On Periodic Solutions of a Nonlinear Reaction-Diffusion System

We consider a system of three parabolic partial differential equations of a special reaction-diffusion type. In this system, the terms that describe diffusion are identical and linear with constants coefficients, whereas reactions are described by homogenous polynomials of degree 3 that depend on three parameters. The desired functions are considered to be dependent on time and an arbitrary number of spatial variables (a multidimensional case). It has been shown that the reaction-diffusion system under study has a whole family of exact solutions that can be expressed via a product of the solution to the Helmholz equations and the solution to a system of ordinary differential equations with homogenous polynomials, taken from the original system, in the right-hand side. We give the two first integrals and construct a general solution to the system of three ordinary differential equations, which is represented by the Jacobi elliptic functions. It has been revealed that all particular solutions derived from the general solution to the system of ordinary differential equations are periodic functions of time with periods depending on the choice of initial conditions. Additionally, it has been shown that this system of ordinary differential equations has blow-up on time solutions that exist only on a finite time interval. The corresponding values of the first integrals and initial data are found through the equality conditions. A special attention is paid to a class of radially symmetric with respect to spatial variables solutions. In this case, the Helmholz equation degenerates into an non-autonomous linear second-order ordinary differential equation, which general solution is found in terms of the power functions and the Bessel functions. In a particular case of three spatial variables the general solution is expressed using trigonometric or hyperbolic functions

Lie point symmetries and ODEs passing the Painlev\'e test

The Lie point symmetries of ordinary differential equations (ODEs) that are candidates for having the Painlev\'e property are explored for ODEs of order $n =2, \dots ,5$. Among the 6 ODEs identifying the Painlev\'e transcendents only $P_{III}$, $P_V$ and $P_{VI}$ have nontrivial symmetry algebras and that only for very special values of the parameters. In those cases the transcendents can be expressed in terms of simpler functions, i.e. elementary functions, solutions of linear equations, elliptic functions or Painlev\'e transcendents occurring at lower order. For higher order or higher degree ODEs that pass the Painlev\'e test only very partial classifications have been published. We consider many examples that exist in the literature and show how their symmetry groups help to identify those that may define genuinely new transcendents

Asymptotics of higher-order Painlevé equations

We undertake an asymptotic study of a second Painlevé hierarchy based on the Jimbo-Miwa Lax pair in the limit as the independent variable approaches infinity. The hierarchy is defined by an infinite sequence of non-linear ordinary differential equations, indexed by order, with the classical second Painlevé equation as the first member. We investigate general and special asymptotic behaviours admitted by each equation in the hierarchy. We show that the general asymptotic behaviour is described by two related hyperelliptic functions, where the genus of the functions increases with each member of the hierarchy, and we prove that there exist special families of solutions which are represented by algebraic formal power series. For specific values of the constants which appear in the higher-order second Painlevé equations, exact solutions are also constructed. Particular attention is given to the fourth-order analogue of the classical second Painlevé equation. In this case, the general asymptotic behaviour is given to leading-order by two related genus-2 hyperelliptic functions. These functions are characterised by four complex parameters which depend on the independent variable through the perturbation terms of the leading-order equations, and we investigate how these parameters change with respect to this variable. We also show that the fourth-order equation admits two classes of algebraic formal power series and that there exist families of true solutions with these behaviours in specified sectors of the complex plane, as well as unique solutions in extended sectors. To complement our asymptotic study of higher-order Painlevé equations, we consider a new setting in which classical Painlevé equations arise. We study reaction-diffusion equations with quadratic and cubic source terms, with a spatio-temporal dependence included in those terms, and show that solutions of these equations are given by first and second Painlevé transcendents

Differential equations for relativistic radiating stars.

Ph. D. University of KwaZulu-Natal, Durban 2013.We consider radiating spherical stars in general relativity when they are conformally flat, geodesic with shear, and accelerating, expanding and shearing. We study the junction conditions relating the pressure to the heat flux at the boundary of the star in each case. The boundary conditions are nonlinear partial differential equations in the metric functions. We transform the governing equations to ordinary differential equations using the geometric method of Lie. The Lie symmetry generators that leave the equations invariant are identified, and we generate the optimal system in each case. Each element of the optimal system is used to reduce the partial differential equations to ordinary differential equations which are further analyzed. As a result, particular solutions to the junction conditions are presented for all types of radiating stars. New exact solutions, which are group invariant under the action of Lie point infinitesimal symmetries, are found. Our solutions contain families of traveling wave solutions, self-similar variables, and other forms with different combinations of the spacetime variables. The gravitational potentials are given in terms of elementary functions, and the line elements can be given explicitly in all cases. We show that the Friedmann dust model is regained as a special case in particular solutions. We can connect our results to earlier investigations and we show explicitly that our models are generalizations. Some of our solutions satisfy a linear equation of state. We also regain previously obtained solutions for the Euclidean star as a special case in our accelerating model. Our results highlight the importance of Lie symmetries of differential equations for problems arising in relativistic astrophysics

Propagation of High-Frequency Electromagnetic Waves Through a Magnetized Plasma in Curved Space-Time. I

This is the first of two papers on the propagation of high-frequency electromagnetic waves through a magnetized plasma in curved space-time. We first show that the nonlinear system of equations governing the plasma and the electromagnetic field in a given, external gravitational field has locally a unique solution for any initial data set obeying the appropriate constraints, and that this system is linearization stable at any of its solutions. Next we prove that the linearized perturbations of a `background' solution are characterized by a third-order (not strictly) hyperbolic, constraint-free system of three partial differential equations for three unknown functions of the four space-time coordinates. We generalize the algorithm for obtaining oscillatory asymptotic solutions of linear systems of partial differential equations of arbitrary order, depending polynomially on a small parameter such that it applies to the previously established perturbation equation when the latter is rewritten in terms of dimensionless variables and a small scale ratio. For hyperbolic systems we then state a sufficient condition in order that asymptotic solutions of finite order, constructed as usual by means of a Hamiltonian system of ordinary differential equations for the characteristic strips and a system of transport equations determining the propagation of the amplitudes along the rays, indeed approximate solutions of the system. The procedure is a special case of a two-scale method, suitable for describing the propagation of locally approximately plane, monochromatic waves through a dispersive, inhomogeneous medium. In the second part we shall apply the general method to the perturbation equation referred to above

Notes on Lie symmetry group methods for differential equations

Fundamentals on Lie group methods and applications to differential equations are surveyed. Many examples are included to elucidate their extensive applicability for analytically solving both ordinary and partial differential equations.Comment: 85 Pages. expanded and misprints correcte

Differential constraints and exact solutions of nonlinear diffusion equations

The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining equations used in the search for classical Lie symmetries