Series Of New Exact Solutions Research Articles

On the exact solutions for a type of nonlinear Schrödinger equations with a harmonic potential

The nonlinear Schrödinger equation with harmonic potential (NLSE) plays an important role in quantum mechanics, so the exact solutions of this equation is studied in this paper. The NLSE is transformed into the classical nonlinear Schrödinger equation by a new class of traveling wave transformation. Next, the problem of exact solutions is changed into the solutions of ordinary differential equation (ODE) by the method of undetermined function. Then, through low-order sub-ODE method and hyperbolic function method, we get two class of solutions of the ODEs. Finally, a series of new exact solutions of the NLSE are obtained. Meanwhile the related numerical simulations are presented.

Traveling magnetic wave motion in ferrites: Impact of inhomogeneous exchange effects

Abstract In this article, we derive new traveling wave solution to a nonlinear evolution equation, describing propagation of short wave in inhomogeneous ferrites. Applying Jacobi elliptic function, we derive as series of new exact solutions to the system of our interest which are either periodic or localized solutions. We point out the influence of the inhomogeneous exchange effects on the dynamics of traveling waves obtained. It appears that the soliton responsible of localization is deformed by the presence of inhomogeneities in particular its structure. We discuss some physical implications of these results.

New exact traveling wave solutions of the Tzitzéica type equations using a novel exponential rational function method

The Tzitzéica type equations arising in nonlinear optics, including the Tzitzéica, Dodd–Bullough–Mikhailov, and Tzitzéica–Dodd–Bullough equations are solved analytically, in the present paper. A solution technique in the area of computational mathematics called the novel exponential rational function method is formally proposed for this purpose. As a consequence, a series of new exact solutions for the Tzitzéica type equations are obtained, proving the super performance of novel exponential rational function method.

RLW-Burgers方程的势对称及其精确解

通过计算RLW-Burgers方程的势对称扩大了其古典对称，并获得了RLW-Burgers方程的一系列新的精确解。首先，基于微分特征列集算法确定了RLW-Burgers方程的古典对称和势对称。其次，利用推广的Tanh函数法构造了RLW-Burgers方程的不变解，这些解分别以含任意参数的双曲函数、三角函数和有理函数表示。最后，分别选择一个势对称和古典对称的Lie变换群，将其作用于RLW-Burgers方程的不变解上获得了新的精确解，重要的是这些解都不能由方程的古典对称得到。 We expanded the classical symmetries of RLW-Burgers equation by calculating the potential symmetries, and we obtained a series of new exact solutions of RLW-Burgers equation. Firstly, we determined the classical symmetries and potential symmetries of RLW-Burgers equation based on differential characteristic set algorithm. Secondly, we constructed the invariant solutions of Burgers equation by using the extended Tanh function method, and these solutions with arbitrary pa-rameters are expressed by the hyperbolic functions, the trigonometric functions and the rational functions, respectively. Finally, new exact solutions for RLW-Burgers equation are obtained by acting Lie transformation group of potential symmetry and the classical symmetry on the invariant solutions. It is important that these solutions can not be obtained from classical symmetries of Burgers equation.

Solitary Wave and Periodic Wave Solutions and Chaotic Behaviors in a (2+1)-Dimensional Nonlinear System

With the help of the symbolic computation system Maple and an improved mapping method and a variable separation method, a series of new exact solutions (including solitary wave solutions and periodic wave solutions) to the (2+1)-dimensional general Nizhnik-Novikov-Veselov (GNNV) system is derived. Based on the derived solitary wave solution, we obtain some chaotic patterns.

EXACT SOLUTIONS OF THE EINSTEIN–MAXWELL EQUATIONS IN CHARGED PERFECT FLUID SPHERES FOR THE GENERALIZED TOLMAN–OPPENHEIMER–VOLKOFF EQUATIONS

Exact solutions of the Einstein–Maxwell equations for spherically symmetric charged perfect fluid have been broadly studied so far. However, the cases with a nonzero cosmological constant are seldom focused. In the present paper, the Tolman–Oppenheimer–Volkoff (TOV) equations have been generalized from the neutral case of hydrostatic equilibrium to the charged case of hydroelectrostatic equilibrium, and base on it, for the first time we find a series of new exact solutions of Einstein–Maxwell's equations with a nonzero cosmological constant for static charged perfect fluid spheres. Moreover, two special TOV equations and two classical constant density interior solutions are also given.

