Analytic solutions of the (2 + 1)-dimensional nonlinear evolution equations using the sine-cosine method

In this paper, we establish exact solutions for (2 + 1)-dimensional nonlinear evolution equations. The sine-cosine method is used to construct exact periodic and soliton solutions of (2 + 1)-dimensional nonlinear evolution equations. Many new families of exact traveling wave solutions of the (2 + 1)-dimensional Boussinesq, breaking soliton and BKP equations are successfully obtained. These solutions may be important of significance for the explanation of some practical physical problems. It is shown that the sine-cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.     

Exact coherent structures in the (2 + 1)-dimensional KdV equations

Different from the (1 + 1)-dimensional nonlinear systems, (2 + 1) or higher dimensional nonlinear systems admit more rich coherent structures. Taking (2 + 1)-dimensional Korteweg de Vries (KdV for short) equations as an example, the singular manifold method is applied to search these coherent structures in an analytical form. With the aid of symbolic computation and plot representation of Maple, some coherent structures expressed in terms of new forms, such as dromions and solitoffs, have been illustrated by means of arbitrary functions in the analytical forms. In the paper, we will show these results by changing some specific choices for three different special cases for singular variable in details.     

A (3 + 1)-dimensional nonlinear evolution equation with multiple soliton solutions and multiple singular soliton solutions

In this work, a (3 + 1)-dimensional nonlinear evolution equation is investigated. The Hirota’s bilinear method is applied to determine the necessary conditions for the complete integrability of this equation. Multiple soliton solutions are established to confirm the compatibility structure. Multiple singular soliton solutions are also derived. The resonance phenomenon does not exist for this model.     

Wronskian determinant solutions of the (3 + 1)-dimensional Jimbo-Miwa equation

A set of sufficient conditions consisting of systems of linear partial differential equations is obtained which guarantees that the Wronskian determinant solves the (3 + 1)-dimensional Jimbo-Miwa equation in the bilinear form. Upon solving the linear conditions, the resulting Wronskian formulations bring solution formulas, which can yield rational solutions, solitons, negatons, positons and interaction solutions.     

A diversity of localized structures in a (2 + 1)-dimensional KdV equation

The singular manifold method is used to solve a (2 + 1)-dimensional KdV equation. An exact solution containing two arbitrary functions is then obtained. A diversity of localized structures, such as generalized dromions and solitoffs, is exposed by making full use of these arbitrary functions. These localized structures are illustrated by graphs.     

Exact solitary wave solutions of the generalized (2 + 1) dimensional Boussinesq equation

Bifurcation method of dynamical systems is employed to investigate bifurcation of solitary waves in the generalized (2 + 1) dimensional Boussinesq equation. Numbers of solitary waves are given for each parameter condition. Under some parameter conditions, exact solitary wave solutions are obtained.     

Interactions of solitary waves under the conditions of the (3 + 1)-dimensional Kadomtsev-Petviashvilli equation

Using an extended mapping method with a linear variable separation process, a new family of the exact solutions of the (3 + 1)-dimensional Kadomtsev-Petviashvilli (KP) equation was derived. By applying the solitary wave solutions, this paper studied some newly localized excitations and the interactions of various solitary waves under the conditions of the (3 + 1)-dimensional KP equation.     

Four (2 + 1)-dimensional integrable extensions of the Kadomtsev-Petviashvili equation

In this work, four (2 + 1)-dimensional nonlinear extensions of the Kadomtsev-Petviashvili (KP) equation are developed. The complete integrability of these models are investigated. Multiple-soliton solutions and multiple singular soliton solutions are determined to demonstrate the compatibility of these models. The resonance phenomenon does not exist for any of the derived models.     

A generalized extended rational expansion method and its application to (1 + 1)-dimensional dispersive long wave equation

In this Letter, a generalized extended rational expansion method is used to construct exact solutions of the (1 + 1)-dimensional dispersive long wave equation. As a result, many new and more general exact solutions are obtained, the solutions obtained in this Letter include rational triangular periodic wave solutions, rational solitary wave solutions.     

New exact solutions to the (2 + 1)-dimensional Ito equation: Extended homoclinic test technique

Exact soliton solutions to the (2 + 1)-dimensional Ito equation are studied based on the idea of extended homoclinic test and bilinear method. Some explicit solutions, such as triangle function solutions, soliton solutions, doubly-periodic wave solutions and periodic solitary wave solutions, are obtained. It shows that the (2 + 1)-dimensional Ito equation has richer solutions. Besides, the elastic interactions of the solutions and their corresponding physical meaning are discussed.     

