V. Novokshenov

The distribution of poles of zero-parameter solution to Painlevé I, specified by P. Boutroux as intégrale tritronquée, is studied in the complex plane. This solution has regular asymptotics $−\sqrt{z/6}+ O(1)$ as $z \to \infty$, $|\arg z| < 4\pi/5$. At the sector $|\arg z| > 4\pi/5$ it is a meromorphic function with regular asymptotic distribution of poles at infinity. This fact together with numeric simulations for $|z| < const$ allowed B. Dubrovin to make a conjecture that all poles of the intégrale tritronquée belong to this sector. As a step to prove this conjecture, we study the Riemann–Hilbert (RH) problem related to the specified solution of the Painlevé I equation. It is "undressed" to a similar RH problem for the Schrödinger equation with cubic potential. The latter determines all coordinates of poles for the intégrale tritronquée via a Bohr–Sommerfeld quantization conditions.

The distribution of poles of zero-parameter solution to Painlevé I, specified by P. Boutroux as intégrale tritronquée, is studied in the complex plane. This solution has regular asymptotics $−\sqrt{z/6} + O(1)$ as $z \to \infty$, $|\arg z| < 4\pi/5$. At the sector $|\arg z| > 4\pi/5$ it is a meromorphic function with regular asymptotic distribution of poles at infinity. This fact together with numeric simulations for $|z| < const$ allowed B. Dubrovin to make a conjecture that all poles of the intégrale tritronquée belong to this sector. As a step to prove this conjecture, we study the Riemann–Hilbert (RH) problem related to the specified solution of the Painlevé I equation. It is "undressed" to a similar RH problem for the Schrödinger equation with cubic potential. The latter determines all coordinates of poles for the intégrale tritronquée via a Bohr–Sommerfeld quantization conditions.

An asymptotic solution of the KdV equation with small dispersion is studied for the case of smooth hump-like initial condition with monotonically decreasing slopes. Despite the well-known approaches by Lax–Levermore and Gurevich–Pitaevskii, a new way of constructing the asymptotics is proposed using the inverse scattering transform together with the dressing chain technique developed by A. Shabat [1]. It provides the Whitham-type approximaton of the leading term by solving the dressing chain through a finite-gap asymptotic ansatz. This yields the Whitham equations on the Riemann invariants together with hodograph transform which solves these equations explicitly. Thus we reproduce an uniform in $x$ asymptotics consisting of smooth solution of the Hopf equation outside the oscillating domain and a slowly modulated cnoidal wave within the domain. Finally, the dressing chain technique provides the proof of an asymptotic estimate for the leading term.

The paper discusses a mechanism of excitation and control of two-frequency oscillations in the integrable Hamiltonian systems. It is close to the autoresonant technique for controlling the amplitude of nonlinear modes. Autoresonance is usually associated with single frequency mode excitations due to the synchronization and phase lock of various nonlinear modes with the driving force. Despite this we propose a model of multifrequency autoresonance which occur in completely integrable systems. This phenomenon is due to a number stable invariant tori governed by integrals of motion of the integrable system. The basic autoresonant effect of phase locking appears here as Whitham deformations of the invariant tori. This provides also a possibility to transfer a certain initial $n$-periodic motion to a given $m$-periodic motion as a final state.