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.

A Truncated Painlevé Expansion and Exact Analytical Solutions for the Nonlinear Schr¨Odinger Equation with Variable Coefficients

2005 ◽  
Vol 60 (11-12) ◽  
pp. 768-774 ◽  
Author(s):  
Biao Li ◽  
Yong Chen
Keyword(s):  
Analytical Solutions ◽  
Bäcklund Transformation ◽  
Variable Coefficients ◽  
Backlund Transformation ◽  
Exact Analytical Solutions ◽  
Interaction Of Solitons ◽  
Varying Dispersion ◽  
Physical Phenomena ◽  
Truncated Painlevé Expansion ◽  
Singular Soliton

By using the truncated Painlevé expansion analysis an auto-Bäcklund transformation is found for the nonlinear Schrödinger equation with varying dispersion, nonlinearity, and gain or absorption. Then, based on the obtained auto-Bäcklund transformation and symbolic computation, we explore some explicit exact solutions including soliton-like solutions, singular soliton-like solutions, which may be useful to explain the corresponding physical phenomena. Further, the formation and interaction of solitons are simulated by computer. - PACS Nos.: 05.45.Yv, 02.30.Jr, 42.65.Tg

Truncated Painlevé Expansion with Symbolic Computation for a General Kadomtsev-Petviashvili Equation with Variable Coefficients

1996 ◽  
Vol 51 (3) ◽  
pp. 175-178
Author(s):  
Bo Tian ◽  
Yi-Tian Gao
Keyword(s):  
Symbolic Computation ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Variable Coefficients ◽  
Explicit Solutions ◽  
Current Interest ◽  
Backlund Transformation ◽  
Petviashvili Equation ◽  
Mathematical Sciences ◽  
Truncated Painlevé Expansion

Able to realistically model various physical situations, the variable-coefficient generalizations of the celebrated Kadmotsev-Petviashvili equation are of current interest in physical and mathematical sciences. In this paper, we make use of both the truncated Painleve expansion and symbolic computation to obtain an auto-Bäcklund transformation and certain soliton-typed explicit solutions for a general Kadomtsev-Petviashvili equation with variable coefficients.

Symbolic Computation Study of a Generalized Variable-Coefficient Two-Dimensional Korteweg-de Vries Model with Various External-Force Terms from Shallow Water Waves, Plasma Physics, and Fluid Dynamics

2009 ◽  
Vol 64 (3-4) ◽  
pp. 222-228 ◽  
Author(s):  
Xing Lü ◽  
Li-Li Li ◽  
Zhen-Zhi Yao ◽  
Tao Geng ◽  
Ke-Jie Cai ◽  
...  
Keyword(s):  
Symbolic Computation ◽  
External Force ◽  
Bilinear Form ◽  
Water Waves ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Two Dimensional ◽  
Backlund Transformation ◽  
Bilinear Method ◽  
De Vries

Abstract The variable-coefficient two-dimensional Korteweg-de Vries (KdV) model is of considerable significance in describing many physical situations such as in canonical and cylindrical cases, and in the propagation of surface waves in large channels of varying width and depth with nonvanishing vorticity. Under investigation hereby is a generalized variable-coefficient two-dimensional KdV model with various external-force terms. With the extended bilinear method, this model is transformed into a variable-coefficient bilinear form, and then a Bäcklund transformation is constructed in bilinear form. Via symbolic computation, the associated inverse scattering scheme is simultaneously derived on the basis of the aforementioned bilinear Bäcklund transformation. Certain constraints on coefficient functions are also analyzed and finally some possible cases of the external-force terms are discussed

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.

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

On the Variable-coefficient Burgers-Hlavaty Equation

1999 ◽  
Vol 54 (12) ◽  
pp. 761-762
Author(s):  
Yi-Tian Gao ◽  
Bo Tian
Keyword(s):  
Analytical Solutions ◽  
Variable Coefficient ◽  
Burgers Equation ◽  
Bäcklund Transformation ◽  
Backlund Transformation ◽  
Exact Analytical Solutions

Abstract We study Hlavaty’s generalization of the Burgers equation containing certain coefficient functions. We obtain a new auto-Bäcklund transformation and a family of exact analytical solutions along with the constraints on those coefficients.

Sign in / Sign up

Export Citation Format

Share Document