Projective Riccati Equation Research Articles

Evolution Properties of the y-Periodic Solitons for the (2+1)-Dimensional Boiti-Leon-Pempinelli System

With the help of the symbolic computation system Maple and an expanded projective Riccati equation approach, we obtain some new rational explicit solutions with three arbitrary functions for the (2+1)-dimensional Boiti-Leon-Pempinelli system, including Weierstrass function solutions, solitary wave solutions and trigonometric function solutions. From these, several y-periodic soliton localized excitations are constructed and some evolution properties of these novel y-periodic localized structures are discussed.

Double-Parameter Solutions of Projective Riccati Equationsand Their Applications

The purpose of the present paper istwofold. First, the projective Riccati equations (PREs for short)are resolved by means of a linearized theorem, which was known inthe literature. Based on the signs and values of coefficients ofPREs, the solutions with two arbitrary parameters of PREs can beexpressed by the hyperbolic functions, the trigonometric functions,and the rational functions respectively, at the same time therelation between the components of each solution to PREs is alsoimplemented. Second, more new travelling wave solutions for somenonlinear PDEs, such as the Burgers equation, the mKdV equation,the NLS+ equation, new Hamilton amplitude equation, and soon, are obtained by using Sub-ODE method, in which PREs are takenas the Sub-ODEs. The key idea of this method is that thetravelling wave solutions of nonlinear PDE can be expressed by apolynomial in two variables, which are the components of eachsolution to PREs, provided that the homogeneous balance betweenthe higher order derivatives and nonlinear terms in the equationis considered.

A modified variable-coefficient projective Riccati equation method and its application to (2 + 1)-dimensional simplified generalized Broer–Kaup system

A modified variable-coefficient projective Riccati equation method is proposed and applied to a (2 + 1)-dimensional simplified and generalized Broer–Kaup system. It is shown that the method presented by Huang and Zhang [Huang DJ, Zhang HQ. Chaos, Solitons & Fractals 2005; 23:601] is a special case of our method. The results obtained in the paper include many new formal solutions besides the all solutions found by Huang and Zhang.

Some Exact Analytical Solutions to the Inhomogeneous Higher-Order Nonlinear Schrödinger Equation Using Symbolic Computation

In this paper, the generalized projective Riccati equation method is extended to investigate the inhomogeneous higher-order nonlinear Schrödinger (IHNLS) equation including not only the group velocity dispersion and self-phase-modulation, but also various higher-order effects, such as the third-order dispersion, self-steepening and self-frequency shift. With the help of symbolic computation, a broad class of analytical solutions of the IHNLS equation is presented, which include bright-like solitary wave solutions, dark-like solitary wave solutions, W-shaped solitary wave solutions, combined bright-like and dark-like solitary wave solutions, and dispersion-managed solitary wave solutions. From our results, many previously known results about the IHNLS equation can be recovered by means of some suitable selections of the arbitrary functions and arbitrary constants. Furthermore, from the soliton management concept, the main soliton-like character of the exact analytical solutions is discussed and simulated by computer under different parameters conditions.

Novel interactions between solitons of the (2 + 1)-dimensional dispersive long wave equation

In this paper, first, the general projective Riccati equation method is applied to derive variable separation solutions of the (2 + 1)-dimensional dispersive long wave equation. By further studying, we find that these variable separation solutions obtained by PREM, which seem independent, actually depend on each other. Based on variable separation solution and by selecting appropriate multivalued functions, novel interactions among special bell-like semi-foldon, special peakon-like semi-foldon and foldon are investigated. Furthermore, the explicit phase shifts for all the local excitations offered by the common formula have been given, and are applied to these novel interactions in detail.

Novel interactions between semi-foldons of the (2 + 1)-dimensional Boiti–Leon–Pempinelli equation

In this paper, firstly, the general projective Riccati equation method (PREM) is applied to derive variable separation solutions of the (2 + 1)-dimensional Boiti–Leon–Pempinelli (BLP) equation. By further studying, we find that these variable separation solutions, which seem independent, actually depend on each other. Based on the variable separation solution and by selecting appropriate multi-valued functions, novel interactions between a special bell-like semi-foldon and a special peakon-like semi-foldon are investigated.

