Conservation laws for a complexly coupled KdV system,coupled Burgers’ system and Drinfeld–Sokolov–Wilson system via multiplier approach

The multiplier approach (variational derivative method) is used to derive the conservation laws for some nonlinear systems of partial differential equations. Firstly, the multipliers (characteristics) are computed and then conserved vectors are obtained for the each multiplier. Examples of the third-order complexly coupled KdV system, second-order coupled Burgers’ system and third-order Drinfeld–Sokolov–Wilson system are considered. For all three systems the local conservation laws are established by utilizing the multiplier approach.     

Conservation laws for Camassa–Holm equation, Dullin–Gottwald–Holm equation and generalized Dullin–Gottwald–Holm equation

In this paper we construct the conservation laws for the Camassa–Holm equation, the Dullin–Gottwald–Holm equation (DGH) and the generalized Dullin–Gottwald–Holm equation (generalized DGH). The variational derivative approach is used to derive the conservation laws. Only first order multipliers are considered. Two multipliers are obtained for the Camassa–Holm equation. For the DGH and generalized DGH equations the variational derivative approach yields two multipliers; thus two conserved vectors are obtained.     

非线性对流扩散方程的守恒律

利用直接方法研究了非线性对流扩散方程的守恒律,得到了关于非线性对流扩散方程的守恒律乘子性质的一个定理.利用这个定理,可以简化守恒律乘子的确定方程.随后通过对确定方程中的变量函数进行分析,发现在四种情况下乘子的确定方程是可解的.最后解出这些守恒律乘子,利用积分公式法分别得到了四种情况下对应于各个守恒律乘子的守恒律.     

New conservation laws of some third-order systems of pdes arising from higher-order multipliers

In this paper, we study conservation laws for some third-order systems of pdes, viz., some versions of the Boussinesq equations and the BBM equation. It is shown that new and interesting conserved quantities arise from ‘multipliers’ that are of order greater than one in derivatives of the dependent variables. Furthermore, the invariance properties of the conserved flows with respect to the Lie point symmetry generators are investigated via the symmetry action on the multipliers.     

Conservation laws and exact solutions of the Whitham-type equations

In this paper, we study conservation laws for some partial differential equations. It is shown that interesting conserved quantities arise from multipliers by using homotopy operator that is a powerful algorithmic tool. Furthermore, the invariance properties of the conserved flows with respect to the Lie point symmetry generators are investigated via the symmetry action on the multipliers. Furthermore, the similarity reductions and some exact solutions are provided.     

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.     

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.     

A non-variational approach to the construction of new ‘higher-order’ conservation laws of the family of nonlinear equations α(ut + 3uux) + β(utxx + 2uxuxx + uuxxx) − γuxxx = 0

In this paper, we study and classify the conservation laws of the combined nonlinear KdV, Camassa–Holm, Hunter–Saxton and the inviscid Burgers equation which arises in, inter alia, shallow water equations. It is shown that these can be obtained by variational methods but the main focus of the paper is the construction of the conservation laws as a consequence of the interplay between symmetry generators and ‘multipliers’, particularly, the higher-order ones.     

Local conservation laws,symmetries, and exact solutions for a Kudryashov‐Sinelshchikov equation

In this paper, we consider a Kudryashov‐Sinelshchikov equation that describes pressure waves in a mixture of a liquid and gas bubbles taking into consideration the viscosity of liquid and the heat transfer between liquid and gas bubbles. We show that this equation is rich in conservation laws. These conservation laws have been found by using the direct method of the multipliers. We apply the Lie group method to derive the symmetries of this equation. Then, by using the optimal system of 1‐dimensional subalgebras we reduce the equation to ordinary differential equations. Finally, some exact wave solutions are obtained by applying the simplest equation method.     

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.     

Conservation laws and conserved quantities for laminar two-dimensional and radial jets

A systematic way to derive the conserved quantities for the liquid jet, free jet and wall jet using conservation laws is presented. Both two-dimensional and radial jets are considered. The jet flows are described by Prandtl’s momentum boundary layer equation and the continuity equation. The multiplier approach (also know as variational derivative approach) is first applied to construct a basis of conserved vectors for the system. The basis consists of two conserved vectors. By integrating the corresponding conservation laws across the jet and imposing the boundary conditions, conserved quantities are derived for the liquid jet and the free jet. The multiplier approach is then applied to construct a basis of conserved vectors for the third-order partial differential equation for the stream function. The basis consists of two local conserved vectors one of which is a non-local conserved vector for the system. The conserved quantities for the free jet and the wall jet are derived from the corresponding conservation laws and boundary conditions. The approach gives a unified treatment to the derivation of conserved quantities for jet flows and may lead to a new classification of jets through conserved vectors and their multipliers.     

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.     

