In spite of enormous success of the theory of integrable systems, at least three important problems are not resolved yet or are resolved only partly. They are the following:
1. The IST in the case of arbitrary bounded initial data.
2. The statistical description of the systems integrable by the IST. Albeit, the development of the theory of integrable turbulence.
3. Integrability of the deep water equations.
These three problems will be discussed in the talk.[-]

This talk is devoted to solitons and wave collapses which can be considered as two alternative scenarios pertaining to the evolution of nonlinear wave systems describing by a certain class of dispersive PDEs (see, for instance, review [1]). For the former case, it suffices that the Hamiltonian be bounded from below (or above), and then the soliton realizing its minimum (or maximum) is Lyapunov stable. The extremum is approached via the radiation of small-amplitude waves, a process absent in systems with finitely many degrees of freedom. The framework of the nonlinear Schrodinger equation, the ZK equation and the three-wave system is used to show how the boundedness of the Hamiltonian H, and hence the stability of the soliton minimizing H can be proved rigorously using the integral estimate method based on the Sobolev embedding theorems. Wave systems with the Hamiltonians unbounded from below must evolve to a collapse, which can be considered as the fall of a particle in an unbounded potential. The radiation of small-amplitude waves promotes collapse in this case.
This work was supported by the Russian Science Foundation (project no. 14-22-00174).[-]

We discuss the 2D Schrödinger equation for periodic potentials with the symmetry of a hexagonal tiling of the plane. We first review joint work with CL Fefferman on the existence of Dirac points, conical singularities in the band structure, and the resulting effective 2D Dirac dynamics of wave-packets. We then focus on periodic potentials which are superpositions of localized potential wells, centered on the vertices of a regular honeycomb structure, corresponding to the single electron model of graphene and its artificial analogues. We prove that for sufficiently deep potentials (strong binding) the lowest two Floquet-Bloch dispersion surfaces, when appropriately rescaled, are uniformly close to those of the celebrated two-band tight-binding model, introduced by PR Wallace (1947) in his pioneering study of graphite. We then discuss corollaries, in the strong binding regime, on (a) spectral gaps for honeycomb potentials with PT symmetry-breaking perturbations, and (b) topologically protected edge states for honeycomb structures with "rational edges. This is joint work with CL Fefferman and JP Lee-Thorp. Extensions to Maxwell equations (with Y Zhu and JP Lee-Thorp) will also be discussed.[-]

We consider the 1-D Schrödinger system with point vortex-type interactions that was derived by R. Klein, A. Majda and K. Damodaran and by V. Zakharov to modelize the dynamics of N nearly parallel vortex filaments in a 3-D incompressible fluid. We first prove a global in time result and display several classes of solutions. Then we consider the problem of collisions. In particular we establish rigorously the existence of a pair of almost parallel vortex filaments, with opposite circulation, colliding at some point in finite time. These results are joint works with E. Faou and E. Miot.[-]

We consider the Derivative Nonlinear Schrödinger equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but exclude spectral singularities). We prove global wellposedness and give a full description of the long-time behavior of the solutions in the form of a finite sum of localized solitons and a dispersive component. Our analysis provides explicit formulae for the multi-soliton component as well as the correction dispersive term. We use the inverse scattering approach and the nonlinear steepest descent method of Deift and Zhou (1993) revisited by the $\bar{\partial}$-analysis of Dieng-McLaughlin (2008) and complemented by the recent work of Borghese-Jenkins-McLaughlin (2016) on soliton resolution for the focusing nonlinear Schrödinger equation. This is a joint work with R. Jenkins, J. Liu and P. Perry.[-]

I will present two cases of strong interactions between solitary waves for the nonlinear Schrödinger equations (NLS). In the mass sub- and super-critical cases, a work by Tien Vinh Nguyen proves the existence of multi-solitary waves with logarithmic distance in time, extending a classical result of the integrable case (1D cubic NLS equation). In the mass-critical case, a work by Yvan Martel and Pierre Raphaël gives a new class of blow up multi-solitary waves blowing up in infinite time with logarithmic rate.
These special behaviours are due to strong interactions between the waves, in contrast with most previous works on multi-solitary waves of (NLS) where interactions do not affect the general behaviour of each solitary wave.[-]