Igor Leite Freire

Nonlinear self-adjointness of a generalized fifth-order KdV equation

The new concepts of self-adjoint equations formulated by Ibragimov and Gandarias are applied to a class of fifth-order evolution equations. Then, from Ibragimov?s theorem on conservation laws, conservation laws for the generalized Kawahara equation, simplified Kahawara equation and modified simplified Kawahara equation are established.

Self-adjoint sub-classes of third and fourth-order evolution equations

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.

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

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)]. In the present communication 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 Ibragimovs theorem on conservation laws [N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., vol. 333, 311--328, (2007)] conservation laws for some of these equations are established.

CONSERVATION LAWS FOR SELF-ADJOINT FIRST-ORDER EVOLUTION EQUATION

We consider the problem on group classification and conservation laws for first-order evolution equations. Subclasses of these general equations which are quasi-self-adjoint and self-adjoint are obtained. By using the recent new conservation theorem due to Ibragimov, conservation laws for equations admiting self-adjoint equations are established. The results are illustrated applying them to the inviscid Burgers equation. In particular an infinite number of new symmetries of this equation are found.

New conservation laws for inviscid Burgers equation

In this paper it is shown that the inviscid Burgers equation is nonlinearly self-adjoint. Then, from Ibragimovs theorem on conservation laws, local conserved quantities are obtained. Mathematical subject classification: Primary: 76M60; Secondary: 58J70.

Group analysis of the Novikov equation

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 equations, we find the conservation law corresponding to the dilation symmetry and show that other symmetries do not provide nontrivial conservation laws. Then we investigate the invariant solutions.

A family of wave-breaking equations generalizing the Camassa-Holm and Novikov equations

A 4-parameter polynomial family of equations generalizing the Camassa-Holm and Novikov equations that describe breaking waves is introduced. A classification of low-order conservation laws, peaked travelling wave solutions, and Lie symmetries is presented for this family. These classifications pick out a 1-parameter equation that has several interesting features: it reduces to the Camassa-Holm and Novikov equations when the polynomial has degree two and three; it has a conserved H1 norm and it possesses N-peakon solutions when the polynomial has any degree; and it exhibits wave-breaking for certain solutions describing collisions between peakons and anti-peakons in the case N = 2.

On the paper “Symmetry analysis of wave equation on sphere” by H. Azad and M.T. Mustafa

Abstract Using the scalar curvature of the product manifold S 2 × R and the complete group classification of nonlinear Poisson equation on (pseudo) Riemannian manifolds, we extend the previous results on symmetry analysis of homogeneous wave equation obtained by H. Azad and M.T. Mustafa [H. Azad, M.T. Mustafa, Symmetry analysis of wave equation on sphere, J. Math. Anal. Appl. 333 (2007) 1180–1188] to nonlinear Klein–Gordon equations on the two-dimensional sphere.

Conservations laws for critical Kohn-Laplace equations on the Heisenberg group

Abstract Using the complete group classification of semilinear differential equations on the three-dimensional Heisenberg group ℍ, carried out in a preceding work, we establish the conservation laws for the critical Kohn-Laplace equations via the Noethers Theorem.

Lie and Noether symmetries for a class of fourth-order Emden?Fowler equations

A group classification of a fourth-order ordinary differential equation is carried out. The Noether symmetries are considered and some first integrals are established. Solutions for special Lane–Emden systems are also obtained from the invariant solutions of the investigated equation.