Weak self-adjointness and conservation laws for a porous medium equation

The concepts of self-adjoint and quasi self-adjoint equations were introduced by Ibragimov (2006, 2007) [4], [7]. In Ibragimov (2007) [6] a general theorem on conservation laws was proved. In Gandarias (2011) [3] we generalized the concept of self-adjoint and quasi self-adjoint equations by introducing the definition of weak self-adjoint equations. In this paper we find the subclasses of weak self-adjoint porous medium equations. By using the property of weak self-adjointness we construct some conservation laws associated with symmetries of the differential equation.     

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 the nonlinear self-adjointness of the Zakharov–Kuznetsov equation

A new general theorem, which does not require the existence of Lagrangians, allows to compute conservation laws for an arbitrary differential equation. This theorem is based on the concept of self-adjoint equations for nonlinear equations. In this paper we show that the Zakharov–Kuznetsov equation is self-adjoint and nonlinearly self-adjoint. This property is used to compute conservation laws corresponding to the symmetries of the equation. In particular the property of the Zakharov–Kuznetsov equation to be self-adjoint and nonlinearly self-adjoint allows us to get more conservation laws.     

Nonlinear self-adjointness of the Krichever–Novikov equation

It is known that the classification of third-order evolutionary equations with the constant separant possessing a nontrivial Lie–Bäcklund algebra (in other words, integrable equations) results in the linear equation, the KdV equation and the Krichever–Novikov equation. The first two of these equations are nonlinearly self-adjoint. This property allows to associate conservation laws of the equations in question with their symmetries. The problem on nonlinear self-adjointness of the Krichever–Novikov equation has not been solved yet. In the present paper we solve this problem and find the explicit form of the differential substitution providing the nonlinear self-adjointness.     

Conservations laws for a porous medium equation through nonclassical generators

In Ibragimov (2007) [13] a general theorem on conservation laws was proved. In Gandarias (2011) and Ibragimov (2011) [7], [15] the concepts of self-adjoint and quasi self-adjoint equations were generalized and the definitions of weak self-adjoint equations and nonlinearly self-adjoint equations were introduced. In this paper, we find the subclasses of nonlinearly self-adjoint porous medium equations. By using the property of nonlinear self-adjointness, we construct some conservation laws associated with classical and nonclassical generators of the differential equation.     

Nonlocal Symmetries of the Camassa-Holm Type Equations

A class of nonlocal symmetries of the Camassa-Holm type equations with bi-Hamiltonian structures, including the Camassa-Holm equation, the modified Camassa-Holm equation, Novikov equation and Degasperis-Procesi equation, is studied. The nonlocal symmetries are derived by looking for the kernels of the recursion operators and their inverse operators of these equations. To find the kernels of the recursion operators, the authors adapt the known factorization results for the recursion operators of ...     

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.     

Symmetries and nonlocal conservation laws of the general magma equation

In this paper the general magma equation modelling a melt flow in the Earth’s mantle is discussed. Applying the new theorem on nonlocal conservation laws [Ibragimov NH. A new conservation theorem. J Math Anal Appl 2007;333(1):311–28] and using the symmetries of the model equation nonlocal conservation laws are computed. In accordance with Ibragimov [Ibragimov NH. Quasi-self-adjoint differential equations. Preprint in Archives of ALGA, vol. 4, BTH, Karlskrona, Sweden: Alga Publications; 2007. p. 55–60, ISSN: 1652-4934] it is shown that the general magma equation is quasi-self-adjoint for arbitrary m and n and self-adjoint for n = ?m. These important properties are used for deriving local conservation laws.     

