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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.
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 ...
37K10 ; 35C07 ; 35C08 ; 35Q53 ; 35Q55 ; 76B15 ; 76Fxx

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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).
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 ...
35Q53 ; 35Q55 ; 37K10 ; 37N10 ; 76B15

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After Fourier series, the quantum Hopf-Burgers equation $v_t +vv_x = 0$ with periodic boundary conditions is equivalent to a system of coupled quantum harmonic oscillators, which may be prepared in Glauber's coherent states as initial conditions. Sending the displacement of each oscillator to infinity at the same rate, we (1) confirm and (2) determine corrections to the quantum-classical correspondence principle. After diagonalizing the Hamiltonian with Schur polynomials, this is equivalent to proving (1) the concentration of profiles of Young diagrams around a limit shape and (2) their global Gaussian fluctuations for Schur measures with symbol $v : T \to R$ on the unit circle $T$. We identify the emergent objects with the push-forward along $v$ of (1) the uniform measure on $T$ and (2) $H^{1/2}$ noise on $T$. Our proofs exploit the integrability of the model as described by Nazarov-Sklyanin (2013). As time permits, we discuss structural connections to the theory of the topological recursion.
After Fourier series, the quantum Hopf-Burgers equation $v_t +vv_x = 0$ with periodic boundary conditions is equivalent to a system of coupled quantum harmonic oscillators, which may be prepared in Glauber's coherent states as initial conditions. Sending the displacement of each oscillator to infinity at the same rate, we (1) confirm and (2) determine corrections to the quantum-classical correspondence principle. After diagonalizing the ...
05E10 ; 20G43 ; 37K10