scholarly journals Derivation of Asymptotic Dynamical Systems with Partial Lie Symmetry Groups

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Masatomo Iwasa
Keyword(s):  
Dynamical System ◽  
Asymptotic Behavior ◽  
Singular Perturbation ◽  
Symmetry Group ◽  
Group Analysis ◽  
Lie Symmetry ◽  
Symmetry Groups ◽  
Lie Group Analysis ◽  
Constraint Equations ◽  
Perturbation Problems

Lie group analysis has been applied to singular perturbation problems in both ordinary differential and difference equations and has allowed us to find the reduced dynamics describing the asymptotic behavior of the dynamical system. The present study provides an extended method that is also applicable to partial differential equations. The main characteristic of the extended method is the restriction of the manifold by some constraint equations on which we search for a Lie symmetry group. This extension makes it possible to find a partial Lie symmetry group, which leads to a reduced dynamics describing the asymptotic behavior.

Lie group analysis, Hamiltonian equations and conservation laws of Born–Infeld equation

2014 ◽  
Vol 07 (03) ◽  
pp. 1450040 ◽  
Author(s):  
Seyed Reza Hejazi
Keyword(s):  
Conservation Laws ◽  
Symmetry Group ◽  
Lie Group ◽  
Group Analysis ◽  
Similarity Solutions ◽  
Lie Symmetry ◽  
Group Method ◽  
Lie Group Analysis ◽  
Hamiltonian Equations

Lie symmetry group method is applied to study the Born–Infeld equation. The symmetry group is given, and similarity solutions associated to the symmetries are obtained. Finally the Hamiltonian equations including Hamiltonian symmetry group and conservation laws are determined.

Lie Symmetry Group of the Nonisospectral Kadomtsev-Petviashvili Equation

2009 ◽  
Vol 64 (1-2) ◽  
pp. 8-14 ◽  
Author(s):  
Yong Chen ◽  
Xiaorui Hu
Keyword(s):  
Symmetry Group ◽  
Discrete Group ◽  
Group Analysis ◽  
Lie Symmetry ◽  
Kp Equation ◽  
Classical Symmetry ◽  
Petviashvili Equation ◽  
Finite Transformation ◽  
Special Case ◽  
Symmetry Method

The classical symmetry method and the modified Clarkson and Kruskal (C-K) method are used to obtain the Lie symmetry group of a nonisospectral Kadomtsev-Petviashvili (KP) equation. It is shown that the Lie symmetry group obtained via the traditional Lie approach is only a special case of the symmetry groups obtained by the modified C-K method. The discrete group analysis is given to show the relations between the discrete group and parameters in the ansatz. Furthermore, the expressions of the exact finite transformation of the Lie groups via the modified C-K method are much simpler than those obtained via the standard approach.

LIE SYMMETRY ANALYSIS TO THE WEAKLY COUPLED KAUP–KUPERSHMIDT EQUATION WITH TIME FRACTIONAL ORDER

Fractals ◽  
2019 ◽  
Vol 27 (04) ◽  
pp. 1950052 ◽  
Author(s):  
ZHENLI WANG ◽  
LIHUA ZHANG ◽  
CHUANZHONG LI
Keyword(s):  
Fractional Order ◽  
Group Analysis ◽  
Nonlinear Ordinary Differential Equation ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Explicit Solutions ◽  
Lie Group Analysis ◽  
Analysis Method ◽  
Lie Symmetry Analysis ◽  
Weakly Coupled

The aim of this paper is to apply the Lie group analysis method to the weakly coupled Kaup–Kupershmidt (KK) equation with time fractional order. We considered the symmetry analysis, explicit solutions to the weakly coupled time fractional KK (TF-KK) equation with Riemann–Liouville (RL) derivative. The weakly coupled TF-KK equation is reduced to a nonlinear ordinary differential equation (ODE) of fractional order. We solve the reduced fractional ODE using the sub-equation method.

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 Asymptotic behavior of the W1∕q,q-norm of mollified BV functions and applications to singular perturbation problems

2020 ◽  
Vol 26 ◽  
pp. 77
Author(s):  
Arkady Poliakovsky
Keyword(s):  
Asymptotic Behavior ◽  
Singular Perturbation ◽  
Singular Perturbation Problems ◽  
Regular Domain ◽  
Link Type ◽  
Bv Functions ◽  
Perturbation Problems ◽  
Funct Anal

Motivated by results of Figalli and Jerison [J. Funct. Anal. 266 (2014) 1685–1701] and Hernández [Pure Appl. Funct. Anal., Preprint https://arxiv.org/abs/1709.08262 (2017)], we prove the following formula: \begin{equation*} %\label{hjhjjggjjjkhkhjjhjhhgghjhjhjljkjk} \lim_{\e\to 0^+}\frac{1}{|\ln{\e}|}\big\|\eta_\e*u\big\|^q_{W^{1/q,q}(\Omega)}= C_0\int_{J_u}\Big|u^+(x)-u^-(x)\Big|^q{\rm d}\mathcal{H}^{N-1}(x), \end{equation*} where Ω ⊂ ℝN is a regular domain, u ∈ BV (Ω) ∩ L∞(Ω), q > 1 and ηε(z) = ε−Nη(z∕ε) is a smooth mollifier. In addition, we apply the above formula to the study of certain singular perturbation problems.

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.

scholarly journals Reduction of Dynamics with Lie Group Analysis

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
M. Iwasa
Keyword(s):  
Dynamical System ◽  
Lie Group ◽  
Perturbation Methods ◽  
Group Analysis ◽  
Lie Symmetries ◽  
Time Behavior ◽  
Lie Group Analysis ◽  
Long Time Behavior ◽  
Singular Perturbation Methods ◽  
Long Time

This paper is mainly a review concerning singular perturbation methods by means of Lie group analysis which has been presented by the author. We make use of a particular type of approximate Lie symmetries in those methods in order to construct reduced systems which describe the long-time behavior of the original dynamical system. Those methods can be used in analyzing not only ordinary differential equations but also difference equations. Although this method has been mainly used in order to derive asymptotic behavior, when we can find exact Lie symmetries, we succeed in construction of exact solutions.

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