127 research outputs found
Pair excitations and the mean field approximation of interacting Bosons, II
We consider a large number of Bosons with interaction potential . In earlier papers we considered a set of equations for
the condensate and pair excitation function and proved that they
provide a Fock space approximation to the exact evolution of the condensate for
. This result was extended to the case
by E. Kuz, where it was also argued informally that the equations of our
earlier work do not provide an approximation for . In 2013,
we introduced a coupled refinement of our original equations and conjectured
that they provide a Fock space approximation in the range . In
the current paper we prove that this is indeed the case for , at least locally in time. In order to do that, we re-formulate
the equations of \cite{GMM} in a way reminiscent of BBGKY and apply harmonic
analysis techniques in the spirit of X. Chen and J. Holmer to prove the
necessary estimates. In turn, these estimates provide bounds for the pair
excitation function
Second-order corrections to mean-field evolution of weakly interacting Bosons, II
We study the evolution of a N-body weakly interacting system of Bosons. Our
work forms an extension of our previous paper I, in which we derived a
second-order correction to a mean-field evolution law for coherent states in
the presence of small interaction potential. Here, we remove the assumption of
smallness of the interaction potential and prove global existence of solutions
to the equation for the second-order correction. This implies an improved
Fock-space estimate for our approximation of the N-body state
Pair excitations and the mean field approximation of interacting Bosons, I
In our previous work \cite{GMM1},\cite{GMM2} we introduced a correction to
the mean field approximation of interacting Bosons. This correction describes
the evolution of pairs of particles that leave the condensate and subsequently
evolve on a background formed by the condensate. In \cite{GMM2} we carried out
the analysis assuming that the interactions are independent of the number of
particles . Here we consider the case of stronger interactions. We offer a
new transparent derivation for the evolution of pair excitations. Indeed, we
obtain a pair of linear equations describing their evolution. Furthermore, we
obtain apriory estimates independent of the number of particles and use these
to compare the exact with the approximate dynamics
Orbital stability of periodic waves for the nonlinear Schroedinger equation
The nonlinear Schroedinger equation has several families of quasi-periodic
travelling waves, each of which can be parametrized up to symmetries by two
real numbers: the period of the modulus of the wave profile, and the variation
of its phase over a period (Floquet exponent). In the defocusing case, we show
that these travelling waves are orbitally stable within the class of solutions
having the same period and the same Floquet exponent. This generalizes a
previous work where only small amplitude solutions were considered. A similar
result is obtained in the focusing case, under a non-degeneracy condition which
can be checked numerically. The proof relies on the general approach to orbital
stability as developed by Grillakis, Shatah, and Strauss, and requires a
detailed analysis of the Hamiltonian system satisfied by the wave profile.Comment: 34 pages, 7 figure
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