1,440 research outputs found

### On integrability of the differential constraints arising from the singularity analysis

Integrability of the differential constraints arising from the singularity
analysis of two (1+1)-dimensional second-order evolution equations is studied.
Two nonlinear ordinary differential equations are obtained in this way, which
are integrable by quadratures in spite of very complicated branching of their
solutions.Comment: arxiv version is already offcia

### Energy transmission in the forbidden bandgap of a nonlinear chain

A nonlinear chain driven by one end may propagate energy in the forbidden
band gap by means of nonlinear modes. For harmonic driving at a given
frequency, the process ocurs at a threshold amplitude by sudden large energy
flow, that we call nonlinear supratransmission. The bifurcation of energy
transmission is demonstrated numerically and experimentally on the chain of
coupled pendula (sine-Gordon and nonlinear Klein-Gordon equations) and
sustained by an extremely simple theory.Comment: LaTex file, 6 figures, published in Phys Rev Lett 89 (2002) 13410

### Ablowitz-Ladik system with discrete potential. I. Extended resolvent

Ablowitz-Ladik linear system with range of potential equal to {0,1} is
considered. The extended resolvent operator of this system is constructed and
the singularities of this operator are analyzed in detail.Comment: To be published in Theor. Math. Phy

### Collisions of solitons and vortex rings in cylindrical Bose-Einstein condensates

Interactions of solitary waves in a cylindrically confined Bose-Einstein
condensate are investigated by simulating their head-on collisions. Slow vortex
rings and fast solitons are found to collide elastically contrary to the
situation in the three-dimensional homogeneous Bose gas. Strongly inelastic
collisions are absent for low density condensates but occur at higher densities
for intermediate velocities. The scattering behaviour is rationalised by use of
dispersion diagrams. During inelastic collisions, spherical shell-like
structures of low density are formed and they eventually decay into depletion
droplets with solitary wave features. The relation to similar shells observed
in a recent experiment [Ginsberg et al. Phys Rev. Lett. 94, 040403 (2005)] is
discussed

### Elliptical instability of a rapidly rotating, strongly stratified fluid

The elliptical instability of a rotating stratified fluid is examined in the
regime of small Rossby number and order-one Burger number corresponding to
rapid rotation and strong stratification. The Floquet problem describing the
linear growth of disturbances to an unbounded, uniform-vorticity elliptical
flow is solved using exponential asymptotics. The results demonstrate that the
flow is unstable for arbitrarily strong rotation and stratification; in
particular, both cyclonic and anticyclonic flows are unstable. The instability
is weak, however, with growth rates that are exponentially small in the Rossby
number. The analytic expression obtained for the growth rate elucidates its
dependence on the Burger number and on the eccentricity of the elliptical flow.
It explains in particular the weakness of the instability of cyclonic flows,
with growth rates that are only a small fraction of those obtained for the
corresponding anticyclonic flows. The asymptotic results are confirmed by
numerical solutions of Floquet problem.Comment: 17 page

### On Dispersive and Classical Shock Waves in Bose-Einstein Condensates and Gas Dynamics

A Bose-Einstein condensate (BEC) is a quantum fluid that gives rise to
interesting shock wave nonlinear dynamics. Experiments depict a BEC that
exhibits behavior similar to that of a shock wave in a compressible gas, eg.
traveling fronts with steep gradients. However, the governing Gross-Pitaevskii
(GP) equation that describes the mean field of a BEC admits no dissipation
hence classical dissipative shock solutions do not explain the phenomena.
Instead, wave dynamics with small dispersion is considered and it is shown that
this provides a mechanism for the generation of a dispersive shock wave (DSW).
Computations with the GP equation are compared to experiment with excellent
agreement. A comparison between a canonical 1D dissipative and dispersive shock
problem shows significant differences in shock structure and shock front speed.
Numerical results associated with the three dimensional experiment show that
three and two dimensional approximations are in excellent agreement and one
dimensional approximations are in good qualitative agreement. Using one
dimensional DSW theory it is argued that the experimentally observed blast
waves may be viewed as dispersive shock waves.Comment: 24 pages, 28 figures, submitted to Phys Rev

### Spectral decomposition for the Dirac system associated to the DSII equation

A new (scalar) spectral decomposition is found for the Dirac system in two
dimensions associated to the focusing Davey--Stewartson II (DSII) equation.
Discrete spectrum in the spectral problem corresponds to eigenvalues embedded
into a two-dimensional essential spectrum. We show that these embedded
eigenvalues are structurally unstable under small variations of the initial
data. This instability leads to the decay of localized initial data into
continuous wave packets prescribed by the nonlinear dynamics of the DSII
equation

### Exact solution of Riemann--Hilbert problem for a correlation function of the XY spin chain

A correlation function of the XY spin chain is studied at zero temperature.
This is called the Emptiness Formation Probability (EFP) and is expressed by
the Fredholm determinant in the thermodynamic limit. We formulate the
associated Riemann--Hilbert problem and solve it exactly. The EFP is shown to
decay in Gaussian.Comment: 7 pages, to be published in J. Phys. Soc. Jp

### A note on the integrable discretization of the nonlinear Schr\"odinger equation

We revisit integrable discretizations for the nonlinear Schr\"odinger
equation due to Ablowitz and Ladik. We demonstrate how their main drawback, the
non-locality, can be overcome. Namely, we factorize the non-local difference
scheme into the product of local ones. This must improve the performance of the
scheme in the numerical computations dramatically. Using the equivalence of the
Ablowitz--Ladik and the relativistic Toda hierarchies, we find the
interpolating Hamiltonians for the local schemes and show how to solve them in
terms of matrix factorizations.Comment: 24 pages, LaTeX, revised and extended versio

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