Elena Medina

Finite-time aggregation into a single point in a reaction - diffusion system

We consider the following system: which has been used as a model for various phenomena, including motion of species by chemotaxis and equilibrium of self-attracting clusters. We show that, in space dimension N = 3, (S) possess radial solutions that blow-up in a finite time. The asymptotic behaviour of such solutions is analysed in detail. In particular, we obtain that the profile of any such solution consists of an imploding, smoothed-out shock wave that collapses into a Dirac mass when the singularity is formed. The differences between this type of behaviour and that known to occur for blowing-up solutions of (S) in the case N = 2 are also discussed.

Self-similar blow-up for a reaction-diffusion system

Abstract This work is concerned with the following system: which is a model to describe several phenomena in which aggregation plays a crucial role as, for instance, motion of bacteria by chemotaxis and equilibrium of self-attracting clusters. When the space dimension N is equal to three, we show here that (S) has radial solutions with finite mass that blow-up in finite time in a self-similar manner. When N = 2, however, no radial solution with finite mass may give rise to self-similar blow-up.

Reductions and hodograph solutions of the dispersionless KP hierarchy

A general scheme for analysing reductions of dispersionless integrable hierarchies is presented. It is based on a method for determining the S-function by means of a system of first-order differential equations. Compatibility systems of nonlinear partial differential equations of Bourlet type characterizing both reductions and hodograph solutions of the dKP hierarchy are obtained. Wide classes of illustrative explicit examples are exhibited.

An N Soliton Resonance Solution for the KP Equation: Interaction with Change of Form and Velocity

A family of solutions of the KP hierarchy obtained by Sato is analyzed for the KP equation. It is found that these solutions describe processes of interaction of an arbitrary number of resonant solitons. The interacting structures correspond to the one observed by Miles within the context of diffraction of solitons. It is seen that in solutions consisting in interaction processes, the Miles structures share certain branches in such a way that they can be interpreted as a multiple diffraction. It is also observed that, under interaction, these resonant solitons change both their forms and their velocities.

Charged free fermions, vertex operators and the classical theory of conjugate nets

We show that the quantum field theoretical formulation of the -function theory has a geometrical interpretation within the classical transformation theory of conjugate nets. In particular, we prove that (i) the partial charge transformations preserving the neutral sector are Laplace transformations, (ii) the basic vertex operators are Levy and adjoint Levy transformations and (iii) the diagonal soliton vertex operators generate fundamental transformations. We also show that the bilinear identity for the multicomponent Kadomtsev-Petviashvili hierarchy becomes, through a generalized Miwa map, a bilinear identity for the multidimensional quadrilateral lattice equations.

New Symmetry Reductions for some Ordinary Differential Equations

Abstract In this work we derive potential symmetries for ordinary differential equations. By using these potential symmetries we find that the order of the ODE can be reduced even if this equation does not admit point symmetries. Moreover, in the case for which the ODE admits a group of point symmetries, we find that the potential symmetries allow us to perform further reductions than its point symmetries. Some diffusion equations admitting an infinite number of potential symmetries and the scaling group as a Lie symmetry are considered and some general results are obtained. For all the equations that we have studied, a set of potential symmetries admitted by the diffusion equation is “inherited” by the ODE that emerges as the reduced equation under the scaling group.

Supersymmetric quantum mechanics for Bianchi class A models

In this work we present cosmological quantum solutions for all Bianchi Class A cosmological models obtained by means of supersymmetric quantum mechanics . We are able to write one general expression for all bosonic components occuring in the Grassmann expansion of the wave function of the Universe for this class of models. These solutions are obtained by means of a more general ansatz for the so-called master equations.

On the Whitham hierarchy: dressing scheme, string equations and additional symmetries

A new description of the universal Whitham hierarchy in terms of a factorization problem in the Lie group of canonical transformations is provided. This scheme allows us to give a natural description of dressing transformations, string equations and additional symmetries for the Whitham hierarchy. We show how to dress any given solution and prove that any solution of the hierarchy may be undressed, and therefore comes from a factorization of a canonical transformation. A particularly important function, related to the tau-function, appears as a potential of the hierarchy. We introduce a class of string equations which extends and contains previous classes of string equations considered by Krichever and by Takasaki and Takebe. The scheme is also applied for a convenient derivation of additional symmetries. Moreover, new functional symmetries of the Zakharov extension of the Bentley gas equations are given and the action of additional symmetries over the potential in terms of linear PDEs is characterized.

Dressing methods for geometric nets: I. Conjugate nets

The formalism of multicomponent KP hierarchies is applied to deriving efficient dressing methods for conjugate nets. The notion of the Cauchy propagator is used for characterizing these nets in terms of spectral data. Explicit examples in dimensions N = 2 and 3 are given. In particular, periodic nets and Cartesian nets with a Gaussian localized deformation are exhibited.

Hodograph solutions of the dispersionless coupled KdV hierarchies, critical points and the Euler-Poisson-Darboux equation

It is shown that the hodograph solutions of the dispersionless coupled KdV (dcKdV) hierarchies describe critical and degenerate critical points of a scalar function which obeys the Euler–Poisson–Darboux equation. Singular sectors of each dcKdV hierarchy are found to be described by solutions of higher genus dcKdV hierarchies. Concrete solutions exhibiting shock-type singularities are presented.