The Soliton Solution of （3＋1）—Dimensional Kadomtsev—Petviashvili Equations

A simple algebraic transformation relation of a special type of solution between the (3 1)-dimensional Kadomtsev-petviashvili(KP) equation and the cubic nonlinear Klein-Gordon equation (NKG) is established.Using known solutions of the NKG equation,we can obtain many soliton solutions and periodic solution of the (3 1)-dimensional KP equation.     

Symmetry Reduction, Exact Solutions, and Conservation Laws of （2＋1）-Dimensional Burgers Korteweg-de Vries Equation

Using the classical Lie method of infinitesimals, we first obtain the symmetry of the （2＋1）-dimensional Burgers-Korteweg-de-Vries （3D-BKdV） equation. Then we reduce the 3D-BKdV equation using the symmetry and give some exact solutions of the 3D-BKdV equation. When using the direct method, we restrict a condition and get a relationship between the new solutions and the old ones. Given a solution of the 3D-BKdV equation, we can get a new one from the relationship. The relationship between the symmetry obtained by using the classical Lie method and that obtained by using the direct method is also mentioned. At last, we give the conservation laws of the 3D-BKdV equation.     

Approximate solution for the Klein-Gordon-SchrSdinger equation by the homotopy analysis method

The Homotopy analysis method is applied to obtain the approximate solution of the Klein-Gordon Schrodinger equation. The Homotopy analysis solutions of the Klein-Gordon Schrodinger equation contain an auxiliary parameter which provides a convenient way to control the convergence region and rate of the series solutions. Through errors analysis and numerical simulation, we can see the approximate solution is very close to the exact solution.     

Approximate solution for the Klein Gordon-Schrdinger equation by the homotopy analysis method

The Homotopy analysis method is applied to obtain the approximate solution of the Klein--Gordon--Schr?dinger equation. The Homotopy analysis solutions of the Klein--Gordon--Schr?dinger equation contain an auxiliary parameter which provides a convenient way to control the convergence region and rate of the series solutions. Through errors analysis and numerical simulation, we can see the approximate solution is very close to the exact solution.     

Symmetry Reduction and Exact Solutions of the （3＋1）-Dimensional Breaking Soliton Equation

By means of the generalized direct method, a relationship is constructed between the new solutions and the old ones of the （3＋1）-dimensional breaking soliton equation. Based on the relationship, a new solution is obtained by using a given solution of the equation. The symmetry is also obtained for the （3＋1）-dimensional breaking soliton equation. By using the equivalent vector of the symmetry, we construct a seven-dimensional symmetry algebra and get the optimal system of group-invariant solutions. To every case of the optimal system, the （3＋1）-dimensional breaking soliton equation is reduced and some solutions to the reduced equations are obtained. Furthermore, some new explicit solutions are found for the （3＋ 1）-dimensional breaking soliton equation.     

Auto-Backlund Transformation and Exact Solutions to the Generalized Kadomtsev-Petviashvili Equation with Variable Coefficients

By using the extended homogeneous balance method, a new auto-Ba^ecklund transformation(BT) to the generalized Kadomtsew-Petviashvili equation with variable coefficients (VCGKP) are obtained. And making use of the auto-BT and choosing a special seed solution, we get many families of new exact solutions of the VCGKP equations, which include single soliton-like solutions, multi-soliton-like solutions, and special-soliton-like solutions. Since the KP equation and cylindrical KP equation are all special cases of the VCGKP equation, and the corresponding results of these equations are also given respectively.     

Solutions to the Complex Korteweg-de Vries Equation: Blow-up Solutions and Non-Singular Solutions

In the paper two kinds of solutions are derived for the complex Korteweg-de Vries equation, includ- ing blow-up solutions and non-singular solutions. We derive blow-up solutions from known 1-soliton solution and a double-pole solution. There is a complex Miura transformation between the complex Korteweg-de Vries equation and a modified Kortcweg-de Vries equation. Using the transformation, solitons, breathers and rational solutions to the com- plex Korteweg-de Vries equation are obtained from those of the modified Korteweg-de Vries equation. Dynamics of the obtained solutions are illustrated.     

