J.L. Romero

First integrals, integrating factors and λ-symmetries of second-order differential equations

For a given second-order ordinary differential equation (ODE), several relationships among first integrals, integrating factors and λ-symmetries are studied. The knowledge of a λ-symmetry of the equation permits the determination of an integrating factor or a first integral by means of coupled first-order linear systems of partial differential equations. If two nonequivalent λ-symmetries of the equation are known, then an algorithm to find two functionally independent first integrals is provided. These methods include and complete other methods to find integrating factors or first integrals that are based on variational derivatives or in the Prelle–Singer method. These results are applied to several ODEs that appear in the study of relevant equations of mathematical physics.

Nonlocal transformations and linearization of second-order ordinary differential equations

The class of nonlinear second-order equations that are linearizable by means of generalized Sundman transformations (S-transformations) is identified as the class of equations admitting first integrals that are polynomials of first degree in the first-order derivative. This class is also characterized in terms of the coefficients of the equations and constructive methods to derive the linearizing S-transformations are presented. Only the equations of a well-defined subclass can also be linearized by invertible point transformations. These invertible point transformations can be constructed by using the algorithms for the calculation of linearizing S-transformations. Several examples illustrate that both types of linearization are strictly different.

Nonclassical symmetry reductions of a porous medium equation with convection

In this paper new symmetry reductions and exact solutions are presented for the porous medium equation with convection Those spatial forms for which the equation can be reduced to an ordinary differential equation are studied. The symmetry reductions and exact solutions presented are derived by using the nonclassical method developed by Bluman and Cole and are unobtainable by Lie classical method. The asymptotic behaviour of some of the new solutions is analysed.

Lie symmetries and multiple solutions in reaction-diffusion systems

Lie theory of transformation groups is applied to the study of ! reaction-diffusion systems in two-dimensional media. Our study proves that they are invariant with respect to a five-parameter symmetry group. Multiple types of invariant solutions with physical interest are possible, and some of them can be found in the literature applied to particular models.

A λ-symmetry-based method for the linearization and determination of first integrals of a family of second-order ordinary differential equations

We present a characterization of the second-order ordinary differential equations (ODEs) that can be linearized by means of certain nonlocal transformations. This characterization is given in terms of the coefficients of the equation and also determines the second-order ODEs that admit λ-symmetries and first integrals of some specific forms. Systematic methods to find these first integrals, λ-symmetries and the linearizing transformations are also derived.

Reductions of PDEs to second order ODEs and symbolic computation

A new method to obtain second-order reductions for ordinary differential equations which are polynomial in the derivatives of the dependent variable is presented. The method is applied to obtain reductions and new solutions to several well-known equations of mathematical physics: a lubrication equation, a thin-film equation, the Zoomeron equation and a family of 5 th - order partial differential equations which includes the Caudrey-Dodd-Gibbon-Sawada-Kotera, Kaup-Kupershmidt, Ito and Lax equations. Some pieces of computer algebra code to derive the reductions are also included.

New classes of solutions for the Schwarzian Korteweg-de Vries equation in (2+1) dimensions *

We have obtained new classes of solutions for the (2+1)-dimensional Schwarzian Korteweg–de Vries equation by considering several types of reductions of a system equivalent to this equation. The first analysis is done by studying the nonclassical reductions of the system. Further reductions are attained by means of other types of symmetry reductions or by ansatz-based reductions. Most of the new classes of solutions depend on Jacobian elliptic functions and solutions of a Riemann wave equation, including the cnoidal waves solutions. The new classes of solutions can display several types of coherent structures and can exhibit the overturning or intertwining phenomena, according to the suitable selection of the functions these solutions depend on.

Intersection of crisis loci in a driven nonlinearly damped oscillator

Abstract We report on a phenomenon observed in a driven nonlinearly damped oscillator when two control parameters, the frequency of the external excitation and the nonlinear damping coefficient, are varied simultaneously. An interior crisis locus and a boundary crisis locus, corresponding to two different chaotic attractors, intersect in a point of the parameter space. There exists an interchange in the type of crisis that each attractor suffers after crossing the intersection point.

Diffusion equations for nonhomogeneous media. Existence of similarity solutions

Abstract We study the invariance of the diffusion equation δP ( x , t )/ δt = ( δ / δx )[ D ( x ) δP ( x , t )/ δx ] under continuous groups of transformations. We show the conditions which D ( x ) must satisfy for the existence of similarity solutions.

Spiral waves solutions in reaction-diffusion equations with symmetries. Analysis through specific models

Symmetries of nonlinear reaction-diffusion equations determine the existence of regular rotating spiral waves. They are only a consequence of kinetics processes and molecular diffusion. We prove the existence of these waves as invariant solutions of reaction-diffusion models with appropiate Lie point symmetries.