Class Of Conservation Laws Research Articles

On the symmetry and conservation law classification of the de Sitter–Schwarzschild metric and the corresponding wave and Klein–Gordon equations

The main purpose of this work is to focus on a discussion of Lie symmetries admitted by de Sitter–Schwarzschild spacetime metric, and the corresponding wave or Klein–Gordon equations constructed in the de Sitter–Schwarzschild geometry. The obtained symmetries are classified and the variational (Noether) conservation laws associated with these symmetries via the natural Lagrangians are obtained. In the case of the metric, we obtain additional variational ones when compared with the Killing vectors leading to additional conservation laws and for the wave and Klein–Gordon equations, the variational symmetries involve less tedious calculations as far as invariance studies are concerned.

Conservation laws and symmetry analysis for a quasi-linear strongly-damped wave equation

In this paper, it is considered a quasi-linear strongly-damped wave equation defined by a non-linear partial differential equation of third order. The equation describes motions of viscoelastic solids. We study the conservation laws of this equation. By applying the multiplier method of Anco and Bluman to the equation, we find the multipliers. Consequently, we obtain a complete classification of conservation laws. Moreover, we use the Lie-group theory to analyse the symmetries of the equation. From the Lie symmetries, all the reductions are determined. Afterwards, we construct exact solutions with physical interest: travelling wave solutions.

Lie symmetries and conservation laws for a generalized (2+1)-dimensional nonlinear evolution equation

This paper considers a generalized (2+1) dimensional nonlinear evolution equation depending on two nonzero arbitrary constants. We derive the Lie point symmetry generators and Lie symmetry groups. This symmetry analysis leads us the reductions equations, through one of which we obtain solutions. We also get the low-order conservation laws of the equation that have been obtained using the corresponding symmetries of the family. We will present a classification of conservation laws for this equation and we will apply Lie symmetry analysis to the equation in order to obtain exact solutions.

Invariant subspaces, exact solutions and classification of conservation laws for a coupled (1+1)-dimensional nonlinear Wu-Zhang equation

In this work, we apply the invariant subspace method to derive a set of invariant subspaces and solutions of the nonlinear Wu–Zhang equation which describes the dynamic behavior of dispersive long waves in fluid dynamics. The method gives logarithmic and polynomial solutions of the equation. Furthermore, the multipliers approach and new conservation theorem are employed to derive a set of conservation laws of the equation which are to the best of our knowledge reported for the first time in this work. The physical structure of the results is shown by figures of some special solutions in order to give us a better interpretation on the evolution of the solutions.

Convergence of the finite volume method on a Schwarzschild background

We introduce a class of nonlinear hyperbolic conservation laws on a Schwarzschild black hole background and derive several properties satisfied by (possibly weak) solutions. Next, we formulate a numerical approximation scheme which is based on the finite volume methodology and takes the curved geometry into account. An interesting feature of our model is that no boundary conditions is required at the black hole horizon boundary. We establish that this scheme converges to an entropy weak solution to the initial value problem and, in turn, our analysis also provides us with a theory of existence and stability for a new class of conservation laws.

Optical solitary waves and conservation laws to the (2 + 1)-dimensional hyperbolic nonlinear Schrödinger equation

This work studies the hyperbolic nonlinear Schrödinger equation (H-NLSE) in (2 + 1)-dimensions. The model describes the evolution of the elevation of water wave surface for slowly modulated wave trains in deep water in hydrodynamics, and also governs the propagation of electromagnetic fields in self-focusing and normally dispersive planar wave guides in optics. A class of gray and black optical solitary wave solutions of the H-NLSE are reported by adopting an appropriate solitary wave ansatz solution. Moreover, classification of conservation laws (Cls) to the H-NLSE is implemented using the multipliers approach. Some physical interpretations and analysis of the results obtained are also presented.

Dynamics of optical solitons, multipliers and conservation laws to the nonlinear schrödinger equation in (2+1)-dimensions with non-Kerr law nonlinearity

ABSTRACTThis work studies the (2 + 1)-dimensional nonlinear Schrödinger equation which arises in optical fibre. Grey and black optical solitons of the model are reported using a suitable complex envelope ansatz solution. The integration lead to some certain conditions which must be satisfied for the solitons to exist. On applying the Chupin Liu's theorem to the grey and black optical solitons, we construct new sets of combined optical soliton solutions of the model. Moreover, classification of conservation laws (Cls) of the model is implemented using the multipliers approach. This is achieved by constructing a set of first-order multipliers of a system of nonlinear partial differential equations acquired by transforming the model into real and imaginary components are derived, which are subsequently used to construct the Cls.

