31 research outputs found
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
KAM for the quantum harmonic oscillator
In this paper we prove an abstract KAM theorem for infinite dimensional
Hamiltonians systems. This result extends previous works of S.B. Kuksin and J.
P\"oschel and uses recent techniques of H. Eliasson and S.B. Kuksin. As an
application we show that some 1D nonlinear Schr\"odinger equations with
harmonic potential admits many quasi-periodic solutions. In a second
application we prove the reducibility of the 1D Schr\"odinger equations with
the harmonic potential and a quasi periodic in time potential.Comment: 54 pages. To appear in Comm. Math. Phy
Normal Forms for Semilinear Quantum Harmonic Oscillators
We consider the semilinear harmonic oscillator i\psi_t=(-\Delta +\va{x}^{2}
+M)\psi +\partial_2 g(\psi,\bar \psi), \quad x\in \R^d, t\in \R where is
a Hermite multiplier and a smooth function globally of order 3 at least. We
prove that such a Hamiltonian equation admits, in a neighborhood of the origin,
a Birkhoff normal form at any order and that, under generic conditions on
related to the non resonance of the linear part, this normal form is integrable
when and gives rise to simple (in particular bounded) dynamics when
. As a consequence we prove the almost global existence for solutions
of the above equation with small Cauchy data. Furthermore we control the high
Sobolev norms of these solutions
Geometric optics and instability for semi-classical Schrodinger equations
We prove some instability phenomena for semi-classical (linear or) nonlinear
Schrodinger equations. For some perturbations of the data, we show that for
very small times, we can neglect the Laplacian, and the mechanism is the same
as for the corresponding ordinary differential equation. Our approach allows
smaller perturbations of the data, where the instability occurs for times such
that the problem cannot be reduced to the study of an o.d.e.Comment: 22 pages. Corollary 1.7 adde
Periodic solutions for a class of nonlinear partial differential equations in higher dimension
We prove the existence of periodic solutions in a class of nonlinear partial
differential equations, including the nonlinear Schroedinger equation, the
nonlinear wave equation, and the nonlinear beam equation, in higher dimension.
Our result covers cases where the bifurcation equation is infinite-dimensional,
such as the nonlinear Schroedinger equation with zero mass, for which solutions
which at leading order are wave packets are shown to exist.Comment: 34 page
Exponential mixing of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noises
International audienceWe prove the strong Feller property and exponential mixing for 3D stochastic Navier-Stokes equation driven by mildly degenerate noises (i.e. all but finitely many Fourier modes are forced) via Kolmogorov equation approach
Universally Coupled Massive Gravity, II: Densitized Tetrad and Cotetrad Theories
Einstein's equations in a tetrad formulation are derived from a linear theory
in flat spacetime with an asymmetric potential using free field gauge
invariance, local Lorentz invariance and universal coupling. The gravitational
potential can be either covariant or contravariant and of almost any density
weight. These results are adapted to produce universally coupled massive
variants of Einstein's equations, yielding two one-parameter families of
distinct theories with spin 2 and spin 0. The theories derived, upon fixing the
local Lorentz gauge freedom, are seen to be a subset of those found by
Ogievetsky and Polubarinov some time ago using a spin limitation principle. In
view of the stability question for massive gravities, the proven non-necessity
of positive energy for stability in applied mathematics in some contexts is
recalled. Massive tetrad gravities permit the mass of the spin 0 to be heavier
than that of the spin 2, as well as lighter than or equal to it, and so provide
phenomenological flexibility that might be of astrophysical or cosmological
use.Comment: 2 figures. Forthcoming in General Relativity and Gravitatio