Asymptotic expansions for some integrals of quotients with degenerated divisors

We study asymptotic expansion as $\nu\to0$ for integrals over ${ \mathbb{R} }^{2d}=\{(x,y)\}$ of quotients $F(x,y) \big/ \big( (x\cdot y)^2+(\nu \Gamma(x,y))^2\big)^{-1}$, where $\Gamma$ is strictly positive and $F$ decays at infinity sufficiently fast. Integrals of this kind appear in description of the four--waves interactions

Damped-driven KdV and effective equation for long-time behaviour of its solutions

For the damped-driven KdV equation $$\dot u-\nu{u_{xx}}+u_{xxx}-6uu_x=\sqrt\nu \eta(t,x), x\in S^1, \int u dx\equiv \int\eta dx\equiv0,$$ with $0<\nu\le1$ and smooth in $x$ white in $t$ random force $\eta$, we study the limiting long-time behaviour of the KdV integrals of motions $(I_1,I_2,...)$, evaluated along a solution $u^\nu(t,x)$, as $\nu\to0$. We prove that %if $u=u^\nu(t,x)$ is a solution of the equation above, for $0\le\tau:= \nu t \lesssim1$ the vector $I^\nu(\tau)=(I_1(u^\nu(\tau,\cdot)),I_2(u^\nu(\tau,\cdot)),...),$ converges in distribution to a limiting process $I^0(\tau)=(I^0_1,I^0_2,...)$. The $j$-th component $I_j^0$ equals \12(v_j(\tau)^2+v_{-j}(\tau)^2), where $v(\tau)=(v_1(\tau), v_{-1}(\tau),v_2(\tau),...)$ is the vector of Fourier coefficients of a solution of an {\it effective equation} for the dam-ped-driven KdV. This new equation is a quasilinear stochastic heat equation with a non-local nonlinearity, written in the Fourier coefficients. It is well posed

Rigorous results in space-periodic two-dimensional turbulence

We survey the recent advance in the rigorous qualitative theory of the 2d stochastic Navier-Stokes system that are relevant to the description of turbulence in two-dimensional fluids. After discussing briefly the initial-boundary value problem and the associated Markov process, we formulate results on the existence, uniqueness and mixing of a stationary measure. We next turn to various consequences of these properties: strong law of large numbers, central limit theorem, and random attractors related to a unique stationary measure. We also discuss the Donsker-Varadhan and Freidlin-Wentzell type large deviations, as well as the inviscid limit and asymptotic results in 3d thin domains. We conclude with some open problems

Vey theorem in infinite dimensions and its application to KdV

We consider an integrable infinite-dimensional Hamiltonian system in a Hilbert space $H=\{u=(u_1^+,u_1^-; u_2^+,u_2^-;....)\}$ with integrals $I_1, I_2,...$ which can be written as $I_j={1/2}|F_j|^2$, where $F_j:H\to \R^2$, $F_j(0)=0$ for $j=1,2,...$ . We assume that the maps $F_j$ define a germ of an analytic diffeomorphism $F=(F_1,F_2,...):H\to H$, such that dF(0)=id$,$(F-id)$is a$\kappa$-smoothing map ($\kappa\geq 0$) and some other mild restrictions on$F$hold. Under these assumptions we show that the maps$F_j$may be modified to maps$F_j^\prime$such that$F_j-F_j^\prime=O(|u|^2)$and each$\frac12|F'_j|^2$still is an integral of motion. Moreover, these maps jointly define a germ of an analytic symplectomorphism$F^\prime: H\to H$, the germ$(F^\prime-id)$is$\kappa$-smoothing, and each$I_j$is an analytic function of the vector$(\frac12|F'_j|^2,j\ge1)$. Next we show that the theorem with$\kappa=1$ applies to the KdV equation. It implies that in the vicinity of the origin in a functional space KdV admits the Birkhoff normal form and the integrating transformation has the form `identity plus a 1-smoothing analytic map'

KdV equation under periodic boundary conditions and its perturbations

In this paper we discuss properties of the KdV equation under periodic boundary conditions, especially those which are important to study perturbations of the equation. Next we review what is known now about long-time behaviour of solutions for perturbed KdV equations

Reducibility of the quantum harmonic oscillator in d-dimensions with polynomial time-dependent perturbation

We prove a reducibility result for a quantum harmonic oscillator in arbitrary dimension with arbitrary frequencies perturbed by a linear operator which is a polynomial of degree 2 in (xj, -i 02,j) with coefficients which depend quasiperiodically on time

The effective equation method

In this chapter we present a general method of constructing the effective equation which describes the behaviour of small-amplitude solutions for a nonlinear PDE in finite volume, provided that the linear part of the equation is a hamiltonian system with a pure imaginary discrete spectrum. The effective equation is obtained by retaining only the resonant terms of the nonlinearity (which may be hamiltonian, or may be not); the assertion that it describes the limiting behaviour of small-amplitude solutions is a rigorous mathematical theorem. In particular, the method applies to the three-- and four--wave systems. We demonstrate that different possible types of energy transport are covered by this method, depending on whether the set of resonances splits into finite clusters (this happens, e.g. in case of the Charney-Hasegawa-Mima equation), or is connected (this happens, e.g. in the case of the NLS equation if the space-dimension is at least two). For equations of the first type the energy transition to high frequencies does not hold, while for equations of the second type it may take place. In the case of the NLS equation we use next some heuristic approximation from the arsenal of wave turbulence to show that under the iterated limit "the volume goes to infinity", taken after the limit "the amplitude of oscillations goes to zero", the energy spectrum of solutions for the effective equation is described by a Zakharov-type kinetic equation. Evoking the Zakharov ansatz we show that stationary in time and homogeneous in space solutions for the latter equation have a power law form. Our method applies to various weakly nonlinear wave systems, appearing in plasma, meteorology and oceanology