Initial-value Problems for Extended KdV-Burgers Equations via Generalized Conditional Symmetries

We classify initial-value problems for extended KdV-Burgers equations via generalized conditional symmetries. These equations can be reduced to Cauchy problems for some systems of first-order ordinary differential equations. The obtained reductions cannot be derived within the framework of the standard Lie approach.     

A note on the Lie symmetries of complex partial differential equations and their split real systems

Folklore suggests that the split Lie-like operators of a complex partial differential equation are symmetries of the split system of real partial differential equations. However, this is not the case generally. We illustrate this by using the complex heat equation, wave equation with dissipation, the nonlinear Burgers equation and nonlinear KdV equations. We split the Lie symmetries of a complex partial differential equation in the real domain and obtain real Lie-like operators. Further, the complex partial differential equation is split into two coupled or uncoupled real partial differential equations which constitute a system of two equations for two real functions of two real variables. The Lie symmetries of this system are constructed by the classical Lie approach. We compare these Lie symmetries with the split Lie-like operators of the given complex partial differential equation for the examples considered. We conclude that the split Lie-like operators of complex partial differential equations are not in general symmetries of the split system of real partial differential equations. We prove a proposition that gives the criteria when the Lie-like operators are symmetries of the split system.     

Integrable ODEs on Associative Algebras

In this paper we give definitions of basic concepts such as symmetries, first integrals, Hamiltonian and recursion operators suitable for ordinary differential equations on associative algebras, and in particular for matrix differential equations. We choose existence of hierarchies of first integrals and/or symmetries as a criterion for integrability and justify it by examples. Using our componentless approach we have solved a number of classification problems for integrable equations on free associative algebras. Also, in the simplest case, we have listed all possible Hamiltonian operators of low order. Received: 22 September 1999 / Accepted: 17 November 1999     

New Symmetries for a Model of Fast Diffusion

The new symmetries for a mathematical model of fast diffusion are determined. A new system method is given to search for new symmetries of differential equations written in a conserved form, several new symmetry generators and exact solutions are presented.     

Invariantization of the Crank-Nicolson method for Burgers’ equation

In this article, a geometric technique to construct numerical schemes for partial differential equations (PDEs) that inherit Lie symmetries is proposed. The moving frame method enables one to adjust the numerical schemes in a geometric manner and systematically construct proper invariant versions of them. To illustrate the method, we study invariantization of the Crank-Nicolson scheme for Burgers’ equation. With careful choice of normalization equations, the invariantized schemes are shown to surpass the standard scheme, successfully removing numerical oscillation around sharp transition layers.     

On kinematical invariances of the equations of motion

Invariance properties of the equations of motion are considered from the differential geometric viewpoint, on making use of vector fields and differential forms. It will be shown that the kinematical symmetries generated from first integrals and the invariance such as the dilation can be treated on an equal footing.     

Conservation laws and hierarchies of potential symmetries for certain diffusion equations

We show that the so-called hidden potential symmetries considered in a recent paper [M.L. Gandarias, New potential symmetries for some evolution equations, Physica A 387 (2008) 2234-2242] are ordinary potential symmetries that can be obtained using the method introduced by Bluman and collaborators [G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, New York, 1989; G.W. Bluman, G.J. Reid, S. Kumei, New classes of symmetries for partial differential equations, J. Math. Phys. 29 (1988) 806-811]. 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.     

Symmetries and conservation laws of lattice Boussinesq equations

Sequences of canonical conservation laws and generalized symmetries for the lattice Boussinesq and the lattice modified Boussinesq systems are successively derived. The interpretation of these symmetries as differential-difference equations leads to corresponding hierarchies of such equations for which conservation laws and Lax pairs are constructed. Finally, using the continuous symmetry reduction approach, an integrable, multidimensionally consistent system of partial differential equations is derived in relation with the lattice modified Boussinesq system.     

Lie and Noether symmetries of systems of complex ordinary differential equations and their split systems

The Lie and Noether point symmetry analyses of a kth-order system of m complex ordinary differential equations (ODEs) with m dependent variables are performed. The decomposition of complex symmetries of the given system of complex ODEs yields Lie- and Noether-like operators. The system of complex ODEs can be split into 2m coupled real partial differential equations (PDEs) and 2m Cauchy–Riemann (CR) equations. The classical approach is invoked to compute the symmetries of the 4m real PDEs and these are compared with the decomposed Lie- and Noether-like operators of the system of complex ODEs. It is shown that, in general, the Lie- and Noether-like operators of the system of complex ODEs and the symmetries of the decomposed system of real PDEs are not the same. A similar analysis is carried out for restricted systems of complex ODEs that split into 2m coupled real ODEs. We summarize our findings on restricted complex ODEs in two propositions.     

Multiple Lax pair for the double Ernst equation

The double-complex function method is used to find a multiple Lax pair for the double Ernst equation. Then multiple hidden symmetries and multiple infinite sequences of nonlocal conserved charges are obtained. The result of Papachristou and Harrison is contained in our result.     

