1,014 research outputs found
On the formation of singularities of solutions of nonlinear differential systems in antistokes directions
We determine the position and the type of spontaneous singularities of
solutions of generic analytic nonlinear differential systems in the complex
plane, arising along antistokes directions towards irregular singular points of
the system. Placing the singularity of the system at infinity we look at
equations of the form with
analytic in a neighborhood of , with genericity
assumptions; is then a rank one singular point. We analyze the
singularities of those solutions which tend to zero for in some sectorial region, on the edges of the maximal region (also
described) with this property. After standard normalization of the differential
system, it is shown that singularities occuring in antistokes directions are
grouped in nearly periodical arrays of similar singularities as ,
the location of the array depending on the solution while the (near-) period
and type of singularity are determined by the form of the differential system.Comment: 61
Analytic linearization of nonlinear perturbations of Fuchsian systems
Nonlinear perturbation of Fuchsian systems are studied in regions including
two singularities. Such systems are not necessarily analytically equivalent to
their linear part (they are not linearizable). Nevertheless, it is shown that
in the case when the linear part has commuting monodromy, and the eigenvalues
have positive real parts, there exists a unique correction function of the
nonlinear part so that the corrected system becomes analytically linearizable
Evolution of a model quantum system under time periodic forcing: conditions for complete ionization
We analyze the time evolution of a one-dimensional quantum system with an
attractive delta function potential whose strength is subjected to a time
periodic (zero mean) parametric variation . We show that for generic
, which includes the sum of any finite number of harmonics, the
system, started in a bound state will get fully ionized as . This
is irrespective of the magnitude or frequency (resonant or not) of .
There are however exceptional, very non-generic , that do not lead to
full ionization, which include rather simple explicit periodic functions. For
these the system evolves to a nontrivial localized stationary state
which is related to eigenfunctions of the Floquet operator
Decay of a Bound State under a Time-Periodic Perturbation: a Toy Case
We study the time evolution of a three dimensional quantum particle,
initially in a bound state, under the action of a time-periodic zero range
interaction with ``strength'' (\alpha(t)). Under very weak generic conditions
on the Fourier coefficients of (\alpha(t)), we prove complete ionization as (t
\to \infty). We prove also that, under the same conditions, all the states of
the system are scattering states.Comment: LaTeX2e, 15 page
Singular normal form for the Painlev\'e equation P1
We show that there exists a rational change of coordinates of Painlev\'e's P1
equation and of the elliptic equation after which these
two equations become analytically equivalent in a region in the complex phase
space where and are unbounded. The region of equivalence comprises all
singularities of solutions of P1 (i.e. outside the region of equivalence,
solutions are analytic). The Painlev\'e property of P1 (that the only movable
singularities are poles) follows as a corollary. Conversely, we argue that the
Painlev\'e property is crucial in reducing P1, in a singular regime, to an
equation integrable by quadratures
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