A number of nonlinear partial differential equations serve as universal physical models, describing a great variety of phenomena. These are soliton-type equations: the Korteweg-de Vries equation, the sine-Gordon equation, the nonlinear Schrödinger equation, the continuum Heisenberg model described by the isotropic Landau-Lifshits equation, the Boussinesq equation, the Toda model etc. They are called completely integrable equations due to infinitely many integrals of motion. Every mentioned equation represents a hierarchy of integrable Hamiltonian systems, and can be constructed on a coadjoint orbit of a loop group by the orbit method. This method gives a powerful apparatus for obtaining periodic and multisoliton solutions for such equations and solving other important problems – one of them is the problem of quantization on a Lagrangian manifold , that is a canonical quantization.
The controversial question how to choose a proper Lagrangian manifold for canonical quantization is evidently solved in terms of variables of separation (Darboux coordinates). For integrable Hamiltonian systems constructed by means of the orbit method we have a definite procedure for obtaining variables of separation, they are points of the spectral curve connected to a system. A half of them parametrizes the Liouville torus of the integrable system, thus the Liouville torus serves as a Lagrangian manifold. And the complexified Lagrangian manifold is a generalized Jacobian of the spectral curve, which coincides with the phase space of the system.
The canonical quantization of an integrable system gives a representation of its phase space symmetry algebra over the space of functions on a Lagrangian manifold. Using the Liouville torus as a Lagrangian manifold guarantees that the representation space consists of holomorphic functions – they are defined on the generalized Jacobian serving as the phase space of the system. The obtained representation is indecomposable and non-exponentiated .
As an example we consider the nonlinear Schrödinger equation and the continuum Heisenberg model, connecting to the same spectral curve, that allows to use the same Lagrangian manifold for canonical quantization. We construct the corresponding phase space symmetry algebras and their representations over the space of holomorphic functions on the Lagrangian manifold. Harmonic analysis on the representation space leads to the Whittaker equation (in some particular case) and its generalization with an irregular singularity.
 M. Karasev, Quantization by membranes and integral representations of wave functions, Quantization and infinite-dimensional systems.
Proceedings of the 12th summer workshop on differential geometric methods in physics, Bialowieza, Poland, 1993, 9–19.
 J. Bernatska, P. Holod, Harmonic analysis on Lagrangian manifolds of Integrable Hamiltonian systems, Journal of Geometry and Symmetry in Physics, 2013, Vol.~29, 39–51 (arXiv:1307.1785 [math-ph]).
General background : modelling of stochastic perturbations and validation of models
The aim of this lecture is to present a set of mathematical tools and results permitting to construct a stochastic viable model from a known deterministic one and second to validate them via numerical simulations. Indeed, many dynamical systems in Biology or Physics are in a first approach modelled by a deterministic dynamical systems. We can cite for example the classical Hodgkin-Huxley model in Biology or the Landau-Lipshitz model in Physics. Then, a set of observations are made and lead to the introduction of a stochastic or random component. For the Hodgkin-Huxley model this comes from the fact that there exists an intrinsic stochastic bioelectrical activity of neurons which is observed experimentally. For the Landau-Lifshitz model this comes from the fact that the electromagnetic fields behaves very randomly. However, to take into account these stochastic effects is in general not an easy task.
The Lecture is made of two parts : the first one deals with the construction of the stochastic model. The second one is devoted to numerical methods designed in order to validate these models. All the mathematical tools and results will be illustrated by numerous examples coming from Biology, Physics, Astronomy and Celestial Mechanics.
Part I – Admissible or viable stochastic models
In a first part of this Lecture, we discuss such a modelling in the context of the theory of stochastic differential equations.
Returning to the initial deterministic model is always interesting. Indeed, it provides a set of constraints which are in general considered as fundamental by the scientists. This can be some fundamental law of Physics like conservation of energy, existence of some symmetries or invariance properties. At least these properties are in general respected by the deterministic model and a natural way to extend such a model in the stochastic framework is to construct a suitable stochastic perturbation respecting such a constraints in an appropriate sense.
In this Lecture, we will provide many results characterizing stochastic perturbations which preserve important properties : invariance of domain (in particular positivity), first integrals, variational structures (Lagrangian or Hamiltonian), symmetries, etc. These results will then be applied in various fields : Biology (behaviour of neurons, HIV population dynamics, Virus transmission models and models for the immune system, Cellular signaling networks, Population growth models, Tumor growth), Physics (Ferromagnetism), Astronomy and Celestial Mechanics (Two-body problem, Orbits of Satellites, Earth’s rotation).
Part II – Numerical methods and validation of models
In a second part of this Lecture, we discuss how to make numerical simulations in order to validate these stochastic models. The main difficulty which is not usually discussed in the literature, is to provide numerical methods respecting the constraints of the models. This problem is well known in Hamiltonian mechanics where the conservation of energy is an important feature of the models and has leaded to the theory of variational integrators. For invariance of domains, symmetries, etc, the state of the arts is not so clear even in the deterministic case.
In this Lecture, we will discuss the construction of variational integrators in the context of the theory of discrete embeddings ans secondly we will discuss the construction of topological numerical scheme which are reminiscent of a general program initiated by R. Mickens around non-standard numerical scheme. These methodes will be discussed in the deterministic and stochastic case with numerous examples.
In recent years we have been working on the formulation an analysis of structured population models for infectious disease dynamics. In contrast to previous models, where for example the age of infection have been used as a structuring variable, we introduce structuring of the population with respect to infection (bacterium/virus) load and/or infectiousness.
In this talk we will focus on the models. First we will introduce the so called Wentzell (or Feller) boundary conditions in a structured population model with diffusion. The diffusion in our equation allows us to model random noise in a deterministic fashion. The power of Wentzell boundary condition is that it allows to incorporate a boundary state, which carries mass, namely the population of the uninfected individuals, in an elegant fashion. First we will consider a general linear model, then we will consider a nonlinear model which arises when modelling Wolbachia infection dynamics. We establish existence of solutions and consider the existence of positive steady states of the model. if time permits we will briefly mention a general framework we developed recently to treat models with 2-dimensional but non-monotone nonlinearities.
In the second part we will introduce and investigate an SIS-type model for the spread of an infectious disease, where the infected population is structured with respect to the different strain of the virus/bacteria they are carrying. Our aim is to capture the interesting scenario when individuals infected with different strains cause secondary (new) infections at different rates. Therefore, we consider a nonlinear infection process, which generalizes the bilinear process arising from the classic mass-action assumption. Our main motivation is to study competition between different strains of a virus/bacteria. From the mathematical point of view, we are interested whether the nonlinear infection process leads to a well-posed model. We use a semilinear formulation to show global existence and positivity of solutions up to a critical value of the exponent in the nonlinearity. Furthermore, we establish the existence of the endemic steady state for particular classes of nonlinearities.
Consider the differential equation
where is a positive parameter and is a -periodic function. This is a simple model frequently employed to illustrate the methods of Nonlinear Analysis. Results on the existence of periodic solutions are usually obtained by a combination of tools coming from Topology and Calculus of Variations. The goal of this talk is to show that these tools are also useful in the study of the stability of periodic solutions. Stability is understood in the Lyapunov sense. We will assume that the parameter satisfies and the function has zero average. The main result says that there exists a stable -periodic solution for almost every periodic function with zero average. The phrase „for almost every periodic function” is understood in the sense of prevalence. For this reason the notion of set of zero measure in a Banach space of infinite dimensions will play a role.The condition is sharp. The conclusion of the theorem is not valid „for all periodic functions”.