Exact solutions of nonlinear Schrodinger’s equation with dual power-law nonlinearity by extended trial equation method

In this paper, we acquire the soliton solutions of the nonlinear Schrodinger’s equation with dual power-law nonlinearity. Primarily, we use the extended trial equation method to find exact solutions of this equation. Then, we attain some exact solutions including soliton solutions, rational and elliptic function solutions of this equation using the extended trial equation method.     

Some exact solutions of generalized Zakharov system

In this paper, we seek exact solutions of generalized Zakharov system. We use extended trial equation method to obtain exact solutions of this system. Consequently, we obtain some exact solutions including soliton solutions, rational, Jacobi elliptic and hyperbolic function solutions of this system by using extended trial equation method.     

试探方程法及其在非线性发展方程中的应用

提出了一种比较系统的求解非线性发展方程精确解的新方法, 即试探方程法. 以一个带5阶 导数项的非线性发展方程为例, 利用试探方程法化成初等积分形式，再利用三阶多项式的完 全判别系统求解，由此求得的精确解包括有理函数型解, 孤波解, 三角函数型周期解, 多项 式型Jacobi椭圆函数周期解和分式型Jacobi椭圆函数周期解 关键词： 试探方程法 非线性发展方程 孤波解 Jacobi椭圆函数 周期解     

A New Trial Equation Method and Its Applications

As an improved version of trial equation method, a new trial equation method is proposed. Using this method, abundant new exact traveling wave solutions to a high-order KdV-type equation are obtained.     

New soliton solutions for Sasa–Satsuma equation

In this work, we survey exact solutions of Sasa–Satsuma equation (SSE). We utilize extended trial equation method (ETEM) and generalized Kudryashov method to acquire exact solutions of SSE. First of all, we gain some exact solutions such as soliton solutions, rational, Jacobi elliptic, and hyperbolic function solutions of SSE by means of ETEM. Furthermore, we procure dark soliton solution of this equation by the help of generalized Kudryashov method. Lastly, for certain parameter values, we draw two- and three-dimensional graphics of imaginary and real values of some exact solutions that we achieved using these methods.     

Dynamics of combined soliton solutions of unstable nonlinear Schrodinger equation with new version of the trial equation method

In this article, a new version of the trial equation method is suggested. With this method, it is possible to find the new exact solutions of the nonlinear partial differential equations. The developed method is applied to unstable nonlinear Schrödinger equation. New exact solutions are expressed with Jacobi elliptic function solutions, 1-soliton solutions and rational function solutions. When the obtained results are examined, we can say the unstable nonlinear Schrödinger equation shows different dynamic behaviors. In addition, the physical behaviors of these new exact solution are given with two and three dimensional graphs.     

New exact solutions of the generalized Zakharov–Kuznetsov modified equal-width equation

In this paper, new exact solutions, including soliton, rational and elliptic integral function solutions, for the generalized Zakharov–Kuznetsov modified equal-width equation are obtained using a new approach called the extended trial equation method. In this discussion, a new version of the trial equation method for the generalized nonlinear partial differential equations is offered.     

Application of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics

In this paper some exact solutions including soliton solutions for the KdV equation with dual power law nonlinearity and the K(m, n) equation with generalized evolution are obtained using the trial equation method. Also a more general trial equation method is proposed.     

Exact wave functions for concentric two-electron systems

We show that the exact solution of the Schrödinger equation for two electrons confined to two distinct concentric rings or spheres can be found in closed form for particular values of the ring or sphere radii. In the case of rings, we report exact polynomial and irrational solutions. In the case of spheres, we report exact polynomial solutions for the ground and excited states of S symmetry.     

Exact solutions to complex Ginzburg–Landau equation

In this paper, the trial equation method and the complete discrimination system for polynomial method are applied to retrieve the exact travelling wave solutions of complex Ginzburg–Landau equation. Both the Kerr and power laws of nonlinearity are considered. All the possible exact travelling wave solutions consisting of the rational function-type solutions, solitary wave solutions, triangle function-type periodic solutions and Jacobian elliptic functions solutions are obtained, and some of them are new solutions. In addition, concrete examples are presented to ensure the existence of obtained solutions. Moreover, four types of representative solutions are depicted to present the nature of the obtained solutions.     

Volterra差分微分方程和KdV差分微分方程新的精确解

辅助方程法和试探函数法为基础,给出函数变换与辅助方程相结合的一种方法,借助符号计算系统Mathematica构造了Volterra差分微分方程和KdV差分微分方程新的精确孤立波解和三角函数解.该方法也适合求解其他非线性差分微分方程的精确解. 关键词： 辅助方程 函数变换 非线性差分微分方程 孤立波解     

New Exact Envelope Traveling Wave Solutions of High-Order Dispersive Cubic-Quintic Nonlinear SchrSdinger Equation

Using trial equation method, abundant exact envelope traveling wave solutions of high-order dispersive cubic-quintic nonlinear Schr6dinger equation, which include envelope soliton solutions, triangular function envelope solutions, and Jacobian elliptic function envelope solutions, are obtained. To our knowledge, all of these results are new. In particular, our proposed method is very simple and can be applied to a lot of similar equations.     

