Darboux Transformation and Soliton Solutions for the (2+1)-Dimensional Generalization of Shallow Water Wave Equation with Symbolic Computation

In this paper, the (2+1)-dimensional generalization of shallow water wave equation, which may be used to describe the propagation of ocean waves, is analytically investigated. With the aid of symbolic computation, we prove that the (2+1)-dimensional generalization of shallow water wave equation possesses the Painlevé property under a certain condition, and its Lax pair is constructed by applying the singular manifold method. Based on the obtained Lax representation, the Darboux transformation (DT) is constructed. The first iterated solution, second iterated solution and a special N-soliton solution with an arbitrary function are derived with the resulting DT. Relevant properties are graphically illustrated, which might be helpful to understanding the propagation processes for ocean waves in shallow water.     

Lax Pair and Darboux Transformation for a Variable-Coefficient Fifth-Order Korteweg-de Vries Equation with Symbolic Computation

In this paper, we put our focus on a variable-coe~cient fifth-order Korteweg-de Vries （fKdV） equation, which possesses a great number of excellent properties and is of current importance in physical and engineering fields. Certain constraints are worked out, which make sure the integrability of such an equation. Under those constraints, some integrable properties are derived, such as the Lax pair and Darboux transformation. Via the Darboux transformation, which is an exercisable way to generate solutions in a recursive manner, the one- and two-solitonic solutions are presented and the relevant physical applications of these solitonic structures in some fields are also pointed out.     

Darboux Transformation and Grammian Solutions for Nonisospectral Modified Kadomtsev-Petviashvili Equation with Symbolic Computation

In the present paper, under investigation is a nonisospectral modified Kadomtsev-Petviashvili equation, which is shown to have two Painleve branches through the Painleve analysis. With symbolic computation, two Lax pairs for such an equation are derived by applying the generalized singular manifold method. Furthermore, based on the two obtained Lax pairs, the binary Darboux transformation is constructed and then the N-th-iterated potential transformation formula in the form of Grammian is also presented.     

Darboux Transformation and Grammian Solutions for Nonisospectral Modified Kadomtsev-Petviashvili Equation with Symbolic Computation

In the present paper, under investigation is a nonisospectral modified Kadomtsev-Petviashvili equation, which is shown to have two Painlevé branches through the Painlevé analysis. With symbolic computation, two Lax pairs for such an equation are derived by applying the generalized singular manifold method. Furthermore, based on the two obtained Lax pairs, the binary Darboux transformation is constructed and then the N-th-iterated potential transformation formula in the form of Grammian is also presented.     

Symbolic Computation and Construction of Soliton-Like Solutions to the(2+1)-Dimensional Breaking Soliton Equation

Based on the computerized symbolic system Maple, a new generalized expansion method of Riccatiequation for constructing non-travelling wave and coefficient functions‘ soliton-like solutions is presented by a new generalansatz. Making use of the method, we consider the (2 1)-dimensional breaking soliton equation, ut buxxy 4buvx 4buxv = 0, uy = vx, and obtain rich new families of the exact solutions of the breaking soliton equation, including thenon-travelling wave and constant function soliton-like solutions, singular soliton-like solutions, and triangular functionsolutions.     

Symbolic Computation and Construction of Soliton－Like Solutions to the（2＋l）－Dimensional Breaking Soliton Equation

Based on the computerized symbolic system Mapte, a new generalized expansion method of Riccati equation for constructing non-travelling wave and coefficient functions‘ soliton-like solutions is presented by a new general ansatz. Making use of the method, we consider the (2 1)-dimensional breaking soliton equation, ut buxxy 4buvx 4buxv = O,uv=vx, and obtain rich new families of the exact solutions of the breaking sofiton equation, including then on-traveilin~ wave and constant function sofiton-like solutions, singular soliton-like solutions, and triangular function solutions.     

Symbolic Computation and q-Deformed Function Solutions of (2+1)-Dimensional Breaking Soliton Equation

In this paper,by using symbolic and algebra computation,Chen and Wang's multiple Riccati equations rational expansion method was further extended.Many double soliton-like and other novel combined forms of exact solutions of the (2 1 )-dimensional Breaking soliton equation are derived by using the extended multiple Riccati equations expansion method.     

Darboux Transformation and Soliton Solutions for Inhomogeneous Coupled Nonlinear Schrodinger Equations with Symbolic Computation

With the aid of computation, we consider the variable-coefficient coupled nonlinear Schrodinger equations with the effects of group-velocity dispersion, self-phase modulation and cross-phase modulation, which have potential applications in the long-distance communication of two-pulse propagation in inhomogeneous optical fibers. Based on the obtained nonisospectral linear eigenvalue problems （i.e. Lax pair）, we construct the Darboux transformation for such a model to derive the optical soliton solutions. In addition, through the one- and two-soliton-like solutions, we graphically discuss the features of picosecond solitons in inhomogeneous optical fibers.     

Painleve Analysis and Darboux Transformation for a Variable-Coefficient Boussinesq System in Fluid Dynamics with Symbolic Computation

The new soliton solutions for the variable-coefficient Boussinesq system, whose applications are seen in fluid dynamics, are studied in this paper with symbolic computation. First, the Painleve analysis is used to investigate its integrability properties. For the identified case we give, the Lax pair of the system is found, and then the Darboux transformation is constructed. At last, some new soliton solutions are presented via the Darboux method. Those solutions might be of some value in fluid dynamics.     

