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

2004 ◽  
Vol 15 (05) ◽  
pp. 741-751 ◽  
Author(s):  
BIAO LI ◽  
HONGQING ZHANG
Keyword(s):  
Exact Solutions ◽  
Symbolic Computation ◽  
Analytical Solutions ◽  
Schrodinger Equation ◽  
Riccati Equations ◽  
Trigonometric Function ◽  
Nls Equation ◽  
Exact Analytical Solutions ◽  
Nonlinear Schrödinger ◽  
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.

A Generalized Sub-Equation Expansion Method and Some Analytical Solutions to the Inhomogeneous Higher-Order Nonlinear Schrödinger Equation

2008 ◽  
Vol 63 (12) ◽  
pp. 763-777 ◽  
Author(s):  
Biao Li ◽  
Yong Chen ◽  
Yu-Qi Li
Keyword(s):  
Solitary Wave ◽  
Analytical Solutions ◽  
Elliptic Function ◽  
Schrodinger Equation ◽  
Expansion Method ◽  
High Order ◽  
Solitary Wave Solutions ◽  
Exact Analytical Solutions ◽  
Nonlinear Schrödinger ◽  
Wave Solutions

On the basis of symbolic computation a generalized sub-equation expansion method is presented for constructing some exact analytical solutions of nonlinear partial differential equations. To illustrate the validity of the method, we investigate the exact analytical solutions of the inhomogeneous high-order nonlinear Schrödinger equation (IHNLSE) including not only the group velocity dispersion, self-phase-modulation, but also various high-order effects, such as the third-order dispersion, self-steepening and self-frequency shift. As a result, a broad class of exact analytical solutions of the IHNLSE are obtained. From our results, many previous solutions of some nonlinear Schrödinger-type equations can be recovered by means of suitable selections of the arbitrary functions and arbitrary constants. With the aid of computer simulation, the abundant structure of bright and dark solitary wave solutions, combined-type solitary wave solutions, dispersion-managed solitary wave solutions, Jacobi elliptic function solutions and Weierstrass elliptic function solutions are shown by some figures.

Exact analytical solutions of higher-order nonlinear Schrödinger equation

Optik ◽  
2017 ◽  
Vol 131 ◽  
pp. 438-445 ◽  
Author(s):  
Hongcheng Wang ◽  
Jianchu Liang ◽  
Guihua Chen ◽  
Ling Zhou ◽  
Dongxiong Ling ◽  
...  
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Analytical Solutions ◽  
Schrodinger Equation ◽  
Higher Order ◽  
Nonlinear Schrodinger Equation ◽  
Exact Analytical Solutions ◽  
Nonlinear Schrödinger

scholarly journals New Soliton-like Solutions of Variable Coefficient Nonlinear Schrödinger Equations

2004 ◽  
Vol 59 (4-5) ◽  
pp. 196-202 ◽  
Author(s):  
Heng-Nong Xuan ◽  
Changji Wang ◽  
Dafang Zhang
Keyword(s):  
Exact Solutions ◽  
Symbolic Computation ◽  
Variable Coefficient ◽  
Schrodinger Equation ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Nonlinear Schrödinger ◽  
System Method ◽  
Nonlinear Schrodinger Equations

The improved projective Riccati system method for solving nonlinear evolution equations (NEEs) is established. With the help of symbolic computation, one can obtain more exact solutions of some NEEs. To illustrate the method, we take the variable coefficient nonlinear Schrödinger equation as an example, and obtain four families of soliton-like solutions. Eight figures are given to illustrate some features of these solutions.

SYMBOLIC COMPUTATION OF BÄCKLUND TRANSFORMATION AND EXACT SOLUTIONS TO THE VARIANT BOUSSINESQ MODEL FOR WATER WAVES

1999 ◽  
Vol 10 (06) ◽  
pp. 983-987
Author(s):  
BO TIAN
Keyword(s):  
Exact Solutions ◽  
Symbolic Computation ◽  
Analytical Solutions ◽  
Water Waves ◽  
Bäcklund Transformation ◽  
Explicit Solutions ◽  
Backlund Transformation ◽  
Boussinesq Model ◽  
Exact Analytical Solutions ◽  
Symbolic Computations

In this paper, we show how computerized symbolic computations can be used to find an auto-Bäcklund transformation and a family of exact analytical solutions to the variant Boussinesq model for water waves. Sample explicit solutions are presented, which are respectively solitonic and rational.

Painlevé Analysis and Symbolic Computation for a Nonlinear Schrödinger Equation with Dissipative Perturbations

1996 ◽  
Vol 51 (3) ◽  
pp. 167-170
Author(s):  
Bo Tian ◽  
Yi-Tian Gao
Keyword(s):  
Exact Solutions ◽  
Symbolic Computation ◽  
Schrodinger Equation ◽  
Bäcklund Transformation ◽  
Dispersive Media ◽  
Painlevé Analysis ◽  
Backlund Transformation ◽  
Weakly Nonlinear ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations

The nonlinear Schrödinger equations with small dissipative perturbations are of current importance in modeling weakly nonlinear dispersive media with dissipation. In this paper, the Painlevé formulation with symbolic computation is presented for one of those equations. An auto-Bäcklund transformation and some exact solutions are explicitly constructed

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

2006 ◽  
Vol 61 (10-11) ◽  
pp. 509-518 ◽  
Author(s):  
Biao Li ◽  
Yong Chen
Keyword(s):  
Solitary Wave ◽  
Symbolic Computation ◽  
Analytical Solutions ◽  
Group Velocity Dispersion ◽  
Broad Class ◽  
Higher Order ◽  
Solitary Wave Solutions ◽  
Exact Analytical Solutions ◽  
Nonlinear Schrödinger ◽  
Wave Solutions

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.

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