New classes of nonlinearly self-adjoint evolution equations of third- and fifth-order

In a recent communication Ibragimov introduced the concept of nonlinearly self-adjoint differential equation [Ibragimov NH. Nonlinear self-adjointness and conservation laws. J Phys A Math Theor 2011;44:432002 (8pp.)]. In this paper a nonlinear self-adjoint classification of a general class of fifth-order evolution equation with time dependent coefficients is presented. As a result five subclasses of nonlinearly self-adjoint equations of fifth-order and four subclasses of nonlinearly self-adjoint equations of third-order are obtained. From the Ibragimov’s theorem on conservation laws [Ibragimov NH. A new conservation theorem. J Math Anal Appl 2007;333:311–28] conservation laws for some of these equations are established.     

耦合KdV方程组的对称,精确解和守恒律

通过利用修正的CK直接方法建立了耦合KdV方程组的对称群理论.利用对称群理论和耦合KdV方程组的旧解得到了它们的新的精确解.基于上述理论和耦合KdV方程组的共轭方程组的理论,得到了耦合KdV方程组的守恒律.     

Some weak self-adjoint Hamilton-Jacobi-Bellman equations arising in financial mathematics

In Gandarias (2011) [12] one of the present authors has introduced the concept of weak self-adjoint equations. This definition generalizes the concept of self-adjoint and quasi self-adjoint equations that were introduced by Ibragimov (2006) [11]. In this paper we find a class of weak self-adjoint Hamilton-Jacobi-Bellman equations which are neither self-adjoint nor quasi self-adjoint. By using a general theorem on conservation laws proved in Ibragimov (2007) [9] and the new concept of weak self-adjointness (Gandarias, 2011) [12] we find conservation laws for some of these partial differential equations.     

Symmetries,Conservation Laws,and Noether's Theorem for Differential‐Difference Equations

This paper mainly contributes to the extension of Noether's theorem to differential‐difference equations. For this purpose, we first investigate the prolongation formula for continuous symmetries, which makes a characteristic representation possible. The relations of symmetries, conservation laws, and the Fréchet derivative are also investigated. For nonvariational equations, because Noether's theorem is now available, the self‐adjointness method is adapted to the computation of conservation laws for differential‐difference equations. Several differential‐difference equations are investigated as illustrative examples, including the Toda lattice and semidiscretizations of the Korteweg–de Vries (KdV) equation. In particular, the Volterra equation is taken as a running example.     

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.     

Type-II hidden symmetry and nonlinear self-adjointness of Boiti–Leon–Pempinelli equation

This paper focuses on two aspects. Firstly, we convert Boiti–Leon–Pempinelli (BLP) equation to (1+1)-dimensional partial differential equation via similarity transformation, and then analyze hidden symmetry of BLP equations via studying classical and nonclassical symmetries of the (1+1)-dimensional equations. As a byproduct, some new invariant solutions of BLP equations are constructed. Secondly, we show that BLP equation is nonlinearly self-adjoint and give the general formula of conservation laws.     

How to find solutions,Lie symmetries,and conservation laws of forced Korteweg–de Vries equations in optimal way

It is shown that the forced Korteweg–de Vries (KdV) equation studied in the recent papers [A.H. Salas, Computing solutions to a forced KdV equation, Nonlinear Anal. RWA 12 (2011) 1314–1320] and [M.L. Gandarias, M.S. Bruzón, Some conservation laws for a forced KdV equation, Nonlinear Anal. RWA 13 (2012) 2692–2700] is reduced to the classical (constant-coefficient) KdV equation by point transformations for all values of variable coefficients. The equivalence-based approach proposed in [R.O. Popovych, O.O. Vaneeva, More common errors in finding exact solutions of nonlinear differential equations: part I, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 3887–3899] allows one to obtain more results in a much simpler way.     

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.     

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.     

Self-adjointness and conservation laws for Kadomtsev–Petviashvili–Burgers equation

In this work we study the Kadomtsev–Petviashvili–Burgers equation, which is a natural model for the propagation of the two-dimensional damped waves. We show that the equation is nonlinear self-adjoint and it will become strict self-adjoint or weak self-adjoint in some equivalent form. By using Ibragimov’s theorem on conservation laws we find some conservation laws for this equation.     

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.