Variable Separation Solutions for the (2+1)-Dimensional Breaking Soliton Equation

With the variable separation approach and based on the general reduction theory, we successfully obtain the variable separation solutions for the (2+1)-dimensional breaking soliton equation using of the projective Riccati equation. Based on one of the variable separation solutions and by selecting appropriate functions, a new type of interaction between the multi-valued and the single-valued solitons, that is a compacton-like semi-foldon and a 4-compacton, is investigated.

Solitons with Fusion and Fission Properties in the (2+1)-Dimensional Modified Dispersive Water-Wave System

In this paper, by means of the general projective Riccati equation method (PREM), the variable separation solutions of the (2+1)-dimensional modified dispersive water-wave system are obtained. By further studying, we find that these variable separation solutions, which seem independent, actually depend on each other. Based on the special variable separation solution and choosing suitable functions p and q, soliton fusion and fission phenomena among peakons, compactons, dromions and semifoldons are firstly investigated. - PACS numbers: 05.45.Yv, 02.30.Jr, 02.03Ik

Solitons, chaos and fractals in the (2+1)-dimensional dispersive long wave equation

Abstract For a higher-dimensional integrable nonlinear dynamical system, there are abundant coherent soliton excitations. With the aid of a projective Riccati equation approach, the paper obtains several types of exact solutions to the (2 + 1)-dimensional dispersive long wave (DLW) equation which include multiple soliton solution, periodic soliton solution and Weierstrass function solution. Subsequently, several multisolitons are derived and some novel features are revealed by introducing lower-dimensional patterns.

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.

Quasi-periodic and non-periodic waves in the (2+1)-dimensional generalized Broer–Kaup system

Abstract For a higher-dimensional integrable nonlinear dynamical system, there are abundant coherent soliton excitations. With the aid of an improved projective Riccati equation approach, the paper obtains several types of exact solutions to the (2 + 1)-dimensional generalized Broer–Kaup system, including multiple soliton solutions, periodic soliton solutions and Weierstrass function solutions. From which, new quasi-periodic and non-periodic excitations for the system are studied by means of the Jacobi elliptic functions with the space–time-dependent modulus.

The subsidiary elliptic-like equation and the exact solutions of the higher-order nonlinear Schrödinger equation

Abstract By using the homogeneous balance principle, the exact solutions of the subsidiary elliptic-like equation are derived with the aid of the coupled projective Riccati equations. The higher-order nonlinear Schrodinger equation describing the propagation of ultrashort pulses in nonlinear optical fibers is researched and its various exact solutions are derived by means of the simple transformation and the subsidiary elliptic-like equation mentioned above.

Some Coherent Excitations for the (2+1)-Dimensional Generalized Broer-Kaup System

Using a projective Riccati equation, several types of solutions of the (2+1)-dimensional generalized Broer-Kaup system are obtained, including multiple soliton solutions, periodic soliton solutions and Weierstrass function solutions. From these, two sets of wave packets are expressed as rational functions of elliptic functions. Especially, peculiar wave patterns that are localized in one direction but periodic in the other direction arise by taking the long wave length limit in one spatial variable. Also exponentially localized wave patterns, which differ from the known dromions, are obtained by taking the long wave length limit in both spatial variables. The interactions of two dromions with inelastic and elastic behaviors are presented

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.

A symbolic computational method for constructing exact solutions to difference-differential equations

In this paper, we extended tanh method to solve difference-differential equations and pure difference equations with the projective Riccati equation. As an example, we applied this method to a (2 + 1)-dimensional Toda lattice equation. As a result, many exact solutions are obtained with the help of symbolic system 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 equations.

A simple method for constructing elliptic function solutions to the nonlinear evolution equations and its applications

Many travelling wave solutions of nonlinear evolution differential equations can be written as a polynomial of two elementary functions which satisfy a projective Riccati equation. From that property, we deduce a method for building these solutions by establishing the relation between the solutions of the general elliptic equation and the projective Riccati equation. The method is effective for both type Ⅰ and type Ⅱ equtions. At the same time, we also answer the question of how to construct the elliptic function solutions in fraction form to the nonlinear evolution equations.

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.