Soil water redistribution and extraction flow models: Conservation laws

We determine all the nontrivial conservation laws for soil water redistribution and extraction flow equations which are modelled by a class of (2+1) nonlinear evolution partial differential equations with three arbitrary elements. It is shown that for arbitrary elements in the model equation there exist trivial conservation laws. We point out that nontrivial conservation laws exist for certain classes of equations which admit point symmetries.     

Conservation Laws and Potential Symmetries of Linear Parabolic Equations

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.     

New exact solutions of a generalised Boussinesq equation with damping term and a system of variant Boussinesq equations via double reduction theory

The conservation laws of a generalised Boussinesq (GB) equation with damping term are derived via the partial Noether approach. The derived conserved vectors are adjusted to satisfy the divergence condition. We use the definition of the association of symmetries of partial differential equations with conservation laws and the relationship between symmetries and conservation laws to find a double reduction of the equation. As a result, several new exact solutions are obtained. A similar analysis is performed for a system of variant Boussinesq (VB) equations.     

Zeroth-order conservation laws of two-dimensional shallow water equations with variable bottom topography

We classify zeroth-order conservation laws of systems from the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The classification is carried out up to equivalence generated by the equivalence group of this class. We find additional point equivalences between some of the listed cases of extensions of the space of zeroth-order conservation laws, which are inequivalent up to transformations from the equivalence group. Hamiltonian structures of systems of shallow water equations are used for relating the classification of zeroth-order conservation laws of these systems to the classification of their Lie symmetries. We also construct generating sets of such conservation laws under action of Lie symmetries.     

Conservation laws for the Black–Scholes equation

In the search for solutions to the important partial differential equation due to Black, Scholes and Merton potential symmetries are very useful as new solutions of the equation can be obtained as a result. These potential symmetries require that the equation be written in conserved form, ie. we need to determine conservation laws for the equation. We calculate the conservation laws utilizing the point symmetries of the equation following the method of Kara and Mahomed [A.H. Kara, F.M. Mahomed, The relationship between symmetries and conservation laws, Int. J. Theor. Phys. 39 (2000) 23–40].     

Symmetries, Conservation Laws, and Cohomology of Maxwell's Equations Using Potentials

New nonlocal symmetries and conservation laws are derived for Maxwell's equations in 3 + 1 dimensional Minkowski space using a covariant system of joint vector potentials for the electromagnetic tensor field and its dual. A key property of this system, as well as of this class of new symmetries and conservation laws, is their invariance under the duality transformation that exchanges the electromagnetic field with its dual. (In contrast the standard potential system using a single vector potential is not duality-invariant.) The nonlocal symmetries of Maxwell's equations come from an explicit classification of all symmetries of a certain natural geometric form admitted by the joint potential system in Lorentz gauge. In addition to scaling and duality-rotation symmetries, and the well-known Poincaré and dilation symmetries which involve homothetic Killing vectors, the classification yields new geometric symmetries involving Killing–Yano tensors related to rotations/boosts and inversions. The nonlocal conservation laws of Maxwell's equations are constructed from these geometric symmetries by applying a conserved current formula that uses the joint potentials and directly generates conservation laws from any (local or nonlocal) symmetries of Maxwell's equations. This formula is shown to arise through a series of mappings that relate, respectively, symmetries/adjoint-symmetries of the joint potential system and adjoint-symmetries/symmetries of Maxwell's equations. The mappings are derived as by-products of the study of cohomology of closed one-forms and two-forms locally constructed from the electromagnetic field and its derivatives to any finite order for all solutions of Maxwell's equations. In particular it is shown that the only nontrivial cohomology consists of the electromagnetic field (two-form) itself as well as its dual (two-form), and that this two-form cohomology is killed by the introduction of corresponding potentials.     

