Potential symmetry generators and associated conservation laws of perturbed nonlinear equations (original) (raw)
Journal of Computational and Applied Mathematics, 2009
We show how one can construct approximate conservation laws of approximate Euler-type equations via approximate Noethertype symmetry operators associated with partial Lagrangians. The ideas of the procedure for a system of unperturbed partial differential equations are extended to a system of perturbed or approximate partial differential equations. These approximate Noether-type symmetry operators do not form a Lie algebra in general. The theory is applied to the perturbed linear and nonlinear (1 + 1) wave equations and the Maxwellian tails equation. We have also obtained new approximate conservation laws for these equations.
This article focuses on developing and applying approximation techniques to derive conservation laws for the Timoshenko-Prescott mixed derivatives perturbed partial differential equations (PDEs). Central to our approach is employing approximate Noether-type symmetry operators linked to a conventional Lagrangian one. Within this framework, this paper highlights the creation of approximately conserved vectors for PDEs with mixed derivatives. A crucial observation is that the integration of these vectors resulted in the emergence of additional terms. These terms hinder the establishment of the conservation law, indicating a potential flaw in the initial approach. In response to this challenge, we embarked on the rectification process. By integrating these additional terms into our model, we could modify the conserved vectors, deriving new modified conserved vectors. Remarkably, these modified vectors successfully satisfy the conservation law. Our findings not only shed light on the intricate dynamics of fourth-order mechanical systems but also pave the way for refined analytical approaches to address similar challenges in PDE-driven systems.
International Journal of Non-linear Mechanics, 2002
Some recent results on the Lie symmetry generators of equations with a small parameter and the relationship between symmetries and conservation laws for such equations are used to construct first integrals and Lagrangians for autonomous weakly non-linear systems, y″+εF(t)y′+y=f(y,y′). An adaptation of a theorem that provides the point symmetry generators that leave the invariant functional involving a Lagrangian for such
NEW APPROXIMATE SYMMETRY THEOREMS AND COMPARISONS WITH EXACT SYMMETRIES
Qeios
Three new approximate symmetry theories are proposed. The approximate symmetries are contrasted with each other and with the exact symmetries. The theories are applied to nonlinear ordinary differential equations for which exact solutions are available. It is shown that from the symmetries, approximate solutions as well as exact solutions in some restricted cases can be retrievable. Depending on the specific approximate theory and the equations considered, the approximate symmetries may expand the Lie Algebra of the exact symmetries, may be a perturbed form of the exact symmetries or may be a subalgebra of the exact symmetries. Exact and approximate solutions are retrieved using the symmetries.
Journal of Mathematics, 2021
The Korteweg–de Vries (KdV) equation is a weakly nonlinear third-order differential equation which models and governs the evolution of fixed wave structures. This paper presents the analysis of the approximate symmetries along with conservation laws corresponding to the perturbed KdV equation for different classes of the perturbed function. Partial Lagrange method is used to obtain the approximate symmetries and their corresponding conservation laws of the KdV equation. The purpose of this study is to find particular perturbation (function) for which the number of approximate symmetries of perturbed KdV equation is greater than the number of symmetries of KdV equation so that explore something hidden in the system.
Symmetry classification and conservation laws for some nonlinear partial differential equations
2019
In this thesis we study some nonlinear partial differential equations which appear in several physical phenomena of the real world. Exact solutions and conservation laws are obtained for such equations using various methods. The equations which are studied in this work are: a hyperbolic Lane-Emden system, a generalized hyperbolic Lane-Emden system, a coupled Jaulent-Miodek system and a (2+1)-dimensional Jaulent-Miodek equation power-law nonlinearity. We carry out a complete Noether and Lie group classification of the radial form of a coupled system of hyperbolic equations. From the Noether symmetries we establish the corresponding conserved vectors. We also determine constraints that the nonlinearities should satisfy in order for the scaling symmetries to be Noetherian. This led us to a critical hyperbola for the systems under consideration. An explicit solution is also obtained for a particular choice of the parameters. We perform a complete Noether symmetry analysis of a generaliz...
Methodical aspects about the numerical computation of the Lie symmetries and conservation laws
Romanian Journal of Physics
This paper proposes an algorithm for the Lie symmetries investigation in the case of a high order evolution equation. General Lie operators are deduced and, in the next step, the associated conservation laws are derived. Due to the large number of equations derived from the symmetries conditions, the use of mathematical software is necessary, and we employed a MAPLE application. Some models arising from physics was chosen to test the method.
