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Upcoming events in Mathematics

(previous see here)
Tue 22/10/19 14:00
Dalhousie 2F13
Mathematics Seminar
Prof. Gennady El (Northumbria University)
Soliton and breather gas in the focusing nonlinear Schrödinger equation: nonlinear spectral theory.
abstract

Abstract

Solitons and breathers are localized solutions of integrable systems that can be viewed as "particles" of complex statistical objects called soliton and breather gases. In view of the growing evidence of their ubiquity in fluids and nonlinear optical media these "integrable" gases present fundamental interest for nonlinear physics. We develop nonlinear spectral theory of a breather gas by considering a special, thermodynamic type limit of multi-phase (finite-gap) solutions of the focusing nonlinear Schrödinger (fNLS) equation. This limit is defined by the locus and the critical scaling of the spectrum of the associated Zakharov-Shabat operator. The family of generalized breather gases includes gas of fundamental solitons and gas of conventional breathers (solitons on finite background) as particular cases. We compute the thermodynamic limit of nonlinear dispersion relations of the finite-gap fNLS solutions and derive the equation of state of a breather gas. For a spatially inhomogeneous breather gas the evolution of the spectral distribution function, termed the density of states, is described by a nonlinear transport equation, which, together with the equation of state, form a nonlinear kinetic equation. We consider several particular cases of the generalised breather gas including the so-called bound state soliton gas which is shown to exhibit the transition to a "soliton condensate" for a certain density of states distribution. The statistical properties of the bound state soliton condensate reveal a remarkable connection with the nonlinear stage of modulational instability.

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Fri 15/11/19 16:30
Dalhousie 3G02
Mathematics Seminar
Prof. Mario Pulvirenti (Universita di Roma "La Sapienza")
Scaling limits and derivation of effective equations for particle systems
abstract

Abstract

Many interesting systems in Physics and Applied Sciences are constituted by a large number of identical components so that they are difficult to be analyzed from a mathematical point of view. On the other hand, quite often, we are not interested in a detailed description of the system but rather to his collective behavior. Therefore it is necessary to look for all procedures leading to simplified models, retaining all the interesting features of the original system, cutting away redundant informations. This is exactly the methodology of the Statistical Mechanics and Kinetic Theory when dealing with large particle systems. Here we want to consider a particle system, as the basic starting point, to outline the limiting procedure leading to a more practical macroscopic description, usually expressed in term of a nonlinear partial differential equation. In this talk I discuss the most popular scalings, namely mean-field and low-density limits, leading to the Vlasov and Boltzmann equation respectively, starting from particle systems.

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