N-soliton solutions of an integrable equation studied by Qiao

In this paper, we studied N-soliton solutions of a new integrable equation studied by Qiao [J. Math. Phys. 48 082701 (2007)]. Firstly, we employed the Darboux matrix method to construct a Darboux transformation for the modified Korteweg-de Vries equation. Then we use the Darboux transformation and a transformation, introduced by Sakovich [J. Math. Phys. 52 023509 (2011)], to derive N-soliton solutions of the new integrable equation from the seed solution. In particular, the multiple soliton solutions are explicitly obtained and shown through some figures.     

New multi—soliton solutions and travelling wave solutions of the dispersive long—wave equations

Using the extended homogeneous balance method,the (1 1)-dimensional dispersive long-wave equations have been solved.Starting from the homogeneous balance method,we have obtained a nonlinear transformation for simplifying a dispersive long-wave equation into a linear partial differential equation.Usually,we can obtain only a type of soliton-like solution.In this paper,we have further found some new multi-soliton solutions and exact travelling solutions of the dispersive long-wave equations from the linear partial equation.     

New explicit exact solutions to a nonlinear dispersive－dissipative equation

Using the first-integral method, we obtain a series of new explicit exact solutions such as exponential function solutions, triangular function solutions, singular solitary wave solution and kink solitary wave solution of a nonlinear dispersive－dissipative equation, which describes weak nonlinear ion-acoustic waves in plasma consisting of cold ions and warm electrons.     

Novel Interaction Solutions to Kadomtsev-Petviashvili Equation

In this paper, based on the forms and structures of Wronskian solutions to soliton equations, a Wronskian form expansion method is presented to find a new class of interaction solutions to the Kadomtsev-Petviashvili equation. One characteristic of the method is that Wronskian entries do not satisfy linear partial differential equation.     

Representations and Classification of Traveling Wave Solutions to sinh-GSrdon Equation

Two concepts named atom solution and combinatory solution are defined. The classification of all single traveling wave atom solutions to sinh-Gordon equation is obtained, and qualitative properties of solutions are discussed. In particular, we point out that some qualitative properties derived intuitively from dynamic system method are not true. Finally, we prove that our solutions to sinh-Gordon equation include all solutions obtained in the paper [Z.T. Fu, et al., Commun. Theor. Phys. （Beijing, China） 45 （2006） 55]. Through an example, we show how to give some new identities on Jacobian elliptic functions.     

New Explicit Solutions of （1 ＋ 1）-Dimensional Variable-Coefficient Broer-Kaup System

By using the compatibility method, many explicit solutions of the （1＋1）-dimensional variable-coefficient Broer Kaup system are constructed, which include new solutions expressed by error function, Bessel function, exponential function, and Airy function. Some figures of the solutions are given by the symbolic computation system Maple.     

New Exact Periodic Solitary-Wave Solutions for New （2＋1）-Dimensional KdV Equation

The extended homoclinic test function method is a kind of classic, efficient and well-developed method to solve nonlinear evolution equations. In this paper, with the help of this approach, we obtain new exact solutions （including kinky periodic solitary-wave solutions, periodic soliton solutions, and cross kink-wave solutions） for the new （2＋1）-dimensional KdV equation. These results enrich the variety of the dynamics of higher-dimensionai nonlinear wave field.     

Casoratian Solutions to Toda Lattice Via Its Backlund Transformation

Generalized Casoratian condition and Casoratian solutions of the Toda lattice are given in terms of its bilinear Bgcklund transformation. By choosing suitable Casoratian entries and parameter in the bilinear Bgcklund transformation, we can give transformations among many kinds of solutions.     

