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.     

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.     

扰动Boussinesq方程的近似守恒律

构造了具有扰动项的Boussinesq方程的近似守恒向量和近似守恒律.在方程允许拉格朗日函数的情况下,利用欧拉方程的部分拉格朗日函数方法,研究了含有一阶线性组合扰动项的Boussineq方程的近似守恒律.给出了该方程的近似守恒向量及近似守恒律的分类结果.     

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.     

Conservation laws of a nonlinear (1+1) wave equation

In this paper, further study of the conservation laws of the nonlinear (1+1) wave equation involving two arbitrary functions of the dependent variable is performed. This equation is not derivable from a variational principle. By writing the equation, admitting a partial Lagrangian, in the partial Euler–Lagrange   form, partial Noether operators associated with the partial Lagrangian are obtained for all possible cases of the functions. These partial Noether operators do not form a Lie algebra in general. Partial Noether operators aid via a formula in the construction of the conservation laws of the equation. We obtain new conservation laws for the equation which have not been presented in the earlier literature.     

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.     

On the conservation laws and invariant solutions of the mKdV equation

In this paper, we consider modified Korteweg-de Vries (mKdV) equation. By using the nonlocal conservation theorem method and the partial Lagrangian approach, conservation laws for the mKdV equation are presented. It is observed that only nonlocal conservation theorem method lead to the nontrivial and infinite conservation laws. In addition, invariant solution is obtained by utilizing the relationship between conservation laws and Lie-point symmetries of the equation.     

Conservation laws of KdV equation with time dependent coefficients

We determine conservation laws of the generalized KdV equation of time dependent variable coefficients of the linear damping and dispersion. The underlying equation is not derivable from a variational principle and hence one cannot use Noether’s theorem here to construct conservation laws as there is no Lagrangian. However, we show that by utilizing the new conservation theorem and the partial Lagrangian approach one can construct a number of local and nonlocal conservation laws for the underlying equation.     

Geometry of conservation laws for a class of parabolic PDE's, II: Normal forms for equations with conservation laws

We consider conservation laws for second-order parabolic partial differential equations for one function of three independent variables. An explicit normal form is given for such equations having a nontrivial conservation law. It is shown that any such equation whose space of conservation laws has dimension at least four is locally contact equivalent to a quasi-linear equation. Examples are given of nonlinear equations that have an infinite-dimensional space of conservation laws parameterized (in the sense of Cartan-K?hler) by two arbitrary functions of one variable. Furthermore, it is shown that any equation whose space of conservation laws is larger than this is locally contact equivalent to a linear equation.     

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.     

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).     

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

We show how one can construct approximate conservation laws of approximate Euler-type equations via approximate Noether-type 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.     

New conservation laws obtained directly from symmetry action on a known conservation law

Two formulas are introduced to directly obtain new conservation laws for any system of partial differential equations from a known conservation law and admitted symmetries. The first formula maps any conservation law of a given system to the corresponding conservation law of the system obtained through a contact transformation. When the contact transformation is a symmetry of the given system, then the corresponding conservation law is a conservation law of the given system. The second formula checks a priori whether or not the action of a symmetry (continuous or discrete) on a conservation law can yield one or more new conservation laws of the given system. Several examples are considered, including the use of a discrete symmetry to obtain a new conservation law and the use of a continuous symmetry to generate two new conservation laws.     

Nonlinear self-adjointness and conservation laws for a generalized Fisher equation

In this work we study a generalization of the well known Fisher equation. We determine the subclasses of these equations which are nonlinear self-adjoint. By using a general theorem on conservation laws proved by Nail Ibragimov and the symmetry generators we find conservation laws for these partial differential equations without classical Lagrangians.     

Self-adjoint sub-classes of generalized thin film equations

In this work we consider a class of fourth-order nonlinear partial differential equation containing several un-specified coefficient functions of the dependent variable which encapsulates various mathematical models used, e.g. for describing the dynamics of thin liquid films. We determine the subclasses of these equations which are self-adjoint. By using a general theorem on conservation laws proved by one of the authors (NHI) we find conservation laws for some of these partial differential equations without classical Lagrangians.     

Characteristics of Conservation Laws for Difference Equations

Each conservation law of a given partial differential equation is determined (up to equivalence) by a function known as the characteristic. This function is used to find conservation laws, to prove equivalence between conservation laws, and to prove the converse of Noether’s Theorem. Transferring these results to difference equations is nontrivial, largely because difference operators are not derivations and do not obey the chain rule for derivatives. We show how these problems may be resolved and illustrate various uses of the characteristic. In particular, we establish the converse of Noether’s Theorem for difference equations, we show (without taking a continuum limit) that the conservation laws in the infinite family generated by Rasin and Schiff are distinct, and we obtain all five-point conservation laws for the potential Lotka–Volterra equation.     

Conservation laws for laminar axisymmetric jet flows with weak swirl

The conservation laws for laminar axisymmetric jet flows with weak swirl are studied here. The multiplier approach is used to derive the conservation laws for the system of three boundary layer equations for the velocity components governing flow in laminar axisymmetric jet flows with weak swirl. Conservation laws for the system of two partial differential equations for the stream function are also derived.     

Essential Conservation Laws for Equations with Infinite Symmetries

We consider partial differential equations of variational problems with infinite symmetry groups. We study local conservation laws associated with arbitrary functions of one variable in the group generators. We show that only symmetries with arbitrary functions of dependent variables lead to an infinite number of conservation laws. We also calculate local conservation laws for the potential Zabolotskaya-Khokhlov equation for one of its infinite subgroups.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 190–198, July, 2005.     

Conservation laws for some compacton equations using the multiplier approach

This paper is an application of the variational derivative method to the derivation of the conservation laws for partial differential equations. The conservation laws for (1+1) dimensional compacton k(2,2) and compacton k(3,3) equations are studied via multiplier approach. Also the conservation laws for (2+1) dimensional compacton Zk(2,2) equation are established by first computing the multipliers.     

Conservation Laws For the (1+2)-Dimensional Wave Equation in Biological Environment

The derivation of conservation laws for the wave equation on sphere, cone and flat space is considered. The partial Noether approach is applied for wave equation on curved surfaces in terms of the coefficients of the first fundamental form (FFF) and the partial Noether operator's determining equations are derived. These determining equations are then used to construct the partial Noether operators and conserved vectors for the wave equation on different surfaces. The conserved vectors for the wave equation on the sphere, cone and flat space are simplified using the Lie point symmetry generators of the equation and conserved vectors with the help of the symmetry conservation laws relation.