Lie symmetries and conservation laws of the Fokker-Planck equation with power diffusion

We concentrate on Lie symmetries and conservation laws of the Fokker-Planck equation with power diffusion describing the growth of cell populations. First, we perform a complete symmetry classification of the equation, and then we find some interesting similarity solutions by means of the symmetries and the variable coefficient heat equation. Local dynamical behaviors are analyzed via the solutions for the growing cell populations. Second, we show that the conservation law multipliers of the equation take the form Λ=Λ(t,x,u), which satisfy a linear partial differential equation, and then give the general formula of conservation laws. Finally, symmetry properties of the conservation law are investigated and used to construct conservation laws of the reduced equations.     

Conservation laws for a variable coefficient nonlinear diffusion–convection–reaction equation

We find the Lie point symmetries of a class of second-order nonlinear diffusion–convection–reaction equations containing an unspecified coefficient function of the independent variable t and determine the subclasses of these equations which are nonlinearly self-adjoint. By using a general theorem on conservation laws proved recently by N.H. Ibragimov we establish conservation laws corresponding to the aforementioned Lie point symmetries, one by one, for the simultaneous system of the original equation together with its adjoint equation through a formal Lagrangian. Particularly, for the nonlinearly self-adjoint subclasses, we construct conservation laws for the corresponding equations themselves.     

Nonlinear self-adjointness through differential substitutions

It is known (Ibragimov, 2011; Galiakberova and Ibragimov, 2013) [14,18] that the property of nonlinear self-adjointness allows to associate conservation laws of the equations under study, with their symmetries. In this paper we show that, even when the equation is nonlinearly self-adjoint with a non differential substitution, finding the explicit form of the differential substitution can provide new conservation laws associated to its symmetries. By using the general theorem on conservation laws (Ibragimov, 2007) [11] and the property of nonlinear self-adjointness we find some new conservation laws for the modified Harry-Dym equation. By using a differential substitution we construct a conservation law for the Harry-Dym equation, which has not been derived before using Ibragimov method.     

On Noether’s Theorem for the Euler–Poincaré Equation on the Diffeomorphism Group with Advected Quantities

We show how Noether conservation laws can be obtained from the particle relabelling symmetries in the Euler–Poincaré theory of ideal fluids with advected quantities. All calculations can be performed without Lagrangian variables, by using the Eulerian vector fields that generate the symmetries, and we identify the time-evolution equation that these vector fields satisfy. When advected quantities (such as advected scalars or densities) are present, there is an additional constraint that the vector fields must leave the advected quantities invariant. We show that if this constraint is satisfied initially then it will be satisfied for all times. We then show how to solve these constraint equations in various examples to obtain evolution equations from the conservation laws. We also discuss some fluid conservation laws in the Euler–Poincaré theory that do not arise from Noether symmetries, and explain the relationship between the conservation laws obtained here, and the Kelvin–Noether theorem given in Sect. 4 of Holm et al. (Adv. Math. 137:1–81, 1998).     

Conservation laws for one-layer shallow water wave systems

The problem of correspondence between symmetries and conservation laws for one-layer shallow water wave systems in the plane flow, axisymmetric flow and dispersive waves is investigated from the composite variational principle of view in the development of the study [N.H. Ibragimov, A new conservation theorem, Journal of Mathematical Analysis and Applications, 333(1) (2007) 311–328]. This method is devoted to construction of conservation laws of non-Lagrangian systems. Composite principle means that in addition to original variables of a given system, one should introduce a set of adjoint variables in order to obtain a system of Euler–Lagrange equations for some variational functional. After studying Lie point and Lie–Bäcklund symmetries, we obtain new local and nonlocal conservation laws. Nonlocal conservation laws comprise nonlocal variables defined by the adjoint equations to shallow water wave systems. In particular, we obtain infinite local conservation laws and potential symmetries for the plane flow case.     

