Study on the Integrable Properties of Two Coupled KdV Equations

2011 ◽  
Vol 267 ◽  
pp. 683-692
Author(s):  
Li Yuan Zhang ◽  
Li Mei Cheng ◽  
Wei Yuan ◽  
Ruo Xia Yao
Keyword(s):  
Conservation Laws ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Symmetry And Conservation Laws ◽  
Lie Symmetry ◽  
Point Symmetry ◽  
Symmetry Groups ◽  
Kdv Equations ◽  
Generalized Symmetries ◽  
Coupled Kdv Equations

Two coupled KdV equations describing the atmospheric and oceanic phenomena are re-analyzed from the view points of Lie point symmetry, Lie symmetry groups, symmetry reductions, infinitely many generalized symmetries and conservation laws armed with the computer algebra system Maple. The results obtained in this paper show that the two coupled KdV equations are completely integrable in the sense of symmetry and conservation laws. Some results obtained by us are new and first reported here.

scholarly journals Lie symmetry reductions and conservation laws for fractional order coupled KdV system

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Hossein Jafari ◽  
Hong Guang Sun ◽  
Marzieh Azadi
Keyword(s):  
Conservation Laws ◽  
Fractional Order ◽  
Lie Symmetry ◽  
Theoretical Physics ◽  
Lie Symmetry Analysis ◽  
Kdv Equations ◽  
Ode System ◽  
Coupled Kdv Equations ◽  
Scientific Phenomena ◽  
New System

AbstractLie symmetry analysis is achieved on a new system of coupled KdV equations with fractional order, which arise in the analysis of several problems in theoretical physics and numerous scientific phenomena. We determine the reduced fractional ODE system corresponding to the governing factional PDE system.In addition, we develop the conservation laws for the system of fractional order coupled KdV equations.

scholarly journals Construction of complex shell models of MHD turbulence in computer algebra systems

2020 ◽  
Vol 196 ◽  
pp. 02008
Author(s):  
Liubov Feshchenko ◽  
Gleb Vodinchar
Keyword(s):  
Conservation Laws ◽  
Computer Algebra ◽  
Nonlinear Interaction ◽  
Power Distribution ◽  
Computer Algebra System ◽  
Stationary Solutions ◽  
Computer Algebra Systems ◽  
Mhd Turbulence ◽  
Shell Models ◽  
Shell Models Of Turbulence

The paper describes the developed by authors technique for construct-ing complex shell models of turbulence. The compilation of the equa-tions of this model and its exactly solution are implemented using by computer algebra system. The technique allows one to vary the sizes of nonlocality of nonlinear interaction in the space of scales, expressions for shell analogues of conservation laws, and the nature of stationary solutions with different power distribution.

scholarly journals Group analysis and variational principle for nonlinear (3+1) schrodinger equation

2010 ◽  
Vol 7 (1) ◽  
pp. 115-122
Author(s):  
Eman Salem A. Alaidarous
Keyword(s):  
Variational Principle ◽  
Schrödinger Equation ◽  
Conservation Laws ◽  
Lie Groups ◽  
Schrodinger Equation ◽  
Group Analysis ◽  
Lie Symmetry ◽  
Functional Integral ◽  
Symmetry Groups ◽  
Conserved Currents

The generators of the admitted variational Lie symmetry groups are derived and conservation laws for the conserved currents are obtained via Noether's theorem. Moreover, the consistency of a functional integral are derived for the nonlinear Schrödinger equation. In addition to this analysis functional integral are studied using Lie groups.

scholarly journals Symmetry Groups, Similarity Reductions, and Conservation Laws of the Time-Fractional Fujimoto–Watanabe Equation Using Lie Symmetry Analysis Method

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Baoyong Guo ◽  
Huanhe Dong ◽  
Yong Fang
Keyword(s):  
Conservation Laws ◽  
Differential Operators ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Symmetry Groups ◽  
Analysis Method ◽  
Lie Symmetry Analysis ◽  
Reduced Equations ◽  
Liouville Fractional Derivative ◽  
Similarity Reductions

In this paper, the time-fractional Fujimoto–Watanabe equation is investigated using the Riemann–Liouville fractional derivative. Symmetry groups and similarity reductions are obtained by virtue of the Lie symmetry analysis approach. Meanwhile, the time-fractional Fujimoto–Watanabe equation is transformed into three kinds of reduced equations and the third of which is based on Erdélyi–Kober fractional integro-differential operators. Furthermore, the conservation laws are also acquired by Ibragimov’s theory.