Exact Solutions to a （3＋l）-Dimensional Variable-Coefficient Kadomtsev-Petviashvilli Equation via the Bilinear Method and Wronskian Technique

By truncating the Painleve expansion at the constant level term, the Hirota bilinear form is obtained for a （3＋1）-dimensional variable-coefficient Kadomtsev Petviashvili equation. Based on its bilinear form, solitary-wave solutions are constructed via the ε-expansion method and the corresponding graphical analysis is given. Furthermore, the exact solution in the Wronskian form is presented and proved by direct substitution into the bilinear equation.     

Preparation and characterization of nanosized GdxBi0.95-xVO4：0.05Eu3＋ solid solution as red phosphor

A complete solid solutions with monophasic zircon-type structure of vanadates of formula GdxBio.95-xVO4：0.05Eu3＋ （x = 04）.95） are synthesized by combined method of co-precipitation and hydrothermal synthesis. Their microstructures and morphologies are characterized by X-ray powder diffraction and transmission electronic microscope, and the results show that each of all the samples has a monophasic zircon-type structure. The absorption spectrum of the prepared phosphor shows a blue-shift of the fundamental absorption band edge with increasing the gadolinium content. Under UV-light and visible-light excitation, all the prepared phosphors show the typical luminescence properties of Eu3＋ in the zircon-type structure. The emission intensity of GdxBi0.95-xVO4：0.05Eu3＋ （x = 0.55） is strongest in all samples under UV-light and visible-light excitations. Finally, the mechanisms of luminescence of Eu3＋ in the GdxBi0.95-xVO4：0.05Eu3＋ （x = 0-0.95） solid solutions are analyzed and discussed.     

Rational and Periodic Solutions for a （2＋1）-Dimensional Breaking Soliton Equation Associated with ZS-AKNS Hierarchy

The double Wronskian solutions whose entries satisfy matrix equation for a （2＋1）-dimensional breaking soliton equation （（2＋ 1）DBSE） associated with the ZS-AKNS hierarchy are derived through the Wronskian technique. Rational and periodic solutions for （2＋1）DBSE are obtained by taking special eases in general double Wronskian solutions.     

The Analytic Solution of SchrSdinger Equation with Potential Function Superposed by Six Terms with Positive-power and Inverse-power Potentials

The analytic solution of the radial Schrodinger equation is studied by using the tight coupling condition of several positive-power and inverse-power potential functions in this article. Furthermore, the precisely analytic solutions and the conditions that decide the existence of analytic solution have been searched when the potential of the radial Schrodinger equation is V（r） =α1r^8 ＋α2r^3 ＋ α3r^2 ＋β3r^-1 ＋β2r^-3 ＋β1r6-4. Generally speaking, there is only an approximate solution, but not analytic solution for SchrSdinger equation with several potentials＇ superposition. However, the conditions that decide the existence of analytic solution have been found and the analytic solution and its energy level structure are obtained for the Schrodinger equation with the potential which is motioned above in this paper. According to the single-value, finite and continuous standard of wave function in a quantum system, the authors firstly solve the asymptotic solution through the radial coordinate r → ∞ and r →0; secondly, they make the asymptotic solutions combining with the series solutions nearby the neighborhood of irregular singularities; and then they compare the power series coefficients, deduce a series of analytic solutions of the stationary state wave function and corresponding energy level structure by tight coupling among the coefficients of potential functions for the radial SchrSdinger equation; and lastly, they discuss the solutions and make conclusions.     

Numerical Complexiton Solutions of Complex KdV Equation

In this paper, we directly extend the applications of the Adomian decomposition method to investigate the complex KdV equation. By choosing different forms of wave functions as the initial values, three new types of realistic numerical solutions： numerical positon, negaton solution, and particularly the numerical analytical complexiton solution are obtained, which can rapidly converge to the exact ones obtained by Lou et al. Numerical simulation figures are used to illustrate the efficiency and accuracy of the proposed method.