扩充偏微分方程(组)守恒律和对称的辅助方程方法及微分形式吴方法的应用

首先,我们给出了引入伴随方程(组)扩充原方程(组)的策略使给定偏微分方程(组)的扩充方程组具有对应泛瓯即,成为Lagrange系统的方法,以此为基础提出了作为偏微分方程(组)传统守恒律和对称概念的一种推广-偏微分方程(组)扩充守恒律和扩充对称的概念;其次,以得到的Lagrange系统为基础给定了确定原方程(组)扩充守恒律和扩充对称的方法,从而达到扩充给定偏微分方程(组)的首恒律和对称的目的;第三,提出了适用于一般形式微分方程(组)的计算固有守恒律的方法;第四,实现以上算法过程中,我们先把计算(扩充)守恒律和对称问题均归结为求解超定线性齐次偏微分方程组(确定方程组)的问题.然后,对此关键问题我们提出了用微分形式吴方法处理的有效算法;最后,作为方法的应用我们计算确定了非线性电报方程组在内的五个发展方程(组)的新守恒律和对称,同时也说明了方法的有效性.     

Evolution systems with constraints in the form of zero-divergence conditions

We study evolution systems of partial differential equations in the presence of consistent constraints having the form of a system of continuity equations. We show that in addition to possible conservation laws of the standard degree equal to the number of spatial variables, each such system has conservation laws whose degree is one less than this number. We begin by completely describing the conservation laws and symmetries of the system of continuity equations. As an example, we calculate the second-degree conservation laws for the classical system of Maxwell’s equations (the number of spatial variables is three here).     

Discrete equation on a square lattice with a nonstandard structure of generalized symmetries

We clarify the integrability nature of a recently found discrete equation on the square lattice with a nonstandard symmetry structure. We find its L-A pair and show that it is also nonstandard. For this discrete equation, we construct the hierarchies of both generalized symmetries and conservation laws. This equation yields two integrable systems of hyperbolic type. The hierarchies of generalized symmetries and conservation laws are also nonstandard compared with known equations in this class.     

Group analysis,exact solutions and conservation laws of a generalized fifth order KdV equation

We study the generalized fifth order KdV equation using group methods and conservation laws. All of the geometric vector fields of the special fifth order KdV equation are presented. By using the nonclassical Lie group method, it is show that this equation does not admit nonclassical type symmetries. Then, on the basis of the optimal system, the symmetry reductions and exact solutions to this equation are constructed. For some special cases, we obtain additional nontrivial conservation laws and scaling symmetries.     

On double reductions from symmetries and conservation laws for a damped Boussinesq equation

In this work, we study a Boussinesq equation with a strong damping term from the point of view of the Lie theory. We derive the classical Lie symmetries admitted by the equation as well as the reduced ordinary differential equations. Some nontrivial conservation laws are derived by using the multipliers method. Taking into account the relationship between symmetries and conservation laws and applying the double reduction method, we obtain a direct reduction of order of the ordinary differential equations and in particular a kink solution.     

Relations between symmetries and conservation laws for difference systems

This paper presents a relation between divergence variational symmetries for difference variational problems on lattices and conservation laws for the associated Euler–Lagrange system provided by Noether's theorem. This inspires us to define conservation laws related to symmetries for arbitrary difference equations with or without Lagrangian formulations. These conservation laws are constrained by partial differential equations obtained from the symmetries generators. It is shown that the orders of these partial differential equations have been reduced relative to those used in a general approach. Illustrative examples are presented.     

Approximate conservation laws of nonlinear perturbed heat and wave equations

We construct approximate conservation laws for non-variational nonlinear perturbed (1+1) heat and wave equations by utilizing the partial Lagrangian approach. These perturbed nonlinear heat and wave equations arise in a number of important applications which are reviewed. Approximate symmetries of these have been obtained in the literature. Approximate partial Noether operators associated with a partial Lagrangian of the underlying perturbed heat and wave equations are derived herein. These approximate partial Noether operators are then used via the approximate version of the partial Noether theorem in the construction of approximate conservation laws of the underlying perturbed heat and wave equations.