A series of new exact solutions of nonlinear evolution equations

AbstractWith the aid of computer symbolic computation system Maple, the generalized auxiliary equation method is first applied to two nonlinear evolution equations, namely, the nonlinear elastic rod equation and (2 + 1)‐dimensional Boiti‐Leon‐Pempinelli equation. As a results, some new types of exact traveling wave solutions are obtained which include bell and kink profile solitary wave solutions, and triangular periodic wave solutions and singular solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

New soliton and periodic solutions of (1 + 2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity

With the aid of symbolic computation, the new generalized algebraic method is extended to the (1+2)-dimensional nonlinear Schrödinger equation (NLSE) with dual-power law nonlinearity for constructing a series of new exact solutions. Because of the dual-power law nonlinearity, the equation cannot be directly dealt with by the method and require some kinds of techniques. By means of two proper transformations, we reduce the NLSE to an ordinary differential equation that is easy to solve and find a rich variety of new exact solutions for the equation, which include soliton solutions, combined soliton solutions, triangular periodic solutions and rational function solutions. Numerical simulations are given for a solitary wave solution to illustrate the time evolution of the solitary creation. Finally, conditional stability of the solution in Lyapunov’s sense is discussed.

Solutions in elementary functions obtained for 3D and axisymmetric intense electron flows

Three-dimensional nonrelativistic electron flows in the presence of an arbitrarily oriented uniform magnetic field are considered. The flows are formed by elliptic or hyperbolic current tubes and have a spatial axis. For such flows, a series of new exact solutions in elementary functions are constructed. In the case of nonmonoenergetic beams, a new type of singularities corresponding to the zero-velocity injection is discovered.

New exact solutions to the generalized Burgers–Huxley equation

Based on He’s Exp-function method, a series of new exact solutions of the generalized Burger–Huxley equation have been obtained. It is shown that the Exp-function method is straightforward and concise, and its applications are promising.

Solitary Wave and Doubly Periodic Wave Solutions to Three-Dimensional Nizhnik–Novikov–Veselov Equation

The generalized transformation method is utilized to solve three-dimensional Nizhnik–Novikov–Veselov equation and construct a series of new exact solutions including kink-shaped and bell-shaped soliton solutions, trigonometric function solutions, and Jacobi elliptic doubly periodic solutions. Among them, the Jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh methods and Jacobi function method, the method we used here gives more general exact solutions without much extra effort.

Extended proposed algorithm with symbolic computation to construct exact solutions for some nonlinear differential equations

In this paper an algebraic method is devised to construct uniformly a series of complete exact solutions for some nonlinear differential equations. For illustration, we apply the extended proposed method to revisit the nonlinear coupled physical system, the nonlinear variant Buossinesq equations (I), (II) and Hirota–Satsuma coupled KdV nonlinear equations. We construct successfully a series of new exact solutions including the soliton solutions and elliptic doubly periodic solutions with the aid of computerized symbolic computation.

New exact solutions to the Fitzhugh–Nagumo equation

By the first integral method, a series of new exact solutions of the Fitzhugh–Nagumo equation have been obtained. It is shown that this method is one of the most effective approaches to obtain the exact solutions of the nonlinear evolution equations, especially for nonintegrable models.

A modified extended method to find a series of exact solutions for a system of complex coupled KdV equations

In this paper an algebraic method is devised to uniformly construct a series of complete new exact solutions for general nonlinear equations. For illustration, we apply the modified proposed method to revisit a complex coupled KdV system and successfully construct a series of new exact solutions including the soliton solutions and elliptic doubly periodic solutions.

A series of new exact solutions to the (2+1)-dimensional Broer-Kau-Kupershmidt equation

With the aid of computer symbolic systems Maple, and by an improved algebraic method which is used to construct a series of more general exact solutions to nonlinear differential equation or coupled equations, we have solved the (2+1)-dimensional Broer-Kau-Kupershmidt equation, and obtained a number of new exact solutions including polynomial solutions,exponential solutions, rational solutions, triangular periodic solutions,hyperbolic solutions, and Jacobi and Weierstrass doubly periodic wave solutions.

A series of new exact solutions for a complex coupled KdV system

In this paper an algebraic method is devised to uniformly construct a series of exact solutions for general nonlinear equations. Compared with the existing tanh methods and Jacobi function method, the proposed method gives more general exact solutions without much extra effort. More importantly, the method provides a guideline for the classification of the solutions based on the given parameters. For illustration, we apply the proposed method to revisit a complex coupled KdV system and successfully construct a series of new exact solutions including the soliton solutions and elliptic doubly periodic solutions.