Invariance Analysis and Variational Conservation Laws for the Wave Equation on Some Manifolds

In this paper we discuss symmetries of a nonlinear wave equation that arises as a consequence of some Riemannian metrics of signature −2. The objective of this study is to show how geometry can be responsible in giving rise to a nonlinear inhomogeneous wave equation rather than assuming nonlinearities in the wave equation from physical considerations. We find Lie point symmetries of the corresponding wave equations and give their solutions in two cases. Some interesting physical conclusions relating to conservation laws such as energy, linear and angular momenta are also determined.     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.     

The symmetries and conservation laws of some Gordon-type equations in Milne space-time

In this letter, the Lie point symmetries of a class of Gordon-type wave equations that arise in the Milne space-time are presented and analysed. Using the Lie point symmetries, it is showed how to reduce Gordon-type wave equations using the method of invariants, and to obtain exact solutions corresponding to some boundary values. The Noether point symmetries and conservation laws are obtained for the Klein–Gordon equation in one case. Finally, the existence of higherorder variational symmetries of a projection of the Klein–Gordon equation is investigated using the multiplier approach.     

Second Order Integrability Conditions for Difference Equations: An Integrable Equation

Integrability conditions for difference equations admitting a second order formal recursion operator are presented and the derivation of symmetries and canonical conservation laws are discussed. In a generic case, some of these conditions yield nonlocal conservation laws. A new integrable equation satisfying the second order integrability conditions is presented and its integrability is established by the construction of symmetries, conservation laws and a 3 × 3 Lax representation. Finally, via the relation of the symmetries of this equation to the Bogoyavlensky lattice, an integrable asymmetric quad equation and a consistent pair of difference equations are derived.     

Equivalence of Conservation Laws and Equivalence of Potential Systems

We study conservation laws and potential symmetries of (systems of) differential equations applying equivalence relations generated by point transformations between the equations. A Fokker–Planck equation and the Burgers equation are considered as examples. Using reducibility of them to the one-dimensional linear heat equation, we construct complete hierarchies of local and potential conservation laws for them and describe, in some sense, all their potential symmetries. Known results on the subject are interpreted in the proposed framework. This paper is an extended comment on the paper of Mei and Zhang [Int. J. Theor. Phys. 45: 2095–2102, 2006].     

Conservation Laws and Non-Lie Symmetries for Linear PDEs

Abstract

We introduce a method to construct conservation laws for a large class of linear partial differential equations. In contrast to the classical result of Noether, the conserved currents are generated by any symmetry of the operator, including those of the non-Lie type. An explicit example is made of the Dirac equation were we use our construction to find a class of conservation laws associated with a 64 dimensional Lie algebra of discrete symmetries that includes CPT.     

Higher-order symmetries and conservation laws of multi-dimensional Gordon-type equations

In this paper a class of multi-dimensional Gordon-type equations are analysed using a multiplier and homotopy approach to construct conservation laws. The main focus is the analysis of the classical versions of the Gordon-type equations and obtaining higher-order variational symmetries and corresponding conserved quantities. The results are extended to the multi-dimensional Gordon-type equations with the two-dimensional Klein–Gordon equation in particular yielding interesting results.     

Nonlocal Symmetries and Associated Conservation Laws for Wave Equations with Variable Speeds

We show that one can generate a class of nontrivial conservation laws for second-orderpartial differential equations using some recent results dealing with theaction of any Lie—Bäklund symmetry generator of the equivalent first-ordersystem on the respective conservation law. These conserved vectors are nonlocal asthey are constructed from associated nonlocal symmetries of the partial differentialequation. We demonstrate the complete procedure on certain classes of waveequations with variable wave speeds. Some of these have been considered in theliterature using alternative methods.     

CONSERVATION LAWS FOR SELF-ADJOINT FIRST-ORDER EVOLUTION EQUATION

We consider the problem on group classification and conservation laws for first-order evolution equations. Subclasses of these general equations which are quasi-self-adjoint and self-adjoint are obtained. By using the recent new conservation theorem due to Ibragimov, conservation laws for equations admiting self-adjoint equations are established. The results are illustrated applying them to the inviscid Burgers equation. In particular an infinite number of new symmetries of this equation are found.     

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.     

一类高阶非线性波方程的李群分析、最优系统、精确解和守恒律

本文运用李群分析的方法研究了一类高阶非线性波方程,得到了五阶非线性波方程的对称以及方程的最优系统,进而运用幂级数的方法,求得了方程的精确幂级数解.最后,给出了五阶非线性波方程的一些守恒律.     

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.     

