A Miura of the Painlevé I Equation and Its Discrete Analogs

We present Miura transformations for the continuous and several discrete Painlev\'e I equations. In the case of the continuous P_I, we use the Hamiltonian formulation of the Painlev\'e equations and show that there exists a Miura transformation between P_I and the binomial, second degree, equation of Cosgrove SD_V. In the case of the discrete P_I's we obtain two different kinds of Miuras. One kind relates a d-P_I to some other d-P_I while the other leads to discrete four-point equations which are the discrete analogs of the derivative of Cosgrove's equation SD_V.     

Contiguity relations for linearisable systems of Gambier type

We introduce the Schlesinger transformations for the Gambier, linearisable, equation and by combining the former construct the contiguity relations of the solutions of the latter. We extend the approach to the discrete domain obtaining thus the Schlesinger transformations and the contiguity relations of the solutions of the Gambier mapping. In all cases the resulting contiguity relation is a linearisable equation, involving free functions, and which can be related to the generic Gambier mapping.     

Semiclassical approximations for nuclear hamiltonians. I. Spin-independent potentials

A systematic procedure for calculating semiclassical expansions of physically interesting quantities is presented. The method is based on the Wigner transform of operators. It is applied specifically to the case of fermions governed by a one-body Hamiltonian. Expansions up to fourth order in ? of the density matrix for various spin-independent potentials are derived. Expressions of the kinetic energy density in terms of the matter density are given, which are of particular interest for nuclear structure calculations.     

Painleve property and integrals of motion for the Henon-Heiles system

Using the Painlevé criterion three integrable cases are identified for the Hénon-Heiles system. A new integral of motion, unknown to date is presented for the third case.     

A study of the discrete Pv equation: Miura transformations and particular solutions

We derive the form of the Miura transformation of the discrete P_v equation and show that it is indeed an auto-Bäcklund transformation, i.e. it relates the discrete P_v to itself. Using this auto-Bäcklund, we obtain the Schlesinger transformations of discrete P_v which relate the solution for one set of the parameters of the equation to that of another set of neighbouring parameters. Finally, we obtain particular solutions of the discrete P_v (i.e. solutions that exist only for some specific values of the parameters). These solutions are of two types: solutions involving the confluent hypergeometric function (on codimension-one submanifold of parameters) and rational solutions (on codimension-two submanifold of parameters).     

Modelling the morphology of migrating bacterial colonies

We present a model which aims at describing the morphology of colonies of Proteus mirabilis and Bacillus subtilis. Our model is based on a cellular automaton which is obtained by the adequate discretisation of a diffusion-like equation, describing the migration of the bacteria, to which we have added rules simulating the consolidation process. Our basic assumption, following the findings of the group of Chuo University, is that the migration and consolidation processes are controlled by the local density of the bacteria. We show that it is possible within our model to reproduce the morphological diagrams of both bacteria species. Moreover, we model some detailed experiments done by the Chuo University group, obtaining a fine agreement.     

A Class of Integrable Non-QRT Mappings and Their Deautonomisation

We study classes of mappings which do not belong to the QRT family. We obtain several integrable non-autonomous forms of these mappings extending previous results where only linearisable cases were found. Using our recently introduced method of singularity confinement with full deautonomisation, we analyse a mapping which, while non-integrable, does possess confined singularities and show that our method makes it possible to obtain the exact value of its algebraic entropy.     

A Class of Integrable and Nonintegrable Mappings and their Dynamics

We analyse a class of mappings which by construction do not belong to the QRT family. We show that some of the members of this class have invariants of high degree. A new linearisable mapping is also identified. A mapping which possesses confined singularities while having nonzero algebraic entropy is presented. Its dynamics are studied in detail and shown to be related intimately to the Fibonacci recurrence.      

Special functions as solutions to discrete Painlevé equations

We present results on special solutions of discrete Painlevé equations. These solutions exist only when one constraint among the parameters of the equation is satisfied and are obtained through the solutions of linear second-order (discrete) equations. These linear equations define the discrete analogues of special functions.