Lie Symmetries, Perturbation to Symmetries and Adiabatic Invariants of a Generalized Birkhoff System

We study the perturbation to symmetries and adiabatic invariants of a generalized Birkhoff system. Based on the invariance of differential equations under infinitesimal transformations, Lie symmetries, laws of conservations, perturbation to the symmetries and adiabatic invariants of the generalized Birkhoff system are presented. First, the concepts of Lie symmetries and higher order adiabatic invariants of the generalized Birkhoff system are proposed. Then, the conditions for the existence of the exact invariants and adiabatic invariants are proved, and their forms are given. Finally, an example is presented to illustrate the method and results.     

Lie symmetries and nonlocally related systems of the continuous and discrete dispersive long waves system by geometric approach

By using the extended Harrison and Estabrook's differential forms approach, in this paper, we investigate the Lie symmetries of the continuous and discrete dispersive long waves system, respectively. Based on this method, two closed ideals written in terms of a set of differential forms are constructed for the dispersive long waves systems. Furthermore, some invariant solutions are presented for such systems. By a direct computation, it is shown that the discrete dispersive long waves system admits a Kac-Moody-Virasoro type and a Virasoro-like type Lie algebra, respectively. Finally, we present an interesting relationship between the continuous case and a modified dispersive long waves system, which can be used to find nonlocal properties for such systems with each other.     

Study of the Lie symmetries of a relativistic variable mass system

The differential equations of motion of a relativistic variable mass system are given.By using the invariance of the differential equations under the infinitesimal transformations of groups,the determining equations and the restriction equations of the Lie symmetries of a relativistic variable mass system are built,and the structure equation and the conserved quantity of the Lie symmetries are obtained.Then the inverse problem of the Lie symmetries is studied.The corresponding Lie symmetries are found according to a known conserved quantity.An example is given to illustrate the application of the result.     

A Note on Nonclassical Symmetries of a Class of Nonlinear Partial Differential Equations and Compatibility

The nonclassical symmetries of a class of nonlinear partial differential equations obtained by the compatibility method is investigated. We show thenonclassical symmetries obtained in [J. Math. Anal. Appl. 289 (2004) 55, J. Math. Anal. Appl. 311 (2005) 479] are not all the nonclassical symmetries. Based on a new assume on the form of invariant surface condition, all the nonclassical symmetries for a class of nonlinear partial differential equations can be obtained through the compatibility method. The nonlinear Klein-Gordonequation and the Cahn-Hilliard equations all serve as examples showing the compatibility method leads quickly and easily to the determining equations for their all nonclassical symmetries for two equations.     

ON NONLOCAL SYMMETRIES,NONLOCAL CONSERVATION LAWS AND NONLOCAL TRANSFORMATIONS OF EVOLUTION EQUATIONS: TWO LINEARISABLE HIERARCHIES

We discuss nonlocal symmetries and nonlocal conservation laws that follow from the systematic potentialisation of evolution equations. Those are the Lie point symmetries of the auxiliary systems, also known as potential symmetries. We define higher-degree potential symmetries which then lead to nonlocal conservation laws and nonlocal transformations for the equations. We demonstrate our approach and derive second degree potential symmetries for the Burgers' hierarchy and the Calogero–Degasperis–Ibragimov–Shabat hierarchy.     

n<Superscript><Emphasis Type="Italic">th</Emphasis></Superscript>-Order Approximate Lagrangians Induced by Perturbative Geometries

A family of perturbative Lagrangians that describe approximate and multidimensional Klein-Gordon equations are studied. We probe the existence of approximate Noether symmetries via generalized geometric conditions for a perturbation of any order. The knowledge of the geometric conditions uncovers that unlike exact symmetries, the approximate Noether symmetries of the Lagrangian which describes the motion of a particle in n-dimensional space under the action of an autonomous force, is inequivalent to the Noether symmetries admitted by the Klein-Gordon Lagrangian in general.     

Symmetries of Vector Exterior Differential Systems and the Inverse Problem in Second-Order Ostrograds’kii Mechanics

Abstract

Symmetries for variational problems are considered as symmetries of vector-valued exterior differential systems. This approach is applied to equations for the classical spinning particle.     

Conditional Symmetry and Exact Solutions of a Nonlinear Galilei-Invariant Spinor Equation

Abstract

Symmetries for variational problems are considered as symmetries of vector-valued exterior differential systems. This approach is applied to equations for the classical spinning particle.     

Geometric Theory of Time-Dependent Singular Lagrangians

The geometry of jet bundles is used for obtaining a geometric approach to time-dependent Lagrangian systems, both in the regular and singular case. Generalized symmetries are introduced as being given by vector fields along the projection π_1,0. This approach is shown to be very useful for giving a one-to-one correspondence between symmetries and constants of motion and allows a geometric version of the Second Noether Theorem. The theory is illustrated by several examples in which the gauge symmetries are explicitly found.     

Noether conserved quantities and Lie point symmetries of difference Lagrange--Maxwell equations and lattices

This paper presents a method to find Noether-type conserved quantities and Lie point symmetries for discrete mechanico-electrical dynamical systems, which leave invariant the set of solutions of the corresponding difference scheme. This approach makes it possible to devise techniques for solving the Lagrange--Maxwell equations in differences which correspond to mechanico-electrical systems, by adapting existing differential equations. In particular, it obtains a new systematic method to determine both the one-parameter Lie groups and the discrete Noether conserved quantities of Lie point symmetries for mechanico-electrical systems. As an application, it obtains the Lie point symmetries and the conserved quantities for the difference equation of a model that represents a capacitor microphone.