Exact Solutions of Generalized Burgers-Fisher Equation with Variable Coefficients

The generalized Riccati equation vational expansion method is extended in this paper. Several exact solutions for the generalized Burgers-Fisher equation with variable coefficients are obtained by this method, and some of which are derived for the first time. It is concluded from the results that this approach is simple and efficient even in solving partial differential equations with variable coefficients.     

Interaction Solutions for Kadomtsev-Petviashvili Equation with Variable Coefficients *

Based on the Hirota's bilinear form and symbolic computation, the Kadomtsev-Petviashvili equation with variable coefficients is investigated. The lump solutions and interaction solutions between lump solution and a pair of resonance stripe solitons are presented. Their dynamical behaviors are described by some three-dimensional plots and corresponding contour plots.     

New Soliton-Like Solutions to Variable Coefficients MKdV Equation

A further improved projective Riccati equation method is proposed. By applying it to solve the variable coefficients MKdV equation, we obtain many new-type soliton-like solutions to this equation.     

New Soliton-like Solutions to Variable Coefficients MKdV Equation

A further improved projective Riccati equation method is proposed. By applying it to solve the variable coeffcients MKdV equation, we obtain many new-type soliton-like solutions to this equation.     

The Method of Fundamental Solutions for Solving Convection-Diffusion Equations with Variable Coefficients

A meshless method based on the method of fundamental solutions (MFS) is proposed to solve the time-dependent partial differential equations with variable coefficients. The proposed method combines the time discretization and the one-stage MFS for spatial discretization. In contrast to the traditional two-stage process, the one-stage MFS approach is capable of solving a broad spectrum of partial differential equations. The numerical implementation is simple since both closed-form approximate particular solution and fundamental solution are easier to find than the traditional approach. The numerical results show that the one-stage approach is robust and stable.     

New Soliton-like Solutions and Multi-soliton Structures for Broer-Kaup System with Variable Coefficients

By using the further extended tanh method [Phys. Lett. A 307 （2003） 269; Chaos, Solitons ＆ Fractals 17 （2003） 669] to the Broer-Kaup system with variable coefficients, abundant new soliton-like solutions and multl-solitonlike solutions are derived. Based on the derived multi-soliton-like solutions which contain arbitrary functions, some interesting multi-soliton structures are revealed.     

复变系数Ginzburg-Landau方程的啁啾组合孤波解

基于描述非均匀光纤系统的复系数Ginzburg-Landau方程,通过拟解法得到了该方程的精确啁啾组合孤波解,并分析了该解的特性.通过大量的数值模拟,发现在有限的初始扰动下这些组合孤波解是在非均匀光纤系统中稳定的.最后,为了进一步研究组合孤波解的稳定性,我们还探讨了组合孤波的相互作用.     

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.     

Auto-B(a)cklund Transformation and Exact Solutions to the Generalized Kadomtsev-Petviashvili Equation with Variable Coefficients

By using the extended homogeneous balance method, a new auto-Backlund transformation(BT) to thegeneralized Kadomtsev-Petviashvili equation with variable coefficients (VCGKP) are obtained. And making use of theauto-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 KPequation and cylindrical KP equation are all special cases of the VCGKP equation, and the corresponding results ofthese equations are also given respectively.     

Analytical Solitary Wave Solutions to a （3＋1）-Dimensional Gross-Pitaevskii Equation with Variable Coefficients

A （3＋1）-dimensional Gross-Pitaevskii （GP） equation with time variable coefficients is considered, and is transformed into a standard nonlinear Schrodinger （NLS） equation. Exact solutions of the （3＋1）D GP equation are constructed via those of the NLS equation. By applying specific time-modulated nonlinearities, dispersions, and potentials, the dynamics of the solutions can be controlled. Solitary and periodic wave solutions with snaking and breathing behavior are reported.     

Analytical Solitary Wave Solutions to a (3+1)-Dimensional Gross-Pitaevskii Equation with Variable Coefficients

A (3+1)-dimensional Gross-Pitaevskii (GP) equation with time variable coefficients is considered, and is transformed into a standard nonlinear Schrödinger (NLS) equation. Exact solutions of the (3+1)D GP equation are constructed via those of the NLS equation. By applying specific time-modulated nonlinearities, dispersions, and potentials, the dynamics of the solutions can be controlled. Solitary and periodic wave solutions with snaking and breathing behavior are reported.     

Solutions of Heat-Like and Wave-Like Equations with Variable Coefficients by Means of the Homotopy Analysis Method

We employ the homotopy analysis method （HAM） to obtain approximate analytical solutions to the heat-like and wave-like equations. The HAM contains the auxiliary parameter h, which provides a convenient way of controlling the convergence region of series solutions. The analysis is accompanied by several linear and nonlinear heat-like and wave-like equations with initial boundary value problems. The results obtained prove that HAM is very effective and simple with less error than the Adomian decomposition method and the variational iteration method.     

Similarity Solutions for Generalized Variable Coefficients Zakharov-Kuznetsov Equation under Some Integrability Conditions

In this paper, the symmetry method has been carried over to the generalized variable coefficients Zakharov- Kuznetsov equation. The infinitesimal symmetries and the optimal system are deduced and from this optimal system seven basic fields are determined, and for every vector field in the optimal system the admissible forms of the coefficients are found and this also leads us to transform the given equation into partial differential equations in two variables. After using some referenced transformations the mentioned partial differential equations eventually reduce to ordinary differential equations. The search for solutions to those equations has yielded many exact solutions in most cases.     

Similarity Solutions for Generalized Variable Coefficients Zakharov-Kuznetsov Equation under Some Integrability Conditions

In this paper, the symmetry method has been carried over to the generalizedvariable coefficients Zakharov-Kuznetsov equation. The infinitesimalsymmetries and the optimal system are deduced and from this optimal systemseven basic fields are determined, and for every vector field in the optimalsystem the admissible forms of the coefficients are found and this also leadsus to transform the given equation into partial differential equations intwo variables. After using some referenced transformations the mentionedpartial differential equations eventually reduce to ordinary differentialequations. The search for solutions to those equations has yielded manyexact solutions in most cases.     

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.     

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.     

Large-Order Strong Coupling Perturbation Coefficients for Anharmonic Oscillators

A large-order formula for the perturbation coefficients of the strong-couplingperturbation expansion for anharmonic oscillators with Hamiltonian H = p ² +x ² + x ^2m is derived and parameters in thisformula are determined for the ground and first excited states of the quartic,sextic, octic, and decadic oscillators (m = 2, 3, 4, 5).     

Group Classification and Exact Solutions of a Class of Variable Coefficient Nonlinear Wave Equations

Complete group classification of a class of variable coefficient （1＋1）-dimensional wave equations is performed. The possible additional equivalence transformations between equations from the class under consideration and the conditional equivalence groups are also investigated. These allow simplification of the results of the classification and further applications of them. The derived Lie symmetries are used to construct exact solutions of special forms of these equations via the classical Lie method. Nonclassical symmetries of the wave equations are discussed.     

New Multiple Soliton-like Solutions to (3+1)-Dimensional Burgers Equation with Variable Coefficients

A new generalized tanh function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the Riccati equation, which has more new solutions. More new multiple soliton-like solutions are obtained for the (3 1 )-dimensional Burgers equation with variable coefficients.