Traveling wave solutions of conformable time-fractional Zakharov–Kuznetsov and Zoomeron equations

To model physical phenomena more accurately, fractional order differential equations have been widely used. Investigating exact solutions of the fractional differential equations have become more important because of the applications in applied mathematics, mathematical physics, and other areas. In this work, by means of the trial solution method and complete discrimination system, exact traveling wave solutions of the conformable time-fractional Zakharov–Kuznetsov equation and conformable time-fractional Zoomeron equation have been obtained and also solutions have been illustrated. Finding exact solutions of these equations that are encountered in plasma physics, nonlinear optics, fluid mechanics, and laser physics can help to understand nature of the complex phenomena.     

Trial Equation Method to Nonlinear Evolution Equations with Rank Inhomogeneous:Mathematical Discussions and Its Applications

A trial equation method to nonlinear evolution equation with rank inhomogeneous is given. As applications, the exact traveling wave solutions to some higher-order nonlinear equations such as generalized Boussinesq equation, generalized Pochhammer-Chree equation, KdV-Burgers equation, and KS equation and so on, are obtained. Among these, some results are new. The proposed method is based on the idea of reduction of the order of ODE. Some mathematical details of the proposed method are discussed.     

Trial Equation Method Based on Symmetry and Applications to Nonlinear Equations Arising in Mathematical Physics

To find exact traveling wave solutions to nonlinear evolution equations, we propose a method combining symmetry properties with trial polynomial solution to nonlinear ordinary differential equations. By the method, we obtain some exact traveling wave solutions to the Burgers-KdV equations and a kind of reaction-diffusion equations with high order nonlinear terms. As a result, we prove that the Burgers-KdV equation does not have the real solution in the form a ₀+a ₁tan ξ+a ₂tan ² ξ, which indicates that some types of the solutions to the Burgers-KdV equation are very limited, that is, there exists no new solution to the Burgers-KdV equation if the degree of the corresponding polynomial increases. For the second equation, we obtain some new solutions. In particular, some interesting structures in those solutions maybe imply some physical meanings. Finally, we discuss some classifications of the reaction-diffusion equations which can be solved by trial equation method.     

A new trial equation method for finding exact chirped soliton solutions of the quintic derivative nonlinear Schrödinger equation with variable coefficients

In this work, we propose an efficient generalization of the trial equation method introduced recently by Liu [Appl. Math. Comput. 217 (2011) 5866] to construct exact chirped traveling wave solutions of complex differential equations with variable coefficients. The effectiveness of the proposed method has been tested by applying it successfully to the quintic derivative nonlinear Schrödinger equation with variable coefficients. As a result, a class of chirped soliton-like solutions including bright and kink solitons is derived for the first time. Compared with previous work of Liu in which unchirped solutions were given, we obtain exact chirped solutions which have nontrivial phase that varies as a function of the wave intensity. These localized structures characteristically exist due to a balance among the group-velocity dispersion, self-steepening and competing cubic-quintic nonlinearity. Parametric conditions for the existence of envelope solutions with nonlinear chirp are also presented. It is shown that the chirping can be effectively controlled through the variable parameters of group-velocity dispersion and self-steepening.     

Optical solitons in $$$$-dimensions under anti-cubic law of nonlinearity by analytical methods

The present study emphasis to look for new closed form exact solitary wave solutions for the \((n+1)\)-dimensional nonlinear Schrödinger equation using the extended trial equation method (ETEM) and the \(\exp (-\Omega (\eta ))\)-expansion method (EEM) with the help of symbolic computation package maple. As a consequence, the ETEM and EEM are successfully employed and acquired some new exact solitary wave solutions in terms of exponential based functions, hyperbolic based functions, trigonometric based functions and rational based functions. All solutions have been verified back into its corresponding equation with the aid of maple package program. Finally, we believe that the executed method is robust and efficient than other methods and the obtained solutions in this paper can help us to understand the variation of solitary waves in the field of nonlinear optic.     

The auxiliary elliptic-like equation and the exp-function method

The auxiliary equation method is very useful for finding the exact solutions of the nonlinear evolution equations. In this paper, a new idea of finding the exact solutions of the nonlinear evolution equations is introduced. The idea is that the exact solutions of the auxiliary elliptic-like equation are derived using exp-function method, and then the exact solutions of the nonlinear evolution equations are derived with the aid of auxiliary elliptic-like equation. As examples, the RKL models, the high-order nonlinear Schrödinger equation, the Hamilton amplitude equation, the generalized Hirota-Satsuma coupled KdV system and the generalized ZK-BBM equation are investigated and the exact solutions are presented using this method.     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.     

New travelling wave solutions for combined KdV-mKdV equation and （2＋1）-dimensional Broer- Kaup- Kupershmidt system

Some new exact solutions of an auxiliary ordinary differential equation are obtained, which were neglected by Sirendaoreji {\it et al in their auxiliary equation method. By using this method and these new solutions the combined Korteweg--de Vries (KdV) and modified KdV (mKdV) equation and (2+1)-dimensional Broer--Kaup--Kupershmidt system are investigated and abundant exact travelling wave solutions are obtained that include new solitary wave solutions and triangular periodic wave solutions.