Painlevé analysis,conservation laws,and symmetry of perturbed nonlinear equations

We consider the Lie-Backlund symmetries and conservation laws of a perturbed KdV equation and NLS equation. The arbitrary coefficients of the perturbing terms can be related to the condition of existence of nontrivial LB symmetry generator. When the perturbed KdV equation is subjected to Painlevé analysisa la Weiss, it is found that the resonance position changes compared to the unperturbed one. We prove the compatibility of the overdetermined set of equations obtained at the different stages of recursion relations, at least for one branch. All other branches are also indicated and difficulties associated them are discussed considering the perturbation parameter to be small. We determine the Lax pair for the aforesaid branch through the use of Schwarzian derivative. For the perturbed NLS equation we determine the conservation laws following the approach of Chen and Liu. From the recurrence of these conservation laws a Lax pair is constructed. But the Painlevé analysis does not produce a positive answer for the perturbed NLS equation. So here we have two contrasting examples of perturbed nonlinear equations: one passes the Painlevé test and its Lax pair can be found from the analysis itself, but the other equation does not meet the criterion of the Painlevé test, though its Lax pair is found in another way.     

Noether versus Killing symmetry of conformally flat Friedmann metric

In a recent study Noether symmetries of some static spacetime metrics in comparison with Killing vectors of corresponding spacetimes were studied. It was shown that Noether symmetries provide additional conservation laws that are not given by Killing vectors. In an attempt to understand how Noether symmetries compare with conformal Killing vectors, we find the Noether symmetries of the flat Friedmann cosmological model. We show that the conformally transformed flat Friedman model admits additional conservation laws not given by the Killing or conformal Killing vectors. Inter alia, these additional conserved quantities provide a mechanism to twice reduce the geodesic equations via the associated Noether symmetries.     

Conservation Laws of Differential Equations in Finance

Conservation laws of some differential equations in fiance are studied in this paper. This method does not involve the use or existence of a variational principle. As an alternative, linearize the given equation and find adjoint equation of the linearized equation, the conservation laws can be constructed directly from the symmetries and adjoint symmetries of the associated linearized equation and its adjoint equation.     

Affine symmetry,geodesics, and homogeneous spacetimes

We show that the conservation laws for the geodesic equation which are associated to affine symmetries can be obtained from symmetries of the Lagrangian for affinely parametrized geodesics according to Noether’s theorem, in contrast to claims found in the literature. In particular, using Aminova’s classification of affine motions of Lorentzian manifolds, we show in detail how affine motions define generalized symmetries of the geodesic Lagrangian. We compute all infinitesimal proper affine symmetries and the corresponding geodesic conservation laws for all homogeneous solutions to the Einstein field equations in four spacetime dimensions with each of the following energy–momentum contents: vacuum, cosmological constant, perfect fluid, pure radiation, and homogeneous electromagnetic fields.     

Conservation laws and symmetries of generalized sine-Gordon equations

We study some systems of non-linear PDE's (Eqs. 1.1 below) which can be regarded either as generalizations of the sine-Gordon equation or as two-dimensional versions of the Toda lattice equations. We show that these systems have an infinite number of non-trivial conservation laws and an infinite number of symmetries. The second result is deduced from the first by a variant of the Hamiltonian formalism for evolution equations. We also consider some specializations of the systems.     

Extended antifield formalism

The antifield formalism is extended so as to incorporate the rigid symmetries of a given theory. To that end, it is necessary to introduce global ghosts not only for the given rigid symmetries, but also for all the higher order conservation laws, associated with conserved antisymmetric tensors j^μ₁3μ_k fulfilling _μ1j^μ₁3μ_k 2˜ 0. Otherwise, one may encounter obstructions of the type discussed in by the authors. These higher order conservation laws are shown to define additional rigid symmetries of the master equation and to form — together with the standard symmetries — an interesting algebraic structure. They lead furthermore to independent Ward identities which are derived in the standard manner, because the resulting master (“Zinn-Justin”) equation capturing both the gauge symmetries and the rigid symmetries of all orders takes a known form. Issues such as anomalies or consistent deformations of the action preserving some set of rigid symmetries can be also systematically analysed in this framework.     

Gauge invariance,charge conservation,and variational principles

We present new results on the correspondence between symmetries, conservation laws and variational principles for field equations in general non-abelian gauge theories. Our main result states that second order field equations possessing translational and gauge symmetries and the corresponding conservation laws are always derivable from a variational principle. We also show by the way of examples that the above result fails in general for third order field equations.     

A simple-looking relative of the Novikov,Hirota-Satsuma and Sawada-Kotera equations

We study the simple-looking scalar integrable equation f_xxt 3( f_x f_t 1) = 0, which is related (in different ways) to the Novikov, Hirota-Satsuma and Sawada-Kotera equations. For this equation we present a Lax pair, a Bäcklund transformation, soliton and merging soliton solutions (some exhibiting instabilities), two infinite hierarchies of conservation laws, an infinite hierarchy of continuous symmetries, a Painlevé series, a scaling reduction to a third order ODE and its Painlevé series, and the Hirota form (giving further multisoliton solutions).