390 research outputs found
On the dispersionless Kadomtsev-Petviashvili equation in n+1 dimensions: exact solutions, the Cauchy problem for small initial data and wave breaking
We study the (n+1)-dimensional generalization of the dispersionless
Kadomtsev-Petviashvili (dKP) equation, a universal equation describing the
propagation of weakly nonlinear, quasi one dimensional waves in n+1 dimensions,
and arising in several physical contexts, like acoustics, plasma physics and
hydrodynamics. For n=2, this equation is integrable, and it has been recently
shown to be a prototype model equation in the description of the two
dimensional wave breaking of localized initial data. We construct an exact
solution of the n+1 dimensional model containing an arbitrary function of one
variable, corresponding to its parabolic invariance, describing waves, constant
on their paraboloidal wave front, breaking simultaneously in all points of it.
Then we use such solution to build a uniform approximation of the solution of
the Cauchy problem, for small and localized initial data, showing that such a
small and localized initial data evolving according to the (n+1)-dimensional
dKP equation break, in the long time regime, if and only if n=1,2,3; i.e., in
physical space. Such a wave breaking takes place, generically, in a point of
the paraboloidal wave front, and the analytic aspects of it are given
explicitly in terms of the small initial data.Comment: 20 pages, 10 figures, few formulas adde
Nekhoroshev theorem for the periodic Toda lattice
The periodic Toda lattice with sites is globally symplectomorphic to a
two parameter family of coupled harmonic oscillators. The action
variables fill out the whole positive quadrant of . We prove that in
the interior of the positive quadrant as well as in a neighborhood of the
origin, the Toda Hamiltonian is strictly convex and therefore Nekhoroshev's
theorem applies on (almost) all parts of phase space.Comment: 28 page
Ground-state hyperfine structure of H-, Li-, and B-like ions in middle-Z region
The hyperfine splitting of the ground state of H-, Li-, and B-like ions is
investigated in details within the range of nuclear numbers Z = 7-28. The
rigorous QED approach together with the large-scale configuration-interaction
Dirac-Fock-Sturm method are employed for the evaluation of the
interelectronic-interaction contributions of first and higher orders in 1/Z.
The screened QED corrections are evaluated to all orders in (\alpha Z)
utilizing an effective potential approach. The influence of nuclear
magnetization distribution is taken into account within the single-particle
nuclear model. The specific differences between the hyperfine-structure level
shifts of H- and Li-like ions, where the uncertainties associated with the
nuclear structure corrections are significantly reduced, are also calculated.Comment: 22 pages, 11 tables, 2 figure
Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking
We have recently solved the inverse spectral problem for integrable PDEs in
arbitrary dimensions arising as commutation of multidimensional vector fields
depending on a spectral parameter . The associated inverse problem, in
particular, can be formulated as a non linear Riemann Hilbert (NRH) problem on
a given contour of the complex plane. The most distinguished examples
of integrable PDEs of this type, like the dispersionless
Kadomtsev-Petviashivili (dKP), the heavenly and the 2 dimensional
dispersionless Toda equations, are real PDEs associated with Hamiltonian vector
fields. The corresponding NRH data satisfy suitable reality and symplectic
constraints. In this paper, generalizing the examples of solvable NRH problems
illustrated in \cite{MS4,MS5,MS6}, we present a general procedure to construct
solvable NRH problems for integrable real PDEs associated with Hamiltonian
vector fields, allowing one to construct implicit solutions of such PDEs
parametrized by an arbitrary number of real functions of a single variable.
Then we illustrate this theory on few distinguished examples for the dKP and
heavenly equations. For the dKP case, we characterize a class of similarity
solutions, a class of solutions constant on their parabolic wave front and
breaking simultaneously on it, and a class of localized solutions breaking in a
point of the plane. For the heavenly equation, we characterize two
classes of symmetry reductions.Comment: 29 page
Symmetric and asymmetric solitons in linearly coupled Bose-Einstein condensates trapped in optical lattices
We study spontaneous symmetry breaking in a system of two parallel
quasi-one-dimensional traps, equipped with optical lattices (OLs) and filled
with a Bose-Einstein condensate (BEC). The cores are linearly coupled by
tunneling. Analysis of the corresponding system of linearly coupled
Gross-Pitaevskii equations (GPEs) reveals that spectral bandgaps of the single
GPE split into subgaps. Symmetry breaking in two-component BEC solitons is
studied in cases of the attractive (AA) and repulsive (RR) nonlinearity in both
traps; the mixed situation, with repulsion in one trap and attraction in the
other (RA), is considered too. In all the cases, stable asymmetric solitons are
found, bifurcating from symmetric or antisymmetric ones (and destabilizing
them), in the AA and RR systems, respectively. In either case, bi-stability is
predicted, with a nonbifurcating stable branch, either antisymmetric or
symmetric, coexisting with asymmetric ones. Solitons destabilized by the
bifurcation tend to rearrange themselves into their stable asymmetric
counterparts. The impact of a phase mismatch, between the OLs in the two cores
is also studied. Also considered is a related model, for a binary BEC in a
single-core trap with the OL, assuming that the two species (representing
different spin states of the same atom) are coupled by linear interconversion.