Group-theoretical framework for potential symmetries of evolution equations
Journal of Mathematical Physics, 2011
We develop algebraic approach to the problem of classification of potential symmetries of nonlinear evolution equations. It is essentially based on the recently discovered fact [R. Zhdanov, J. Math. Phys. 50, 053522 (2009)], that any such symmetry is mapped into a contact symmetry. The approach enables using the classical results on classification of contact symmetries of nonlinear evolution equations by Sokolov and Magadeev to classify evolution equations admitting potential symmetries. We construct several examples of new nonlinear fourth-order evolution equations admitting potential symmetries. Since the symmetries obtained depend on nonlocal variables, they cannot be derived by the infinitesimal Lie approach. C
Symmetries and conservation laws
Quantum Mechanics, 2009
Conservation laws play an important role in science. The aim of this thesis is to provide an overview and develop new methods for constructing conservation laws using Lie group theory. The derivation of conservation laws for invariant variational problems is based on Noether's theorem. It is shown that the use of Lie-Bäcklund transformation groups allows one to reduce the number of basic conserved quantities for differential equations obtained by Noether's theorem and construct a basis of conservation laws. Several examples on constructing a basis for some well-known equations are provided. Moreover, this approach allows one to obtain new conservation laws even for equations without Lagrangians. A formal Lagrangian can be introduced and used for computing nonlocal conservation laws. For self-adjoint or quasi-self-adjoint equations nonlocal conservation laws can be transformed into local conservation laws. One of the fields of applications of this approach is electromagnetic theory, namely, nonlocal conservation laws are obtained for the generalized Maxwell-Dirac equations. The theory is also applied to the nonlinear magma equation and its nonlocal conservation laws are computed.
Approximate Hamiltonian Symmetry Groups and Recursion Operators for Perturbed Evolution Equations
Advances in Mathematical Physics, 2013
The method of approximate transformation groups, which was proposed by Baikov et al. (1988 and 1996), is extended on Hamiltonian and bi-Hamiltonian systems of evolution equations. Indeed, as a main consequence, this extended procedure is applied in order to compute the approximate conservation laws and approximate recursion operators corresponding to these types of equations. In particular, as an application, a comprehensive analysis of the problem of approximate conservation laws and approximate recursion operators associated to the Gardner equation with the small parameters is presented.
Conservation laws and potential symmetries for certain evolution equations
arXiv (Cornell University), 2008
We show that the so-called hidden potential symmetries considered in a recent paper [18] are ordinary potential symmetries that can be obtained using the method introduced by Bluman and collaborators [7, 8]. In fact, these are simplest potential symmetries associated with potential systems which are constructed with single conservation laws having no constant characteristics. Furthermore we classify the conservation laws for classes of porous medium equations and then using the corresponding conserved (potential) systems we search for potential symmetries. This is the approach one needs to adopt in order to determine the complete list of potential symmetries. The provenance of potential symmetries is explained for the porous medium equations by using potential equivalence transformations. Point and potential equivalence transformations are also applied to deriving new results on potential symmetries and corresponding invariant solutions from known ones. In particular, in this way the potential systems, potential conservation laws and potential symmetries of linearizable equations from the classes of differential equations under consideration are exhaustively described. Infinite series of infinite-dimensional algebras of potential symmetries are constructed for such equations.
Symmetry in Perturbation Problems
1997
The work is devoted to a new branch of application of continuous group's techniques in the investigation of nonlinear differential equations. The principal stages of the development of perturbation theory of nonlinear diffe- rential equations are considered in short. It is shown that its characteristic features make it possible a fruitful usage of continuous group's techniques in problems of per- turbation theory.