Exact travelling wave solutions for the dissipative (2 + 1)-dimensional AKNS equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation. By qualitative analysis, global phase portraits of the dynamic system corresponding to the equation are obtained under different parameter conditions. Furthermore, the relations between the properties of travelling wave solutions and the dissipation coefficient r of the equation are investigated. In addition, the possible bell profile solitary wave solution, kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful. Based on above studies, a main contribution in this paper is to reveal the dissipation effect on travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation.     

Four (2 + 1)-dimensional integrable extensions of the KdV equation: Multiple-soliton and multiple singular soliton solutions

In this work, four (2 + 1)-dimensional nonlinear completely integrable equations, generated by extending the KdV equation are developed. The necessary condition for the complete integrability of these equation are formally derived. Multiple-soliton solutions and multiple singular soliton solutions are determined to emphasize the compatability of these models. The dispersion relations of these models are characterized by distinct physical structures. The resonance phenomenon for these equations does not exist for any model.     

Soliton solutions for (2 + 1)-dimensional and (3 + 1)-dimensional K(m, n) equations

We consider the nonlinear dispersive K(m,n) equation with the generalized evolution term and derive analytical expressions for some conserved quantities. By using a solitary wave ansatz in the form of sech^p function, we obtain exact bright soliton solutions for (2 + 1)-dimensional and (3 + 1)-dimensional K(m,n) equations with the generalized evolution terms. The results are then generalized to multi-dimensional K(m,n) equations in the presence of the generalized evolution term. An extended form of the K(m,n) equation with perturbation term is investigated. Exact bright soliton solution for the proposed K(m,n) equation having higher-order nonlinear term is determined. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients.     

Some new fractals behaviors in (2 + 1)-dimensional integrable system

By introducing the extended homogeneous balance approach into the (2 + 1)-dimensional integrable system, a linearized form of this physical model is established in this paper. Subsequently, after applied the Bäcklund transformation in the system, a variable separation solution with the entrance of different arbitrary functions is obtained. Furthermore, by using the Weierstrass, Bessel and Jacobian elliptic functions, some interesting fractal structures are produced.     

An algorithm for mean residual life computation of (n − k + 1)-out-of-n systems: An application of exponentiated Weibull distribution

In this paper, we establish an algorithm for the computation of the mean residual life of a (n − k + 1)-out-of-n system in the case of independent but not necessarily identically distributed lifetimes of the components. An application for the exponentiated Weibull distribution is given to study the effect of various parameters on the mean residual life of the system. Also the relationship between the mean residual life for the system and that of its components is investigated.     

Positive definite solutions of the nonlinear matrix equation X + AXA = Q (q > 0)

In this paper, the nonlinear matrix equation X + A^∗X^qA = Q (q > 0) is investigated. Some necessary and sufficient conditions for existence of Hermitian positive definite solutions of the nonlinear matrix equations are derived. An effective iterative method to obtain the positive definite solution is presented. Some numerical results are given to illustrate the effectiveness of the iterative methods.     

Comments on some recent results concerning {2, 3} and {2, 4}-generalized inverses

We give a comment on some recent results concerning the representations of generalized {2, 3} and {2, 4}-inverses. Shorter proofs of some previous results are presented.     

Explicit solutions of (2 + 1)-dimensional nonlinear KP-BBM equation by using Exp-function method

This paper is devoted to studying the (2 + 1)-dimensional KP-BBM wave equation. Exp-function method is used to conduct the analysis. The generalized solitary solutions, periodic solutions and other exact solutions for the (2 + 1)-dimensional KP-BBM wave equation are obtained via this method with the aid of symbolic computational system. It is also shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving other nonlinear evolution equations arising in mathematical physics.     

Generalized solitary and periodic wave solutions to a (2 + 1)-dimensional Zakharov-Kuznetsov equation

In this paper, the Exp-function method is employed to the Zakharov-Kuznetsov equation as a (2 + 1)-dimensional model for nonlinear Rossby waves. The observation of solitary wave solutions and periodic wave solutions constructed from the exponential function solutions reveal that our approach is very effective and convenient. The obtained results may be useful for better understanding the properties of two-dimensional coherent structures such as atmospheric blocking events.     

Construction of new exact rational form non-travelling wave solutions to the (2 + 1)-dimensional generalized Broer-Kaup system

With the aid of symbolic computation Maple, several new families of rational form variable separation solutions with three arbitrary functions to the (2 + 1)-dimensional generalized Broer-Kaup system are derived by using an improved mapping approach and a variable separation approach. These solutions include rational solitary wave solutions, periodic wave solutions and rational wave solutions. The properties of the novel localized excitation are revealed by some figures.