4,707 research outputs found

### Exact real-time dynamics of the quantum Rabi model

We use the analytical solution of the quantum Rabi model to obtain absolutely
convergent series expressions of the exact eigenstates and their scalar
products with Fock states. This enables us to calculate the numerically exact
time evolution of and for all regimes of the
coupling strength, without truncation of the Hilbert space. We find a
qualitatively different behavior of both observables which can be related to
their representations in the invariant parity subspaces.Comment: 8 pages, 7 figures, published versio

### Nonlocal symmetries of Riccati and Abel chains and their similarity reductions

We study nonlocal symmetries and their similarity reductions of Riccati and
Abel chains. Our results show that all the equations in Riccati chain share the
same form of nonlocal symmetry. The similarity reduced $N^{th}$ order ordinary
differential equation (ODE), $N=2, 3,4,...$, in this chain yields $(N-1)^{th}$
order ODE in the same chain. All the equations in the Abel chain also share the
same form of nonlocal symmetry (which is different from the one that exist in
Riccati chain) but the similarity reduced $N^{th}$ order ODE, $N=2, 3,4,$, in
the Abel chain always ends at the $(N-1)^{th}$ order ODE in the Riccati chain.
We describe the method of finding general solution of all the equations that
appear in these chains from the nonlocal symmetry.Comment: Accepted for publication in J. Math. Phy

### Construction of classical superintegrable systems with higher order integrals of motion from ladder operators

We construct integrals of motion for multidimensional classical systems from
ladder operators of one-dimensional systems. This method can be used to obtain
new systems with higher order integrals. We show how these integrals generate a
polynomial Poisson algebra. We consider a one-dimensional system with third
order ladders operators and found a family of superintegrable systems with
higher order integrals of motion. We obtain also the polynomial algebra
generated by these integrals. We calculate numerically the trajectories and
show that all bounded trajectories are closed.Comment: 10 pages, 4 figures, to appear in j.math.phys

### Extension of Nikiforov-Uvarov Method for the Solution of Heun Equation

We report an alternative method to solve second order differential equations
which have at most four singular points. This method is developed by changing
the degrees of the polynomials in the basic equation of Nikiforov-Uvarov (NU)
method. This is called extended NU method for this paper. The eigenvalue
solutions of Heun equation and confluent Heun equation are obtained via
extended NU method. Some quantum mechanical problems such as Coulomb problem on
a 3-sphere, two Coulombically repelling electrons on a sphere and hyperbolic
double-well potential are investigated by this method

### Helical Magnetorotational Instability in Magnetized Taylor-Couette Flow

Hollerbach and Rudiger have reported a new type of magnetorotational
instability (MRI) in magnetized Taylor-Couette flow in the presence of combined
axial and azimuthal magnetic fields. The salient advantage of this "helical''
MRI (HMRI) is that marginal instability occurs at arbitrarily low magnetic
Reynolds and Lundquist numbers, suggesting that HMRI might be easier to realize
than standard MRI (axial field only). We confirm their results, calculate HMRI
growth rates, and show that in the resistive limit, HMRI is a weakly
destabilized inertial oscillation propagating in a unique direction along the
axis. But we report other features of HMRI that make it less attractive for
experiments and for resistive astrophysical disks. Growth rates are small and
require large axial currents. More fundamentally, instability of highly
resistive flow is peculiar to infinitely long or periodic cylinders: finite
cylinders with insulating endcaps are shown to be stable in this limit. Also,
keplerian rotation profiles are stable in the resistive limit regardless of
axial boundary conditions. Nevertheless, the addition of toroidal field lowers
thresholds for instability even in finite cylinders.Comment: 16 pages, 2 figures, 1 table, submitted to PR

### Periodic solutions of a many-rotator problem in the plane. II. Analysis of various motions

Various solutions are displayed and analyzed (both analytically and
numerically) of arecently-introduced many-body problem in the plane which
includes both integrable and nonintegrable cases (depending on the values of
the coupling constants); in particular the origin of certain periodic behaviors
is explained. The light thereby shone on the connection among
\textit{integrability} and \textit{analyticity} in (complex) time, as well as
on the emergence of a \textit{chaotic} behavior (in the guise of a sensitive
dependance on the initial data) not associated with any local exponential
divergence of trajectories in phase space, might illuminate interesting
phenomena of more general validity than for the particular model considered
herein.Comment: Published by JNMP at http://www.sm.luth.se/math/JNMP

### Transformations of Heun's equation and its integral relations

We find transformations of variables which preserve the form of the equation
for the kernels of integral relations among solutions of the Heun equation.
These transformations lead to new kernels for the Heun equation, given by
single hypergeometric functions (Lambe-Ward-type kernels) and by products of
two hypergeometric functions (Erd\'elyi-type). Such kernels, by a limiting
process, also afford new kernels for the confluent Heun equation.Comment: This version was published in J. Phys. A: Math. Theor. 44 (2011)
07520

### Out of equilibrium dynamics of coherent non-abelian gauge fields

We study out-of-equilibrium dynamics of intense non-abelian gauge fields.
Generalizing the well-known Nielsen-Olesen instabilities for constant initial
color-magnetic fields, we investigate the impact of temporal modulations and
fluctuations in the initial conditions. This leads to a remarkable coexistence
of the original Nielsen-Olesen instability and the subdominant phenomenon of
parametric resonance. Taking into account that the fields may be correlated
only over a limited transverse size, we model characteristic aspects of the
dynamics of color flux tubes relevant in the context of heavy-ion collisions.Comment: 12 pages, 10 figures; PRD version, minor change

### On convergence towards a self-similar solution for a nonlinear wave equation - a case study

We consider the problem of asymptotic stability of a self-similar attractor
for a simple semilinear radial wave equation which arises in the study of the
Yang-Mills equations in 5+1 dimensions. Our analysis consists of two steps. In
the first step we determine the spectrum of linearized perturbations about the
attractor using a method of continued fractions. In the second step we
demonstrate numerically that the resulting eigensystem provides an accurate
description of the dynamics of convergence towards the attractor.Comment: 9 pages, 5 figure

### Lagrangian Formalism for nonlinear second-order Riccati Systems: one-dimensional Integrability and two-dimensional Superintegrability

The existence of a Lagrangian description for the second-order Riccati
equation is analyzed and the results are applied to the study of two different
nonlinear systems both related with the generalized Riccati equation. The
Lagrangians are nonnatural and the forces are not derivable from a potential.
The constant value $E$ of a preserved energy function can be used as an
appropriate parameter for characterizing the behaviour of the solutions of
these two systems. In the second part the existence of two--dimensional
versions endowed with superintegrability is proved. The explicit expressions of
the additional integrals are obtained in both cases. Finally it is proved that
the orbits of the second system, that represents a nonlinear oscillator, can be
considered as nonlinear Lissajous figuresComment: 25 pages, 7 figure

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