Projective Riccati Equation Method Research Articles

Non-travelling wave solutions to a (3+1)-dimensional potential-YTSF equation and a simplified model for reacting mixtures

In the paper, the generalized projective Riccati equation method is extended to construct some non-travelling wave solutions to a (3 + 1)-dimensional potential-YTSF equation and a simplified model for reacting mixtures. When some arbitrary functions included in these solutions are taken as some special functions, these solutions possess abundant structures.

Symbolic computation and solitons of the nonlinear Schrödinger equation in inhomogeneous optical fiber media

Abstract In this paper, the inhomogeneous nonlinear Schrodinger equation with the loss/gain and the frequency chirping is investigated. With the help of symbolic computation, three families of exact analytical solutions are presented by employing the extended projective Riccati equation method. From our results, many previous known results of nonlinear Schrodinger equation obtained by some authors can be recovered by means of some suitable selections of the arbitrary functions and arbitrary constants. Of optical and physical interests, soliton propagation and soliton interaction are discussed and simulated by computer, which include snake-soliton propagation and snake-solitons interaction, boomerang-like soliton propagation and boomerang-like solitons interaction, dispersion managed (DM) bright (dark) soliton propagation and DM solitons interaction.

Soliton-like Solutions to Wick-type Stochastic mKdV Equation**The projectsupported by the State Key Basic Research Development Program of China underGrant No. 2004CB318000 and National NaturalScience Foundation of China under Grant No. 10072013

In this paper, the Wick-type stochastic mKdV equation is researched. ManyWick-type stochastic soliton-like solutions are given via Hermitetransformation and further generalized projective Riccati equation method.

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.

EXACT SOLITON SOLUTIONS FOR THE HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION

Based on the complex envelope ansatz method, the projective Riccati equation method and q-deformed hyperbolic functions, a method is developed for constructing a series of exact analytical solutions for higher-order nonlinear Schrödinger (HNLS) equation, which describes propagation of femtosecond light pulse in optical fiber under certain parametric conditions. With the help of symbolic computation, six families of new solitary wave solutions are obtained. The solitary wave solutions obtained by Li et al.18 are special cases of our solutions. The novel soliton solutions can describe W-shaped, bright and dark soliton properties in the same expression and their amplitude may approach nonzero when the time variable approaches infinity. Furthermore, the soliton propagation and solitons interaction scenario are discussed and simulated by computer.

New expansion algorithm of three Riccati equations and its applications in nonlinear mathematical physics equations

Based on the study of tanh function method and the coupled projective Riccati equation method, we propose a new algorithm to search for explicit exact solutions of nonlinear evolution equations. We use the higher-order Schrodinger equation and mKdV equation to illustrate this algorithm. As a result, more new solutions are obtained, which include new solitary solutions, periodic solutions and singular solutions. Some new solutions are illustrated in figures.

Relationship Between Soliton-like Solutions and Soliton Solutionsto a Class of Nonlinear Partial Differential Equations

By using the generally projective Riccati equation method, a series ofdoubly periodic solutions (Jacobi elliptic function solution) for a classof nonlinear partial differential equations are obtained in aunified way. When the module m→1, these solutions exactlydegenerate to the soliton solutions of the equations. Then wereveal the relationship between the soliton-like solutionsobtained by other authors and these soliton solutions of theequations.

New exact solutions to the generalized coupled Hirota–Satsuma Kdv system☆

Abstract In this paper, the projective Riccati equations method is improved. The improved method can be applied to find more and new exact solutions to the nonlinear evolution equations with the aid of symbolic computation system like Maple. Some of the solutions are found for the first time.

Exact travelling wave solutions of nonlinear evolution equations in [formula omitted] and [formula omitted] dimensions

Based upon the symbolic computation system Maple, the projective Riccati equation method is improved to seek for travelling wave solutions in ( 1 + 1 ) and ( 2 + 1 ) -dimensional nonlinear evolution equations by introducing two new elementary bell-shaped and kink-shaped functions. The validity and reliability of the method is tested by applying it to the ( 2 + 1 ) -dimensional generalized of the shallow water wave equation, potential KdV equation and Burgers–KdV equation.

Exact solutions for the higher-order nonlinear Schördinger equation in nonlinear optical fibres

First, by using the generally projective Riccati equation method, many kinds of exact solutions for the higher-order nonlinear Schördinger equation in nonlinear optical fibres are obtained in a unified way. Then, some relations among these solutions are revealed.