In that case, the symmetry-breaking bifurcations in the AA and RR models switch
their character, if the inter-species nonlinear interaction becomes stronger
than the intra-species nonlinearity.Comment: 21 pages + 24 figs, accepted to Phys. Rev.
Stochastic Pulse Switching in a Degenerate Resonant Optical Medium
Using the idealized integrable Maxwell-Bloch model, we describe random
optical-pulse polarization switching along an active optical medium in the
Lambda-configuration with disordered occupation numbers of its lower energy
sub-level pair. The description combines complete integrability and stochastic
dynamics. For the single-soliton pulse, we derive the statistics of the
electric-field polarization ellipse at a given point along the medium in closed
form. If the average initial population difference of the two lower sub-levels
vanishes, we show that the pulse polarization will switch intermittently
between the two circular polarizations as it travels along the medium. If this
difference does not vanish, the pulse will eventually forever remain in the
circular polarization determined by which sub-level is more occupied on
average. We also derive the exact expressions for the statistics of the
polarization-switching dynamics, such as the probability distribution of the
distance between two consecutive switches and the percentage of the distance
along the medium the pulse spends in the elliptical polarization of a given
orientation in the case of vanishing average initial population difference. We
find that the latter distribution is given in terms of the well-known arcsine
law
Bright-Dark Soliton Complexes in Spinor Bose-Einstein Condensates
We present bright-dark vector solitons in quasi-one-dimensional spinor (F=1)
Bose-Einstein condensates. Using a multiscale expansion technique, we reduce
the corresponding nonintegrable system of three coupled Gross-Pitaevskii
equations (GPEs) to a completely integrable Yajima-Oikawa system. In this way,
we obtain approximate solutions for small-amplitude vector solitons of
dark-dark-bright and bright-bright-dark types, in terms of the
spinor components, respectively. By means of numerical simulations of the full
GPE system, we demonstrate that these states indeed feature soliton properties,
i.e., they propagate undistorted and undergo quasi-elastic collisions. It is
also shown that, in the presence of a parabolic trap of strength , the
bright component(s) is (are) guided by the dark one(s), and, as a result, the
small-amplitude vector soliton as a whole performs harmonic oscillations of
frequency in the shallow soliton limit. We investigate
numerically deviations from this prediction, as the depth of the solitons is
increased, as well as when the strength of the spin-dependent interaction is
modified.Comment: 10 pages, 4 figures. Submitted to PRA, May 200
Time and temperature dependent correlation functions of the 1D impenetrable electron gas
We consider the one-dimensional delta-interacting electron gas in the case of
infinite repulsion. We use determinant representations to study the long time,
large distance asymptotics of correlation functions of local fields in the gas
phase. We derive differential equations which drive the correlation functions.
Using a related Riemann-Hilbert problem we obtain formulae for the asymptotics
of the correlation functions, which are valid at all finite temperatures. At
low temperatures these formulae lead to explicit asymptotic expressions for the
correlation functions, which describe power law behavior and exponential decay
as functions of temperature, magnetic field and chemical potential.Comment: minor changes, to appear in Int. J. Mod. Phys.
Electron shielding of the nuclear magnetic moment in hydrogen-like atom
The correction to the wave function of the ground state in a hydrogen-like
atom due to an external homogenous magnetic field is found exactly in the
parameter . The projection of the correction to the wave
function of the state due to the external homogeneous magnetic field
is found for arbitrary . The projection of the correction to the
wave function of the state due to the nuclear magnetic moment is
also found for arbitrary . Using these results, we have calculated the
shielding of the nuclear magnetic moment by the electron.Comment: 15 page
Discrete Lagrangian systems on the Virasoro group and Camassa-Holm family
We show that the continuous limit of a wide natural class of the
right-invariant discrete Lagrangian systems on the Virasoro group gives the
family of integrable PDE's containing Camassa-Holm, Hunter-Saxton and
Korteweg-de Vries equations. This family has been recently derived by Khesin
and Misiolek as Euler equations on the Virasoro algebra for
-metrics. Our result demonstrates a universal nature of
these equations.Comment: 6 pages, no figures, AMS-LaTeX. Version 2: minor changes. Version 3:
minor change
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