M. S. Bruzón

Classical and Nonclassical Symmetries of a Generalized Boussinesq Equation

We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation, which has the classical Boussinesq equation as an special case. We study the class of functions f (u) for which this equation admit either the classical or the nonclassical method. The reductions obtained are derived. Some new exact solutions can be derived.

THE CALOGERO-BOGOYAVLENSKII-SCHIFF EQUATION IN 2+1 DIMENSIONS

We use the classical and nonclassical methods to obtain symmetry reductions and exact solutions of the (2+1)-dimensional integrable Calogero–Bogoyavlenskii–Schiff equation. Although this (2+1)-dimensional equation arises in a nonlocal form, it can be written as a system of differential equations and, in potential form, as a fourth-order partial differential equation. The classical and nonclassical methods yield some exact solutions of the (2+1)-dimensional equation that involve several arbitrary functions and hence exhibit a rich variety of qualitative behavior.

The Schwarzian Korteweg–de Vries equation in (2 + 1) dimensions

In this paper, the complete Lie group classification of a (2 + 1)-dimensional integrable Schwarzian Korteweg–de Vries equation is obtained. The reduction to systems of partial differential equations in (1 + 1) dimension is derived from the optimal system of subalgebras. The invariance study of these systems leads to second-order ODEs. These ODEs provide several classes of solutions; all of them are expressible in terms of known functions, some of them are expressible in terms of the second and third Painleve transcendents. The corresponding solutions of the (2 + 1)-dimensional equation involve up to three arbitrary smooth functions. They even appear in the form ρ(z)f(x + (t)). Consequently, the solutions exhibit a rich variety of qualitative behaviour. Indeed, by making appropriate choices for the arbitrary functions, we are able to exhibit large families of solitary waves, coherent structures and different types of bound states.

On the nonlinear self-adjointness of a class of fourth-order evolution equations

In this paper we study the property of nonlinear self-adjointness for a class of nonlinear fourth-order equation. It is shown that if an equation itself is a conservation law then it possesses the property of nonlinear self-adjointness and hence it can be rewritten in an equivalent strictly self-adjoint form. The property of nonlinear self-adjointness is used to obtain all nontrivial conservation laws for the class under consideration.

Symmetry Analysis and Solutions for a Generalization of a Family of BBM Equations

Abstract In this paper, the family of BBM equation with strong nonlinear dispersive B(m,n) is considered. We apply the classical Lie method of infinitesimals. The symmetry reductions are derived from the optimal system of subalgebras and lead to systems of ordinary differential equations. We obtain for special values of the parameters of this equation, many exact solutions expressed by various single and combined nondegenerative Jacobi elliptic function solutions and their degenerative solutions (soliton, kink and compactons).

Symmetries and Conservation Laws for Some Compacton Equation

We consider some equations with compacton solutions and nonlinear dispersion from the point of view of Lie classical reductions. The reduced ordinary differential equations are suitable for qualitative analysis and their dynamical behaviour is described. We derive by using the multipliers method some nontrivial conservation laws for these equations.

Symmetry analysis and solutions for a family of Cahn-Hilliard equations

Abstract In this paper we find some new classes of solutions for a family of Cahn-Hilliard equations. For some equations of this family several solutions have already been obtained by using several methods: the Lie method, the direct method and the singular manifold method. We make full analysis of the symmetry reductions of the family of Cahn-Hilliard equations by using the classical Lie method of infinitesimals and the nonclassical method. New classes of nonlocal symmetries for the family of Cahn-Hilliard equations are obtained. These nonclassical potential symmetries are realized as local nonclassical symmetries of a related integrated equation. For an equation of the Cahn-Hilliard family with the conditional Painleve condition, we also compare symmetry reductions by using the nonclassical method with those obtained elsewhere by the singular manifold method. For this equation, we obtain nonclassical symmetries that reduce the original equation to ordinary differential equations with the Painleve property. Such symmetries have not been derived elsewhere neither by the direct method nor by the singular manifold method.

Conservation laws for a class of quasi self-adjoint third order equations

Abstract In this work we consider a class of third-order nonlinear partial differential equation containing two un-specified coefficient functions of the dependent variable which include various integrable and nonintegrable equations. We determine the subclasses of these equations which are self-adjoint and quasi self-adjoint. By using a general theorem on conservation laws proved by Nail Ibragimov we find conservation laws for some of these partial differential equations without classical Lagrangians.

Nonclassical symmetries for a family of Cahn–Hilliard equations

Abstract In this Letter we make a full analysis of the symmetry reductions of the family of Cahn–Hilliard equations by using the classical Lie method of infinitesimals, the functional forms of the diffusion coefficients for which the Cahn–Hilliard equations can be fully reduced to ordinary differential equations by classical Lie symmetries are derived. We prove that by using the nonclassical method, we obtain several solutions which are not invariant under any Lie group admitted by the equation and consequently which are not obtainable through the Lie classical method. For this Cahn–Hilliard equation, we obtain nonclassical symmetries that reduce the original equation to ordinary differential equations with the Painleve property. We remark that these symmetries have not been derived elsewhere by the singular manifold method.

On symmetries and conservation laws of a Gardner equation involving arbitrary functions

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 not require the use of a variational principle. We also construct conservation laws by using Ibragimov theorem which is based on the concept of adjoint equation for nonlinear differential equations.