear evolution equations.

Exact solitary and periodic wave solutions for a generalized nonlinear Schrödinger equation

The generalized nonlinear Schrödinger equation (GNLS) iu t + u xx + β∣ u∣ 2 u + γ∣ u∣ 4 u + iα (∣ u∣ 2 u) x + iτ(∣ u∣ 2) x u = 0 is studied. Using the bifurcation of travelling waves of this equation, some exact solitary wave solutions were obtained in [Wang W, Sun J,Chen G, Bifurcation, Exact solutions and nonsmooth behavior of solitary waves in the generalized nonlinear Schrödinger equation. Int J Bifucat Chaos 2005:3295–305.]. In this paper, more explicit exact solitary wave solutions and some new smooth periodic wave solutions are obtained.

Bifurcations of travelling wave solutions for Jaulent-Miodek equations

By using the theory of bifurcations of planar dynamic systems to the coupled Jaulent-Miodek equations, the existence of smooth solitary travelling wave solutions and uncountably infinite many smooth periodic travelling wave solutions is studied and the bifurcation parametric sets are shown. Under the given parametric conditions, all possible representations of explicit exact solitary wave solutions and periodic wave solutions are obtained.

New solitary wave solutions for nonlinear wave equation with fifth-order stronger nonlinear term

A new function-transformation method is proposed and four kinds of explicit exact solitary wave solutions of the Lienard equation, the generalized PC equation, the important nonlinear wave equation in mechanics, which have a fifth-order stronger nonlinear term, are obtained by a combination of the function-transformation method and the direct integral method. The method proposed in this paper can also be applied to other nonlinear equations with higher-order nonlinear terms.

A note on “On explicit exact solutions for the Liénard equation and its applications”

Feng [Phys. Lett. A 293 (2002) 50] obtained a kind of explicit exact solutions to the Liénard equation, and applied these results to find some explicit exact solitary wave solutions to the nonlinear Schrödinger equation and the Pochhammer–Chree equation. In this Letter, more explicit exact solitary wave solutions for the generalized Pochhammer–Chree equation are given by seeking qualitatively the homoclinic or heteroclinic orbits for this class of Liénard equation. Our results extended or improved the results in [Phys. Lett. A 293 (2002) 50; Acta Math. Appl. Sinica 21 (2) (1998) 249; Comput. Phys. Commun. 13 (1977) 149; Phys. Lett. A 196 (1995) 301; Stud. Appl. Math. 75 (1986) 95].

Explicit exact solitary wave solutions for generalized symmetric regularized long-wave equations with high-order nonlinear terms

In this paper, we have obtained the bell-type and kink-type solitary wave solutions of the generalized symmetric regularized long-wave equations with high-order nonlinear terms by means of proper transformation and undetermined assumption method.

Explicit Exact Solitary Wave Solutions for the Kundu Equation and the Derivative Schrödinger Equation

Studying explicit exact solutions of nonlinear evolution equations is a vastly open research field. In this paper, we are giving a new approach called the algebraic curve method to the study of the Kundu equation and the derivative Schrödinger equation. Using this technique we obtain the explicit exact solitary solutions for the Kundu equation and two new explicit exact solitary solutions for the derivative Schrödinger equation, which are different from the previous ones. Applications of this technique to some other nonlinear evolution equations are also presented.

New explicit exact solitary wave solutions for a new Hamiltonian amplitude equation

In this Letter, new explicit exact solitary wave solutions for a new Hamiltonian amplitude equation are obtained.