Darboux Transformation and Soliton Solutions for InhomogeneousCoupled Nonlinear Schrödinger Equations with Symbolic Computation

With the aid of computation, we consider the variable-coefficient coupled nonlinear Schrödinger equations with the effects of group-velocity dispersion, self-phase modulation and cross-phase modulation, which have potential applications in the long-distance communication of two-pulse propagation in inhomogeneous optical fibers. Based on the obtained nonisospectral linear eigenvalue problems (i.e. Lax pair), we construct the Darboux transformation for such a model to derive the optical soliton solutions. In addition, through the one- and two-soliton-like solutions, we graphically discuss the features ofpicosecond solitons in inhomogeneous optical fibers.     

Comment on ＂Symbolic Computation and q-Deformed Function Solutions of the （2＋1）-Dimensional Breaking Soliton Equation＂

In a recent article [Commun. Theor. Phys. （Beijing, China） 47 （2007） 270], Cao et al. gave some nontrivial solutions of a Riccati equation by using symbolic and algebra computation. They took these solutions, which are in the form of q-deformed hyperbolic and triangular functions as new solutions. In this comment, we will show that these solutions are just the special cases of some known solutions of the Riccati equation and thus they are not new solutions.     

On Triply Periodic Wave Solutions for(2+1)-Dimensional Boussinesq Equation

By employing Hirota bilinear method and Riemann theta functions of genus one,explicit triply periodic wave solutions for the(2+1)-dimensional Boussinesq equation are constructed under the Backlund transformation u =(1 /6)(u0 1) + 2[ln f(x,y,t)] xx,four kinds of triply periodic wave solutions are derived,and their long wave limit are discussed.The properties of one of the solutions are shown in Fig.1.     

A New (2+1)-Dimensional KdV Equation and Its Localized Structures

A new (2+1)-dimensional KdV equation is constructed by using Lax pair generating technique. Exact solutions of the new equation are studied by means of the singular manifold method. Bäcklund transformation in terms ofthe singular manifold is obtained. And localized structures are also investigated.     

New Double-Periodic Soliton Solutions for the (2+1)-Dimensional Breaking Soliton Equation

Under investigation is the (2+1)-dimensional breaking soliton equation. Based on a special ansätz functions and the bilinear form, some entirely new double-periodic soliton solutions for the (2+1)-dimensional breaking soliton equation are presented. With the help of symbolic computation software Mathematica, many important and interesting properties for these obtained solutions are revealed with some figures.     

Two Classes of New Exact Solutions to (2+1)-Dimensional Breaking Soliton Equation

The singular manifold method is used to obtain two general solutions to a (2 1)-dimensional breaking soliton equation, each of which contains two arbitrary functions. Then the new periodic wave solutions in terms of the Jacobi elliptic functions are generated from the general solutions. The long wave limit yields the new types of dromion and solitary structures.     

Integrability and Solutions of the (2+1)-Dimensional Broer-Kaup Equation with Variable Coefficients

The integrability of the(2+1)-dimensional Broer-Kaup equation with variable coefficients(VCBK) is verified by finding a transformation mapping it to the usual(2+1)-dimensional Broer-Kaup equation(BK).Thus the solutions of the(2+1)-dimensional VCBK are obtained by making full use of the known solutions of the usual(2+1)dimensional BK.Two new integrable models are given by this transformation,their dromion-like solutions and rogue wave solutions are also obtained.Further,the velocity of the dromion-like solutions can be designed and the center of the rogue wave solutions can be controlled artificially because of the appearance of the four arbitrary functions in the transformation.     

Double Elliptic Equation Expansion Approach and Novel Solutions of （2＋1）-Dimensional Break Soliton Equation

In this paper, by means of double elliptic equation expansion approach, the novel double nonlinear wave solutions of the （2＋1）-dimensional break soliton equation are obtained. These double nonlinear wave solutions contain the double Jacobi elliptic function-like solutions, the double solitary wave-like solutions, and so on. The method is also powerful to some other nonlinear wave equations in （2＋1） dimensions.     

Darboux Transformations and N-soliton Solutions of Two （2＋1）-Dimensional Nonlinear Equations

Two Darboux transformations of the （2＋1）-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawaka （ CDGKS） equation and （2＋1）-dimensional modified Korteweg-de Vries （mKdV） equation are constructed through the Darboux matrix method, respectively. N-soliton solutions of these two equations are presented by applying the Darboux trans- formations N times. The right-going bright single-soliton solution and interactions of two and three-soliton overtaking collisions of the （2＋1）-dimensional CDGKS equation are studied. By choosing different seed solutions, the right-going bright and left-going dark single-soliton solutions, the interactions of two and three-soliton overtaking collisions, and kink soliton solutions of the （2＋1）-dimensional mKdV equation are investigated. The results can be used to illustrate the interactions of water waves in shallow water.     

Darboux Transformation and New Soliton-like Solutions for (1+1)-Dimensional Higher-Order Broer-Kaup System

In this letter, we construct a kind of new Darboux transformation for the (1+1)-dimensional higher-order Broer-Kaup (HBK) system with the help of a gauge transformation of a spectral problem. By applying this new Darboux transformation, some new soliton-like solutions of the (1+1)-dimensional HBK system are obtained.