scholarly journals ReLie: A Reduce Program for Lie Group Analysis of Differential Equations

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1826
Author(s):  
Francesco Oliveri
Keyword(s):  
Differential Equations ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Source Code ◽  
Group Analysis ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Lie Group Analysis ◽  
Lie Symmetry Analysis ◽  
Source Program

Lie symmetry analysis provides a general theoretical framework for investigating ordinary and partial differential equations. The theory is completely algorithmic even if it usually involves lengthy computations. For this reason, along the years many computer algebra packages have been developed to automate the computation. In this paper, we describe the program ReLie, written in the Computer Algebra System Reduce, since 2008 an open source program for all platforms. ReLie is able to perform almost automatically the needed computations for Lie symmetry analysis of differential equations. Its source code is freely available too. The use of the program is illustrated by means of some examples; nevertheless, it is to be underlined that it proves effective also for more complex computations where one has to deal with very large expressions.

FINITE SYMMETRY GROUP AND COHERENT SOLITON SOLUTIONS FOR THE BROER–KAUP–KUPERSHMIDT SYSTEM

2013 ◽  
Vol 23 (09) ◽  
pp. 1350156 ◽  
Author(s):  
JUN YU ◽  
HANWEI HU
Keyword(s):  
Symmetry Group ◽  
Direct Method ◽  
Soliton Solutions ◽  
Mathematical Physics ◽  
Lie Symmetry ◽  
Point Symmetry ◽  
Symmetry Groups ◽  
Localized Structures ◽  
Lie Point Symmetry

A modified CK direct method is generalized to find finite symmetry groups of nonlinear mathematical physics systems. For the (2 + 1)-dimensional Broer–Kaup–Kupershmidt (BKK) system, both the Lie point symmetry and the non-Lie symmetry groups are obtained by this method. While using the traditional Lie approach, one can only find the Lie symmetry groups. Furthermore, abundant localized structures of the BKK equation are also obtained from the non-Lie symmetry group.

Generation of Correlated Logistic-Normal Random Variates for Medical Decision Trees

1998 ◽  
Vol 37 (03) ◽  
pp. 235-238 ◽  
Author(s):  
M. El-Taha ◽  
D. E. Clark
Keyword(s):  
Monte Carlo ◽  
Decision Trees ◽  
Computer Algebra ◽  
Taylor Series ◽  
Computer Algebra System ◽  
Random Variable ◽  
Monte Carlo Analysis ◽  
Normal Random Variable ◽  
Medical Decision ◽  
Theoretical Results

AbstractA Logistic-Normal random variable (Y) is obtained from a Normal random variable (X) by the relation Y = (ex)/(1 + ex). In Monte-Carlo analysis of decision trees, Logistic-Normal random variates may be used to model the branching probabilities. In some cases, the probabilities to be modeled may not be independent, and a method for generating correlated Logistic-Normal random variates would be useful. A technique for generating correlated Normal random variates has been previously described. Using Taylor Series approximations and the algebraic definitions of variance and covariance, we describe methods for estimating the means, variances, and covariances of Normal random variates which, after translation using the above formula, will result in Logistic-Normal random variates having approximately the desired means, variances, and covariances. Multiple simulations of the method using the Mathematica computer algebra system show satisfactory agreement with the theoretical results.

scholarly journals On Galilean Invariant and Energy Preserving BBM-Type Equations

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 878
Author(s):  
Alexei Cheviakov ◽  
Denys Dutykh ◽  
Aidar Assylbekuly
Keyword(s):  
Wave Equation ◽  
Conservation Laws ◽  
Numerical Simulations ◽  
Solitary Waves ◽  
Local Symmetry ◽  
Symmetry And Conservation Laws ◽  
Higher Order ◽  
Special Equation ◽  
Long Wave ◽  
Local Symmetries

We investigate a family of higher-order Benjamin–Bona–Mahony-type equations, which appeared in the course of study towards finding a Galilei-invariant, energy-preserving long wave equation. We perform local symmetry and conservation laws classification for this family of Partial Differential Equations (PDEs). The analysis reveals that this family includes a special equation which admits additional, higher-order local symmetries and conservation laws. We compute its solitary waves and simulate their collisions. The numerical simulations show that their collision is elastic, which is an indication of its S−integrability. This particular PDE turns out to be a rescaled version of the celebrated Camassa–Holm equation, which confirms its integrability.

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