221 research outputs found
Time-averaging for weakly nonlinear CGL equations with arbitrary potentials
Consider weakly nonlinear complex Ginzburg--Landau (CGL) equation of the
form: under the periodic boundary conditions, where and
is a smooth function. Let be
the -basis formed by eigenfunctions of the operator . For a
complex function , write it as and
set . Then for any solution of the linear
equation we have . In this work it is
proved that if equation with a sufficiently smooth real potential
is well posed on time-intervals , then for any its
solution , the limiting behavior of the curve
on time intervals of order , as
, can be uniquely characterized by a solution of a certain
well-posed effective equation:
where is a resonant averaging of the nonlinearity . We
also prove a similar results for the stochastically perturbed equation, when a
white in time and smooth in random force of order is added
to the right-hand side of the equation.
The approach of this work is rather general. In particular, it applies to
equations in bounded domains in under Dirichlet boundary conditions
Asymptotic Properties of Integrals of Quotients when the Numerator Oscillates and the Denominator Degenerates
We study asymptotic expansion as ν→0 for integrals over ℝ²d={(x,y)} of quotients of the form F(x,y)cos(λx∙y)/((x∙y)²+ν²), where λ≥0 and F decays at infinity sufficiently fast. Integrals of this kind appear in the theory of wave turbulence.Ми вивчаємо асимптотичне поводження при ν→0 iнтегралiв в ℝ²d = {(x,y)} вiд виразiв вигляду F(x,y)cos(λx∙y)/((x∙y)²+ν²), де λ≥0 i F досить швидко спадає на нескiнченностi. Подiбнi iнтеграли виникають в теорi ї хвильової турбулентностi
Qualitative features of periodic solutions of KdV
In this paper we prove new qualitative features of solutions of KdV on the
circle. The first result says that the Fourier coefficients of a solution of
KdV in Sobolev space , admit a WKB type expansion up to first
order with strongly oscillating phase factors defined in terms of the KdV
frequencies. The second result provides estimates for the approximation of such
a solution by trigonometric polynomials of sufficiently large degree
On finite-dimensional projections of distributions for solutions of randomly forced 2D Navier-Stokes equations
The paper is devoted to studying the image of probability measures
on a Hilbert space under finite-dimensional analytic maps. We establish
sufficient conditions under which the image of a measure has a density
with respect to the Lebesgue measure and continuously depends on the
map. The results obtained are applied to the 2D Navier\u2013Stokes equations
perturbed by various random forces of low dimension
Derivation of a wave kinetic equation from the resonant-averaged stochastic NLS equation
We suggest a new derivation of a wave kinetic equation for the spectrum of the weakly nonlinear Schr\uf6dinger equation with stochastic forcing. The kinetic equation is obtained as a result of a double limiting procedure. Firstly, we consider the equation on a finite box with periodic boundary conditions and send the size of the nonlinearity and of the forcing to zero, while the time is correspondingly rescaled; then, the size of the box is sent to infinity (with a suitable rescaling of the solution). We report here the results of the first limiting procedure, analysed with full rigour in Kuksin and Maiocchi (0000), and show how the second limit leads to a kinetic equation for the spectrum, if some further hypotheses (commonly employed in the weak turbulence theory) are accepted. Finally we show how to derive from these equations the Kolmogorov-Zakharov spectra
Random data Cauchy theory for supercritical wave equations II : A global existence result
We prove that the subquartic wave equation on the three dimensional ball
, with Dirichlet boundary conditions admits global strong solutions for
a large set of random supercritical initial data in .
We obtain this result as a consequence of a general random data Cauchy theory
for supercritical wave equations developed in our previous work \cite{BT2} and
invariant measure considerations which allow us to obtain also precise large
time dynamical informations on our solutions
Quasi-periodic solutions of completely resonant forced wave equations
We prove existence of quasi-periodic solutions with two frequencies of
completely resonant, periodically forced nonlinear wave equations with periodic
spatial boundary conditions. We consider both the cases the forcing frequency
is: (Case A) a rational number and (Case B) an irrational number.Comment: 25 pages, 1 figur
Near-linear dynamics in KdV with periodic boundary conditions
Near linear evolution in Korteweg de Vries (KdV) equation with periodic
boundary conditions is established under the assumption of high frequency
initial data. This result is obtained by the method of normal form reduction
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