Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II

We describe a list of open problems in random matrix theory and the theory of integrable systems that was presented at the conference Asymptotics in Integrable Systems, Random Matrices and Random Processes and Universality, Centre de Recherches Mathematiques, Montreal, June 7-11, 2015. We also describe progress that has been made on problems in an earlier list presented by the author on the occasion of his 60th birthday in 2005 (see [Deift P., Contemp. Math., Vol. 458, Amer. Math. Soc., Providence, RI, 2008, 419-430, arXiv:0712.0849]).Comment: for Part I see arXiv:0712.084

Quasi-classical approximation in vortex filament dynamics. Integrable systems, gradient catastrophe and flutter

Quasiclassical approximation in the intrinsic description of the vortex filament dynamics is discussed. Within this approximation the governing equations are given by elliptic system of quasi-linear PDEs of the first order. Dispersionless Da Rios system and dispersionless Hirota equation are among them. They describe motion of vortex filament with slow varying curvature and torsion without or with axial flow. Gradient catastrophe for governing equations is studied. It is shown that geometrically this catastrophe manifests as a fast oscillation of a filament curve around the rectifying plane which resembles the flutter of airfoils. Analytically it is the elliptic umbilic singularity in the terminology of the catastrophe theory. It is demonstrated that its double scaling regularization is governed by the Painleve' I equation.Comment: 25 pages, 5 figures, minor typos correcte

Nonholonomic deformation of KdV and mKdV equations and their symmetries, hierarchies and integrability

Recent concept of integrable nonholonomic deformation found for the KdV equation is extended to the mKdV equation and generalized to the AKNS system. For the deformed mKdV equation we find a matrix Lax pair, a novel two-fold integrable hierarchy and exact N-soliton solutions exhibiting unusual accelerating motion. We show that both the deformed KdV and mKdV systems possess infinitely many generalized symmetries, conserved quantities and a recursion operator.Comment: Latex, 2 figures, 16 pages. Revised with more explanations after Referees' feedback.To be published in J. Phys.

Geometry of Integrable Lattice Equations and their Reductions

Modern research into discrete integrable systems has provided new insights into a wide variety of fields, including generalisations of special functions, orthogonal polynomials and dynamical systems theory. In this thesis, we extend one of the most productive insights in this area to higher dimensions. In particular, we show how to apply ideas from resolution of singularities and birational geometry to discrete systems in higher dimensions. The most widely studied setting for these ideas lies in spaces of dimension two. By blowing up at certain points to resolve singularities found in maps on surfaces, new surfaces are constructed on which the map becomes an isomorphism, a so-called space of initial conditions. This has led to new developments in the field, including the discovery of new examples of integrable maps by Sakai with solutions that have unexpectedly rich properties. On the other hand, this geometric approach has never been applied to integrable partial difference equations (often called lattice equations), which share other properties with the maps in dimension two. In this thesis, we overcome this gap. In particular, we examine spaces of initial conditions for integrable lattice equations, which are members of the equations classified by Adler et al, known as ABS equations. By explicitly calculating the induced map on their resolved initial value spaces, we find transformations to new lattice equations, and hence find novel reductions to discrete Painlevé equations. We also show that an equation arising from the geometry of ABS equations is satisfied by the coefficients of a cluster algebra associated with a form of the discrete mKdV