Generalized Hyperbolic-function Method with Computerized Symbolic Computation to the Nizhnik-Novikov-Veselov Equation

2012 ◽  
Author(s):  
Liu Jianguo ◽  
Li Fei
Keyword(s):  
Symbolic Computation ◽  
Function Method ◽  
Hyperbolic Function ◽  
Hyperbolic Function Method

SPATIAL XPM-PAIRED SOLITONS IN NONLINEAR METAMATERIALS

2013 ◽  
Vol 22 (01) ◽  
pp. 1350009
Author(s):  
ANLE FANG ◽  
YUANJIANG XIANG ◽  
BINXIAN ZHUANG ◽  
LEYONG JIANG ◽  
XIAOYU DAI ◽  
...  
Keyword(s):  
Refractive Index ◽  
Optical Fibers ◽  
Function Method ◽  
Hyperbolic Function ◽  
Nonlinear Polarization ◽  
Dark Solitons ◽  
Nonlinear Metamaterials ◽  
Theoretical Predictions ◽  
Hyperbolic Function Method ◽  
Propagation Properties

We investigate spatial XPM-paired solitons in nonlinear metamaterials (MMs) based on the (1 + 1)-dimensional coupled nonlinear Schrodinger equation (NLSE) describing the co-propagation of two optical beams of different frequencies in the MM with a Kerr-type nonlinear polarization. Three types of XPM-paired solitons including bright-bright, bright-dark and dark-dark solitons for different combination of the signs of refractive index experienced by the two beams, respectively, are obtained by using a generalized hyperbolic function method, which makes the temporal XPM-paired solitons in optical fibers find their spatial counterparts in MMs. Numerical simulations are performed to confirm the theoretical predictions and further identify the propagation properties of the spatial XPM-paired solitons in MMs described by Drude model.

A modified hyperbolic function method and its application to the Toda lattice

2006 ◽  
Vol 172 (2) ◽  
pp. 938-945 ◽  
Author(s):  
Hongyan Zhi ◽  
Xueqin Zhao ◽  
Zhongyuan Yang ◽  
Hongqing Zhang
Keyword(s):  
Function Method ◽  
Toda Lattice ◽  
Hyperbolic Function ◽  
Hyperbolic Function Method

scholarly journals Improved hyperbolic function method and exact solutions for variable coefficient Benjamin-Bona-Mahony-Burgers equation

2015 ◽  
Vol 19 (4) ◽  
pp. 1183-1187
Author(s):  
Hong-Cai Ma ◽  
Xiao-Fang Peng ◽  
Dan-Dan Yao
Keyword(s):  
Fluid Mechanics ◽  
Exact Solutions ◽  
Variable Coefficient ◽  
Function Method ◽  
Burgers Equation ◽  
Periodic Wave ◽  
Hyperbolic Function ◽  
Hyperbolic Function Method

By using the improved hyperbolic function method, we investigate the variable coefficient Benjamin-Bona-Mahony-Burgers equation which is very important in fluid mechanics. Some exact solutions are obtained. Under some conditions, the periodic wave leads to the kink-like wave.

ON A VARIABLE-COEFFICIENT MODIFIED KP EQUATION AND A GENERALIZED VARIABLE-COEFFICIENT KP EQUATION WITH COMPUTERIZED SYMBOLIC COMPUTATION

2001 ◽  
Vol 12 (06) ◽  
pp. 819-833 ◽  
Author(s):  
YI-TIAN GAO ◽  
BO TIAN
Keyword(s):  
Symbolic Computation ◽  
Variable Coefficient ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Analytic Solutions ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Kp Equation ◽  
Exact Analytic Solutions ◽  
Hyperbolic Function Method

The variable-coefficient nonlinear evolution equations, although realistically modeling various mechanical and physical situations, often cause some well-known powerful methods not to work efficiently. In this paper, we extend the power of the generalized hyperbolic-function method, which is based on the computerized symbolic computation, to a variable-coefficient modified Kadomtsev–Petviashvili (KP) equation and a generalized variable-coefficient KP equation. New exact analytic solutions thus come out.

Approximate symmetry and solutions of the nonlinear Klein–Gordon equation with a small parameter

2017 ◽  
Vol 14 (03) ◽  
pp. 1750046
Author(s):  
Mohammad Rahimian ◽  
Megerdich Toomanian ◽  
Mehdi Nadjafikhah
Keyword(s):  
Small Parameter ◽  
Analytical Solutions ◽  
Function Method ◽  
Gordon Equation ◽  
Hyperbolic Function ◽  
Approximate Symmetry ◽  
Nonlinear Phenomena ◽  
Hyperbolic Function Method ◽  
Klein Gordon ◽  
Klein Gordon Equation

In this paper, the Lie approximate symmetry analysis is applied to investigate new solutions of the nonlinear Klein–Gordon equation with a small parameter. The nonlinear Klein–Gordon equation is used to model many nonlinear phenomena. The hyperbolic function method and Riccati equation method are employed to solve some of the obtained reduced ordinary differential equations. We construct new analytical solutions with a small parameter which is effectively obtained by the proposed method.

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