Exact Solutions of Forced Schrödinger Equation and How to Choose the External Forces

Schrödinger equations are very common equations in physics and mathematics for nonlinear physics to model the dynamics of wave propagation in waveguides such as power lines, atomic chains, optical fibers, and even in quantum mechanics. But all these equations are most often studied without worrying about what would happen if this equation were maintained, that is to say, had a second member synonymous with an external force. It is true that on a physical level, such equations can be considered as describing the generation of waves on a waveguide using an external force. However, the in-depth analysis of this aspect is not at the center of our reflection in this article, but for us, it is a question of proposing exact solutions to this type of equation and above all proposing the general form of the external force so that the obtaining exact solutions is possible.

Some Evolution Hierarchies Derived from Self-dual Yang-Mills Equations

We develop in this paper a new method to construct two explicit Lie algebras E and F. By using a loop algebra E of the Lie algebra E and the reduced self-dual Yang-Mills equations, we obtain an expanding integrable model of the Giachetti-Johnson （G J） hierarchy whose Hamiltonian structure can also be derived by using the trace identity. This provides a much simplier construction method in comparing with the tedious variational identity approach. Furthermore, the nonlinear integrable coupling of the GJ hierarchy is readily obtained by introducing the Lie algebra gN. As an application, we apply the loop algebra E of the Lie algebra E to obtain a kind of expanding integrable model of the Kaup-Newell （KN） hierarchy which, consisting of two arbitrary parameters a and β, can be reduced to two nonlinear evolution equations. In addition, we use a loop algebra F of the Lie algebra F to obtain an expanding integrable model of the BT hierarchy whose Hamiltonian structure is the same as using the trace identity. Finally, we deduce five integrable systems in R3 based on the self-dual Yang-Mills equations, which include Poisson structures, irregular lines, and the reduced equations.

A SOLUTION OF YANG-MILLS EQUATION ON BDS SPACE

In [6], a global solution of Yang-Mills equation on de-Sitter spacetime with conformal fiat metric was given by Prof. Lu. In this article, Yang-Mills equation on ndimensional de-Sitter space with Beltrami-Hua-Lu metric is discussed and a global solution is obtained.

On New Solutions of Classical Yang-Mills Equations with Cylindrical Sources

Strong fields generated by big electric currents are examined within the framework of the Yang-Mills nonlinear generalization of the Maxwell electrodynamics proposed in our earlier papers. First we consider the case of stationary currents and find a new exact solution to the Yang-Mills equations. Then we study a Yang-Mills field inside a thin circular cylinder with nonstationary plasma and find expressions for field strengths in it. Obtained results are applied to interpret several puzzling natural phenomena.

Fractional Breaking Soliton Equation Reduced from a Linear Spectral Problem Associated with Fractional Self-Dual Yang-Mills Equations

Fractional or fractal calculus is everywhere and very important.It is reported that the fractal approach is suitable for insight into the effect of porous structure on thermo-properties of cloth.A novel local fractional breaking soliton equation is derived from the reduction of the linear spectral problem associated with the local fractional non-isospectral self-dual Yang-Mills equations.More specifically,the employed linear spectral problem is first reduced to the(2+1)-dimensional local fractional zero-curvature equation through variable transformations.Based on the reduced local fractional zero-curvature equation,the fractional breaking soliton equation is then constructed by the method of undetermined coefficients.This paper shows that some other local fractional models can be obtained by generalizing the existing methods of generating nonlinear partial differential equations with integer orders.

Explicit and Exact Solutions for Kadomtsev-Petviashvili Equation

借助Ｗｅｉｓｓ等所得到的解的变换，找到五种ＫＰ方程的显示和精确解。其中也包含了孤波解。

New exact solitary wave solutions to generalized mKdV equation and generalized Zakharov-Kuzentsov equation

In this paper, based on hyperbolic tanh-function method and homogeneous balance method, and auxiliary equation method, some new exact solitary solutions to the generalized mKdV equation and generalized Zakharov-Kuzentsov equation are constructed by the method of auxiliary equation with function transformation with aid of symbolic computation system Mathematica. The method is of important significance in seeking new exact solutions to the evolution equation with arbitrary nonlinear term.

A method for constructing exact solutions and application to Benjamin Ono equation

By using an improved projective Riccati equation method, this paper obtains several types of exact travelling wave solutions to the Benjamin Ono equation which include multiple soliton solutions, periodic soliton solutions and Weierstrass function solutions. Some of them are found for the first time. The method can be applied to other nonlinear evolution equations in mathematical physics.

The Jacobi elliptic function-like exact solutions to two kinds of KdV equations with variable coefficients and KdV equation with forcible term

By use of an auxiliary equation and through a function transformation, the Jacobi elliptic function wave-like solutions, the degenerated soliton-like solutions and the triangle function wave solutions to two kinds of Korteweg de Vries （KdV） equations with variable coefficients and a KdV equation with a forcible term are constructed with the help of symbolic computation system Mathematica, where the new solutions are also constructed.

Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics Using the Modified Simple Equation Method

The modified simple equation method is employed to construct the exact solutions involving parameters of nonlinear evolution equations via the (1+1)-dimensional modified KdV equation,and the (1+1)-dimensional reaction-diffusion equation.When these parameters are taken to be special values,the solitary wave solutions are derived from the exact solutions.It is shown that the proposed method provides a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

EXACT SOLUTIONS FOR GENERAL VARIABLE-COEFFICIENT KdV EQUATION

By asing the nonclassical method of symmetry reductions, the exact solutions for general variable coefficient KdV equation with dissipative loss and nonuniformity terms are obtained. When the dissipative loss and nonuniformity terms don't exist, the multisoliton solutions are found and the corresponding Painleve II type equation for the variable coefficient KdV equation is given.

New Exact Travelling Wave Solutions for Generalized Zakharov-Kuzentsov EquationsUsing General Projective Riccati Equation Method

Applying the generalized method, which is a direct and unified algebraic method for constructing multipletravelling wave solutions of nonlinear partial differential equations (PDEs), and implementing in a computer algebraicsystem, we consider the generalized Zakharov-Kuzentsov equation with nonlinear terms of any order. As a result, wecan not only successfully recover the previously known travelling wave solutions found by existing various tanh methodsand other sophisticated methods, but also obtain some new formal solutions. The solutions obtained include kink-shapedsolitons, bell-shaped solitons, singular solitons, and periodic solutions.

Infinite Sequence Soliton-Like Exact Solutions of (2 + 1)-Dimensional Breaking Soliton Equation

To seek new infinite sequence soliton-like exact solutions to nonlinear evolution equations （NEE（s））, by developing two characteristics of construction and mechanization on auxiliary equation method, the second kind of elliptie equation is highly studied and new type solutions and Backlund transformation are obtained. Then （2＋ l ）-dimensional breaking soliton equation is chosen as an example and its infinite sequence soliton-like exact solutions are constructed with the help of symbolic computation system Mathematica, which include infinite sequence smooth soliton-like solutions of Jacobi elliptic type, infinite sequence compact soliton solutions of Jacobi elliptic type and infinite sequence peak soliton solutions of exponential function type and triangular function type.

REDUCTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATION AND EXACT SOLUTIONS

Nonlinear partial differetial equation(NLPDE) is converted into ordinary differential equation(ODE) via a new ansatz.Using undetermined function method,the ODE obtained above is replaced by a set of algebraic equations which are solved out with the aid of Mathematica.The exact solutions and solitary solutions of NLPDE are obtained.

Invariant Sets and Exact Solutions to Higher-Dimensional Wave Equations

The invariant sets and exact solutions of the （1 ＋ 2）-dimensional wave equations are discussed. It is shown that there exist a class of solutions to the equations which belong to the invariant set E0 = {u ： ux = vxF（u）,uy = vyF（u） }. This approach is also developed to solve （1 ＋ N）-dimensional wave equations.

An Extension of Mapping Deformation Method and New Exact Solution for Three Coupled Nonlinear Partial Differential Equations

In this paper, we extend the mapping deformation method proposed by Lou. It is used to find new exacttravelling wave solutions of nonlinear partial differential equation or coupled nonlinear partial differential equations(PDEs). Based on the idea of the homogeneous balance method, we construct the general mapping relation betweenthe solutions of the PDEs and those of the cubic nonlinear Klein-Gordon (NKG) equation. By using this relation andthe abundant solutions of the cubic NKG equation, many explicit and exact travelling wave solutions of three systemsof coupled PDEs, which contain solitary wave solutions, trigonometric function solutions, Jacobian elliptic functionsolutions, and rational solutions, are obtained.

Exact solutions of multi-term fractional difusion-wave equations with Robin type boundary conditions

General exact solutions in terms of wavelet expansion are obtained for multi- term time-fractional diffusion-wave equations with Robin type boundary conditions. By proposing a new method of integral transform for solving boundary value problems, such fractional partial differential equations are converted into time-fractional ordinary differ- ential equations, which are further reduced to algebraic equations by using the Laplace transform. Then, with a wavelet-based exact formula of Laplace inversion, the resulting exact solutions in the Laplace transform domain are reversed to the time-space domain. Three examples of wave-diffusion problems are given to validate the proposed analytical method.

EXACT TRAVELING WAVE SOLUTIONS OF MODIFIED ZAKHAROV EQUATIONS FOR PLASMAS WITH A QUANTUM CORRECTION

In this article, the authors study the exact traveling wave solutions of modified Zakharov equations for plasmas with a quantum correction by hyperbolic tangent function expansion method, hyperbolic secant expansion method, and Jacobi elliptic function ex- pansion method. They obtain more exact traveling wave solutions including trigonometric function solutions, rational function solutions, and more generally solitary waves, which are called classical bright soliton, W-shaped soliton, and M-shaped soliton.

A new method of new exact solutions and solitary wave-like solutions for the generalized variable coefficients Kadomtsev-Petviashvili equation

Using the solution of general Korteweg-de Vries （KdV） equation, the solutions of the generalized variable coefficient Kadomtsev-Petviashvili （KP） equation are constructed, and then its new solitary wave-like solution and Jacobi elliptic function solution are obtained.

An Algebraic Method for Constructing Exact Solutions to Difference-Differential Equations

In this paper, we present a method to solve difference differential equation（s）. As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation（s）.