Related papers: Conservation laws for self-adjoint first order evo…
In this work a class of self-adjoint quasilinear third-order evolution equations is determined. Some conservation laws of them are established and a generalization on a self-adjoint class of fourth-order evolution equations is presented.
In a recent communication Nail Ibragimov introduced the concept of nonlinearly self-adjoint differential equation [N. H. Ibragimov, Nonlinear self-adjointness and conservation laws, J. Phys. A: Math. Theor., vol. 44, 432002, 8 pp., (2011)].…
The self-adjoint sub-classes of nonlinear evolution equations of fourth-order with time dependent coefficients are determined, generalizing some recent results. Using the new conservation theorem recently proved by Nail Ibragimov some…
Conservation laws are formulated for systems of differential equations by using symmetries and adjoint symmetries, and an application to systems of evolution equations is made, together with illustrative examples. The formulation does not…
By the Cole-Hopf transformation, with any linear evolution equation in 1+1 dimensions a generalized Burgers equation is associated. We describe local conservation laws of these equations. It turns out that any generalized Burgers equation…
A general theorem on conservation laws for arbitrary difference equations is proved. The theorem is based on an introduction of an adjoint system related with a given difference system, and it does not require the existence of a difference…
This work extends the Ibragimov's conservation theorem for partial differential equations [{\it J. Math. Anal. Appl. 333 (2007 311-328}] to under determined systems of differential equations. The concepts of adjoint equation and formal…
It is well known that the Camassa-Holm equation possesses numerous remarkable properties characteristic for KdV type equations. In this paper we show that it shares one more property with the KdV equation. Namely, Ibragimov has shown that…
We study local conservation laws for evolution equations in two independent variables. In particular, we present normal forms for the equations admitting one or two low-order conservation laws. Examples include Harry Dym equation,…
A conservation law theorem stated by N. Ibragimov along with its subsequent extensions are shown to be a special case of a standard formula that uses a pair consisting of a symmetry and an adjoint-symmetry to produce a conservation law…
In this work we study a generalized variable-coefficient Gardner equation from the point of view of Lie symmetries in partial differential equations. We find conservation laws by using the multipliers method of Anco and Bluman which does…
The notions of generating sets of conservation laws of systems of differential equations with respect to symmetry groups and equivalence groups are introduced and applied. This allows us to generalize essentially the procedure of finding…
The general concept of nonlinear self-adjointness of differential equations is introduced. It includes the linear self-adjointness as a particular case. Moreover, it embraces the previous notions of self-adjoint and quasi self-adjoint…
A fifth-order KdV equation with time dependent coefficients and linear damping has been studied. Symmetry groups have several different applications in the context of nonlinear differential equations. For instance, they can be used to…
We find the Lie point symmetries of the Novikov equation and demonstrate that it is strictly self-adjoint. Using the self-adjointness and the recent technique for constructing conserved vectors associated with symmetries of differential…
In this paper, the method of approximate transformation groups which was proposed by Baikov, Gazizov and Ibragimov, is extended on Hamiltonian and bi-Hamiltonian systems of evolution equations. Indeed, as a main consequence, this extended…
In the context of adjoint-based optimization, nonlinear conservation laws pose significant problems regarding the existence and uniqueness of both direct and adjoint solutions, as well as the well-posedness of the problem for sensitivity…
The conservation laws of the third order quasilinear scalar evolution equations are considered via differential system and characteristic cohomology. We find a subspace of 2 forms in the infinite prolonged space in which every conservation…
We determine the Lie point symmetries of a Gardner type system and establish its nonlinear self-adjointness. We then construct conservation laws via Ibragimov's Theorem.
We study local conservation laws of variable coefficient diffusion-convection equations of the form $f(x)u_t=(g(x)A(u)u_x)_x+h(x)B(u)u_x$. The main tool of our investigation is the notion of equivalence of conservation laws with respect to…