Integrable systems, random matrices and applications

Dell'Atti, Marta (2021) Integrable systems, random matrices and applications. Diploma thesis, Northumbria University.

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Abstract

Nonlinear integrable systems emerge in a broad class of different problems in Mathematics and Physics.
One of the most relevant characterisation of integrable systems is the existence of an infinite number of conservation laws, associated to integrable hierarchies of equations.
When nonlinearity is involved, critical phenomena may occur. A solution to a nonlinear partial differential equation may develop a gradient catastrophe and the consequent formation of a shock at the critical point. The approach of differential identities provides a convenient description of systems affected by phase transitions, identifying a suitable nonlinear equation for the order parameter of the system.
This thesis is aimed to give a contribution to the perspective offered by the approach of differential identities. We discuss how this method is particularly useful in treating mean-field theories, with some explicit application. The core of the work concerns the Hermitian matrix ensemble and the symmetric matrix ensemble, analysed in the context of integrable systems. They both underlie a discrete integrable structure in form of a lattice, satisfying a discrete integrable hierarchy. We have studied a particular reduction of both system and determined the continuum limit of the dynamics of the field variables at the leading order.
Particular emphasis has been given to the study of the symmetric matrix ensemble. We have unveiled an unobserved double-chain structure shared by the field variables populating the lattice structure associated to the ensemble. In the continuum limit of a particular reduction of the lattice, we have found a new hydrodynamic chain, a hydrodynamic system with infinitely many components. We have shown that the hydrodynamic chain is integrable and we have conjectured the form of the associated hierarchy. The new integrable hydrodynamic chain constitutes per se an interesting object of study. Indeed, it presents some properties that are different from those shared by the standard integrable hydrodynamic chains studied in literature.

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