V. Novokshenov
Publications:
Novokshenov V. Y.
Poles of tritronquée solution to the Painlevé I equation and cubic anharmonic oscillator
2010, vol. 15, no. 6, pp. 390403
Abstract
The distribution of poles of zeroparameter 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.

Novokshenov V. Y.
Poles of tritronquée solution to the Painlevé I equation and cubic anharmonic oscillator
2010, vol. 15, no. 23, pp. 390403
Abstract
The distribution of poles of zeroparameter 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.

Novokshenov V. Y.
ZeroDispersion Limit to the Kortewegde Vries Equation: a Dressing Chain Approach
2008, vol. 13, no. 5, pp. 424430
Abstract
An asymptotic solution of the KdV equation with small dispersion is studied for the case of smooth humplike initial condition with monotonically decreasing slopes. Despite the wellknown 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 Whithamtype approximaton of the leading term by solving the dressing chain through a finitegap 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.

Novokshenov V. Y.
Adiabatic deformations of integrable twofrequency Hamiltonians
2006, vol. 11, no. 2, pp. 299310
Abstract
The paper discusses a mechanism of excitation and control of twofrequency 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.