Entropy stable shock capturing space–time discontinuous Galerkin schemes for systems of conservation laws

We present a streamline diffusion shock capturing spacetime discontinuous Galerkin (DG) method to approximate nonlinear systems of conservation laws in several space dimensions. The degrees of freedom are in terms of the entropy variables and the numerical flux functions are the entropy stable finite volume fluxes. We show entropy stability of the (formally) arbitrarily high order accurate method for a general system of conservation laws. Furthermore, we prove that the approximate solutions converge to the entropy measure valued solutions for nonlinear systems of conservation laws. Convergence to entropy solutions for scalar conservation laws and for linear symmetrizable systems is also shown. Numerical experiments are presented to illustrate the robustness of the proposed schemes.     

On first-order conservation laws for systems of hydrodynamic-type equations

First-order conservation laws quadratic in derivatives are considered for systems of hydrodynamic-type equations. Defining relationships for the densities of such conservation laws are derived in a form that is invariant with respect to pointwise changes of the variables. Examples of nondiagonalizable systems admitting quadratic conservation laws are given.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 1, pp. 109–128, July, 1996.     

Construction of Approximate Entropy Measure-Valued Solutions for Hyperbolic Systems of Conservation Laws

Entropy solutions have been widely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in De Lellis and Székelyhidi Jr (Ann Math 170(3):1417–1436, 2009) and Chiodaroli et al. (2013) have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that state-of-the-art numerical schemes need not converge to an entropy solution of systems of conservation laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of conservation laws, particularly in several space dimensions. We advocate entropy measure-valued solutions, first proposed by DiPerna, as the appropriate solution paradigm for systems of conservation laws. To this end, we present a detailed numerical procedure which constructs stable approximations to entropy measure-valued solutions, and provide sufficient conditions that guarantee that these approximations converge to an entropy measure-valued solution as the mesh is refined, thus providing a viable numerical framework for systems of conservation laws in several space dimensions. A large number of numerical experiments that illustrate the proposed paradigm are presented and are utilized to examine several interesting properties of the computed entropy measure-valued solutions.     

Self-adjointness and conservation laws of two variable coefficient nonlinear equations of Schrödinger type

In this paper, some recent concepts and results on self-adjointness and conservation laws are applied to two variable coefficient nonlinear equations of Schrödinger type: the generalized variable coefficient nonlinear Schrödinger (GVCNLS) equation and the cubic-quintic nonlinear Schrödinger (CQNLS) equation with variable coefficients. The two equations are changed to two real systems by a proper transformation. To obtain the formal Lagrangians of the two systems, we discuss their self-adjointness and find that the GVCNLS system is weak self-adjoint and the CQNLS system is quasi self-adjoint. Having performed Lie symmetry analysis for the two systems, we find five nontrivial conservation laws for the GVCNLS system and four nontrivial conservation laws for the CQNLS system by using a general theorem on conservation laws given by Ibragimov.     

Conservation laws and exact solutions of a Generalized Benjamin–Bona–Mahony–Burgers equation

The concept of nonlinear self-adjointness given by Ibragimov is applied to a Generalized Benjamin–Bona–Mahony–Burgers equation. Then, a nonlinear self-adjoint classification has been achieved. Moreover, some nontrivial conservation laws are constructed by using the multipliers method which does not require the use of a variational principle. Finally, by applying the modified simplest equation method we derive new travelling wave solutions.     

Finite-difference methods for one-dimensional hyperbolic conservation laws

This article contains a survey of some important finite-difference methods for one-dimensional hyperbolic conservation laws. Weak solutions of hyperbolic conservation laws are introduced and the concept of entropy stability is discussed. Furthermore, the Riemann problem for hyperbolic conservation laws is solved. An introduction to finite-difference methods is given for which important concepts such as, e.g., conservativity, stability, and consistency are introduced. Godunov-type methods are elaborated for general systems of hyperbolic conservation laws. Finally, flux limiter methods are developed for the scalar nonlinear conservation law. © 1994 John Wiley & Sons, Inc.     

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.     

Multi‐dimensional conservation laws and integrable systems

In this paper, we introduce a new property of two‐dimensional integrable hydrodynamic chains—existence of infinitely many local three‐dimensional conservation laws for pairs of integrable two‐dimensional commuting flows. Infinitely many local three‐dimensional conservation laws for the Benney commuting hydrodynamic chains are constructed. As a by‐product, we established a new method for computation of local conservation laws for three‐dimensional integrable systems. The Mikhalëv equation and the dispersionless limit of the Kadomtsev‐Petviashvili equation are investigated. All known local and infinitely many new quasilocal three‐dimensional conservation laws are presented. Also four‐dimensional conservation laws are considered for couples of three‐dimensional integrable quasilinear systems and for triplets of corresponding hydrodynamic chains.