On the classification of conservation laws and soliton solutions of the long short-wave interaction system

In this paper, the classification of conservation laws (Cls) of the long short-wave interaction system (LSWS) which appears in fluid mechanics as well as plasma physics is implemented using two Cls theorems, namely, the multipliers approach and the new conservation theorem. The LSWS describes the interaction between one long longitudinal wave and one short transverse wave propagating in a generalized elastic medium. The zeroth-order multipliers and the nonlinear self-adjoint substitutions of the model are derived. Considering the fact that the new conservation theorem needs Lie point symmetries in constructing Cls, we derive the point symmetries of a system of nonlinear partial differential equations (NPDEs) acquired by transforming the model into real and imaginary components. Moreover, we derive some kink-type, bell-shaped, singular and combined soliton solutions to the model using the powerful sine-Gordon expansion method (SGEM). Some figures are presented to show the physical interpretations of the acquired results.

Conservative parameterization schemes

Parameterization (closure) schemes in numerical weather and climate prediction models account for the effects of physical processes that cannot be resolved explicitly by these models. Methods for finding physical parameterization schemes that preserve conservation laws of systems of differential equations are introduced. These methods rest on the possibility to regard the problem of finding conservative parameterization schemes as a conservation law classification problem for classes of differential equations. The relevant classification problems can be solved using the direct or inverse classification procedures. In the direct approach, one starts with a general functional form of the parameterization scheme. Specific forms are then found so that corresponding closed equations admit conservation laws. In the inverse approach, one seeks parameterization schemes that preserve one or more pre-selected conservation laws of the initial model. The physical interpretation of both classification approaches is discussed. Special attention is paid to the problem of finding parameterization schemes that preserve both conservation laws and symmetries. All methods are illustrated by finding conservative and invariant conservative parameterization schemes for systems of one-dimensional shallow-water equations.

SYMMETRY AND CONSERVATION LAW CLASSIFICATION AND EXACT SOLUTIONS TO THE GENERALIZED KdV TYPES OF EQUATIONS

In this paper, complete geometric symmetry and conservation law classification of the generalized KdV types of equations are investigated. All of the geometric vector fields and second-order multipliers for the equations are obtained, and the corresponding conservation laws of the equations are presented explicitly. These comprise all of the second-order conservation laws for the equations. Furthermore, an analytic method is developed for dealing with the exact solutions to the generalized nonlinear partial differential equations with composite function terms.

Conservation Law Classification and Integrability of Generalized Nonlinear Second-Order Equation

This paper is concerned with the generalized nonlinear second-order equation. By the direct construction method, all of the first-order multipliers of the equation are obtained, and the corresponding complete conservation laws (CLs) of such equations are provided. Furthermore, the integrability of the equation is considered in terms of the conservation laws. In addition, the relationship of multipliers and symmetries of the equations is investigated.

Conservation laws for a vortex in a compressible nonequilibrium medium

The problem of conservation of magnitudes is considered for a vortex in a relaxing compressible medium. Heat release due to the relaxation of a nonequilibrium medium leads to the propagation of compression waves, which remove material. Traditional integrals of motion are inapplicable in this case. We pro-pose the concept of integral quantity, which is conserved with an arbitrary degree of accuracy despite the fact that waves cross the boundary of the integration domain. Based on this concept, a broad class of conservation laws is derived for axisymmetric disturbances of columnar vortices, including conservation of the circulation and total angular momentum of the vortex. For nonaxisymmetric disturbances, it is shown that the total angular momentum and properly defined energy integral are conserved. Numerical verification of the derived conservation laws is performed and the perspectives for using these conservation laws in numerical simulations are discussed.