New soliton-like and periodic-like solutions for the KdV equation

In this paper based on a system of Riccati equations, a newly generally projective Riccati equation expansion method is presented. It is direct and more powerful than the tanh-function method, generally projective Riccati equation method and can be used to construct more new exact solutions of nonlinear differential equations in mathematical physics. The well-known KdV equation is chosen to illustrate the algorithm such that more families of new explicit solutions are obtained, which contain soliton-like and periodic-like solutions. This algorithm can also be applied to many other nonlinear differential equations.

Extended projective Riccati equations method and its application

The projective Riccati equations method is extended and applied to find new exac t solutions to the Whitham-Broer-Kaup(WBK) equation with the aid of symbolic com putation system Maple.

The stochastic soliton-like solutions of stochastic mKdV equations

In this paper, the generally projective Riccati equations method is improved by means of a generalized transformation. The improved method can be applied to find not only some exact travelling wave solutions but also some soliton-like solutions with the aid of symbolic computation system — Maple. We choose Wick-type stochastic mKdV equations to illustrate the method. As a result, some stochastic soliton-like solutions are obtained.

Exact Soliton Solutions To An Averaged Dispersion-Managed Fiber System Equation

By introducing a set of ordinary differential equations which possess q-deformed hyperbolic function solutions, and a new ansatz, a method is developed for constructing a series of exact analytical solutions of some nonlinear evolution equations. The proposed method is more powerful than various tanh methods, the secq-tanhq-method, generalized hyperbolic-function method, generalized Riccati equation expansion method, generalized projective Riccati equations method and other sophisticated methods. As an application of the method, an averaged dispersion-managed (DM) fiber system equation, which governs the dynamics of the core of the DM soliton, is chosen to illustrate the method. With the help of symbolic computation, rich new soliton solutions are obtained. From these solutions, some previously known solutions obtained by some authors can be recovered by means of some suitable choices of the arbitrary functions and arbitrary constants. Further, the soliton propagation and solitons interaction scenario are discussed and simulated by computer.

New Soliton-like Solutions to Variable Coefficients MKdV Equation**The project supported by the State Key Basic Research Development Program of China under Grant No. 1998030600 and National Natural Science Foundation of China under Grant No. 10072013

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 families of non-travelling wave solutions to the (2+1)-dimensional modified dispersive water-wave system

In this paper, we introduce a further generalized projective Riccati equation method and apply it to solve the (2+1)-dimensional modified dispersive water-wave system. Many new types of non-travelling wave solutions are obtained for this system.

Exact solutions for the higher-order nonlinear Schördinger equation in nonlinear optical fibres

Abstract First, by using the generally projective Riccati equation method, many kinds of exact solutions for the higher-order nonlinear Schordinger equation in nonlinear optical fibres are obtained in a unified way. Then, some relations among these solutions are revealed.

Variable-coefficient projective Riccati equation method and its application to a new (2+1)-dimensional simplified generalized Broer–Kaup system

Abstract In this paper, based on a new intermediate transformation, a variable-coefficient projective Riccati equation method is proposed. Being concise and straightforward, it is applied to a new (2 + 1)-dimensional simplified generalized Broer–Kaup (SGBK) system. As a result, several new families of exact soliton-like solutions are obtained, beyond the travelling wave. When imposing some condition on them, the new exact solitary wave solutions of the (2 + 1)-dimensional SGBK system are given. The method can be applied to other nonlinear evolution equations in mathematical physics.

New Exact Travelling Wave Solutions to Hirota Equation and (1+1)-Dimensional Dispersive Long Wave Equation**The project supported by National Natural Science Foundation of China under Grant No. 10072013 and the State Key Basic Research Development Program under Grant No. G1998030600

Based on the computerized symbolic Maple, we study two important nonlinear evolution equations, i.e., the Hirota equation and the (1+1)-dimensional dispersive long wave equation by use of a direct and unified algebraic method named the general projective Riccati equation method to find more exact solutions to nonlinear differential equations. The method is more powerful than most of the existing tanh method. New and more general form solutions are obtained. The properties of the new formal solitary wave solutions are shown by some figures.

CONSTRUCTING FAMILIES OF EXACT SOLUTIONS TO A (2+1)-DIMENSIONAL CUBIC NONLINEAR SCHRÖDINGER EQUATION

The projective Riccati equations method is extended to find some novel exact solutions of a (2+1)-dimensional cubic nonlinear Schrödinger (NLS) equation. Applying the extended method and symbolic computation, six families of exact analytical solutions for this NLS equation are reported, which include some new and more general exact soliton-like solutions, trigonometric function forms solutions and rational forms solutions.