Potential symmetry generators and associated conservation laws of perturbed nonlinear equations (original) (raw)

Comparison of Approximate Symmetry Methods for Differential Equations

Acta Applicandae Mathematicae, 2000

Two current approximate symmetry methods and a modified new one are contrasted. Approximate symmetries of potential Burgers equation and non-Newtonian creeping flow equations are calculated using different methods. Approximate solutions corresponding to the approximate symmetries are derived for each method. Symmetries and solutions are compared and advantages and disadvantages of each method are discussed in detail. : 35B20, 35C05, 35K22, 35Q35, 35Q53, 76A05.

Approximate Noether-type symmetries and conservation laws via partial Lagrangians for PDEs with a small parameter

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.

Approximate Symmetries and Conservation Laws for Mechanical Systems Described by Mixed Derivative Perturbed PDEs

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.

Group theoretic methods for approximate invariants and Lagrangians for some classes of y″+ εF( t) y′+ y= f( y, y′)

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

Approximate Symmetries Analysis and Conservation Laws Corresponding to Perturbed Korteweg–de Vries Equation

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.