The Noether Conservation Laws of Some Vaidiya Metrics

In this paper, we show that a large amount information can be extracted from a knowledge of the vector fields that leave the action integral invariant, viz., Noether symmetries. In addition to a larger class of conservation laws than those given by the isometries or Killing vectors, we may conclude what the isometries are and that these form a Lie subalgebra of the Noether symmetry algebra. We perform our analysis on versions of the Vaidiya metric yielding some previously unknown information regarding the corresponding manifold. Lastly, with particular reference to this metric, we show that the only variations on m(u) that occur are m=0, m=constant, m=u and m=m(u).

Physical consequences of action conservation laws and their applications

A class of conservation laws containing Hamilton's action integral is introduced for Lagrangian dynamical systems with a single degree of freedom and for the case when the Lagrangian function depends on the second time derivative of the coordinate. The action conservation laws are derived from the invariant properties of the Lagrange-D'Alembert differential variational principle with respect to infinitesimal transformations of the generalized coordinate and time by supposing that the generators of infinitesimal transformations depend on time, a generalized coordinate, and its first and second derivatives with respect to time. These action integral conservation laws are applied to the stability of columns, heat transfer, Thomas-Fermi problems, and other physical phenomena. A direct method for the approximate solution of these problems is combined with the Ritz variational method in order to obtain results of high accuracy.

Conservation Laws and Non-Lie Symmetries for Linear PDEs

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.

Conservation laws of multidimensional diffusion--convection equations

All possible linearly independent local conservation laws for n-dimensional diffusion–convection equations u t=(A(u)) ii +(B i(u)) i were constructed using the direct method and the composite variational principle. Application of the method of classification of conservation laws with respect to the group of point transformations [R.O.~Popovych, N.M. Ivanova, J. Math. Phys. 46, 2005, 043502 (math-ph/0407008)] allows us to formulate the result in explicit closed form. Action of the symmetry groups on the conservation laws of diffusion equations is investigated and generating sets of conservation laws are constructed.

Conservation laws for nonlinear telegraph equations

A complete conservation law classification is given for nonlinear telegraph (NLT) systems with respect to multipliers that are functions of independent and dependent variables. It turns out that a very large class of NLT systems admits four nontrivial local conservation laws. The results of this work are summarized in tables which display all multipliers, fluxes and densities for the corresponding conservation laws. A physical example is considered for possible applications.

On the symmetry approach to polynomial conservation laws of one-dimensional Lagrangian systems

In this paper we analyze polynomial conservation laws of one-dimensional non-autonomous Lagrangian dynamical systems x ̈ =−∂ Π(t,x)/∂x . The analysis is based upon application of Noether's theorem which relates the existence of conservation laws to the symmetries of Hamilton's action integral. It is shown that the existence of first integrals depends on the solution of the system of first-order partial differential equations — generalized Killing's equations. General solution of the problem is formally determined. It is demonstrated that the final form of dynamical system and corresponding conservation law depends on the solution of the so-called potential equation. However, the structure of symmetry transformations, which generate particular class of conservation laws, could be prescribed independent of the solution of potential equation. This fact is used to underline phenomenological aspect of symmetry approach. Its pragmatic value is confirmed through several concrete examples.

Conservation integrals in couple stress elasticity

Noether’s theorem on invariant variational principles is applied in the case of infinitesimal couple stress elasticity, thereby extending the analysis of Knowles and Sternberg (1972. On a class of conservation laws in linearized and finite elastostatics. Arch. Ration. Mech. Anal. 44, 187–211) beyond the range of classical elasticity. Two conserved integral quantities are deduced which generalize the J-integral and L-integral in the notation of Budiansky and Rice (1973: Budiansky, B. and Rice, J. R. (1973) Conservation laws and energy-release rates. J. Appl. Mech. 40, 201–203). An expression for an M-integral is also obtained, but it is demonstrated that there is no corresponding conservation law for this integral. Relationships of the derived path integrals to other similar quantities for couple stress elasticity which have appeared in the literature are discussed.

Classification of conservation laws for KdV-like equations

We prove that the computation of the conservation laws of (2 n + 1)th-order KdV-like equations (i.e. higher order evolution equations with the same scaling as KdV) can be restricted to polynomials with constant terms, except when the order of the conservation law equals n − 1, in which case the density has linear t dependence. This shows that existing computer algebra programs which assume the conservation law to be of this form are providing the complete answer.