Conservation Laws and Potential Symmetries of Linear Parabolic Equations
Acta Applicandae Mathematicae, 2008
We carry out an extensive investigation of conservation laws and potential symmetries for the class of linear (1 + 1)-dimensional second-order parabolic equations. The group classification of this class is revised by employing admissible transformations, the notion of normalized classes of differential equations and the adjoint variational principle. All possible potential conservation laws are described completely. They are in fact exhausted by local conservation laws. For any equation from the above class the characteristic space of local conservation laws is isomorphic to the solution set of the adjoint equation. Effective criteria for the existence of potential symmetries are proposed. Their proofs involve a rather intricate interplay between different representations of potential systems, the notion of a potential equation associated with a tuple of characteristics, prolongation of the equivalence group to the whole potential frame and application of multiple dual Darboux transformations. Based on the tools developed, a preliminary analysis of generalized potential symmetries is carried out and then applied to substantiate our construction of potential systems. The simplest potential symmetries of the linear heat equation, which are associated with single conservation laws, are classified with respect to its point symmetry group. Equations possessing infinite series of potential symmetry algebras are studied in detail.
Approximate Symmetry Analysis of a Class of Perturbed Nonlinear Reaction-Diffusion Equations
Abstract and Applied Analysis, 2013
The problem of approximate symmetries of a class of nonlinear reaction-diffusion equations called Kolmogorov-Petrovsky-Piskounov (KPP) equation is comprehensively analyzed. In order to compute the approximate symmetries, we have applied the method which was proposed by Fushchich and Shtelen (1989) and fundamentally based on the expansion of the dependent variables in a perturbation series. Particularly, an optimal system of one-dimensional subalgebras is constructed and some invariant solutions corresponding to the resulted symmetries are obtained.
Approximate symmetries and conservation laws for Itô and Stratonovich dynamical systems
Journal of mathematical analysis and …, 2004
The relationship between the approximate Lie±BaÈ cklund symmetries and the approximate conserved forms of a perturbed equation is studied. It is shown that a hierarchy of identities exists by which the components of the approximate conserved vector or the associated approximate Lie±BaÈ cklund symmetries are determined by recursive formulas. The results are applied to certain classes of linear and nonlinear wave equations as well as a perturbed Korteweg±de Vries equation. We construct approximate conservation laws for these equations without regard to a Lagrangian.
Determination of approximate symmetries of differential equations
Group theory and …, 2005
Centre de Recherches Mathematiques CRM Proceedings and Lecture Notes Vo1ume 39, 2005 Determination of Approximate Symmetries of Differential Equations J. Bonasia, Frangois Lemaire, GREG REID, Robin Scott, and Lihong Zhi ABSTRACT. There has been ...
14 Approximate Symmetry Analysis ofa Class of Perturbed Nonlinear Reaction-Diffusion Equations
2014
In this paper, the problem of approximate symmetries of a class of nonlinear reaction-diffusion equations called Kolmogorov-Petrovsky-Piskounov (KPP) equation is comprehensively analyzed. In order to compute the approximate symmetries, we have applied the method which was proposed by Fushchich and Shtelen [8] and fundamentally based on the expansion of the dependent variables in a perturbation series. Particularly, an optimal system of one dimensional subalgebras is constructed and some invariant solutions corresponding to the resulted symmetries are obtained.
Nonlinear Engineering, 2021
Studies on Non-linear evolutionary equations have become more critical as time evolves. Such equations are not far-fetched in fluid mechanics, plasma physics, optical fibers, and other scientific applications. It should be an essential aim to find exact solutions of these equations. In this work, the Lie group theory is used to apply the similarity reduction and to find some exact solutions of a (3+1) dimensional nonlinear evolution equation. In this report, the groups of symmetries, Tables for commutation, and adjoints with infinitesimal generators were established. The subalgebra and its optimal system is obtained with the aid of the adjoint Table. Moreover, the equation has been reduced into a new PDE having less number of independent variables and at last into an ODE, using subalgebras and their optimal system, which gives similarity solutions that can represent the dynamics of nonlinear waves.
New conservation laws obtained directly from symmetry action on a known conservation law
Journal of Mathematical Analysis and Applications, 2006
Two formulas are introduced to directly obtain new conservation laws for any system of partial differential equations from a known conservation law and admitted symmetries. The first formula maps any conservation law of a given system to the corresponding conservation law of the system obtained through a contact transformation. When the contact transformation is a symmetry of the given system, then the corresponding conservation law is a conservation law of the given system. The second formula checks a priori whether or not the action of a symmetry (continuous or discrete) on a conservation law can yield one or more new conservation laws of the given system. Several examples are considered, including the use of a discrete symmetry to obtain a new conservation law and the use of a continuous symmetry to generate two new conservation laws.