52 research outputs found
New geometries associated with the nonlinear Schr\"{o}dinger equation
We apply our recent formalism establishing new connections between the
geometry of moving space curves and soliton equations, to the nonlinear
Schr\"{o}dinger equation (NLS).
We show that any given solution of the NLS gets associated with three
distinct space curve evolutions. The tangent vector of the first of these
curves, the binormal vector of the second and the normal vector of the third,
are shown to satisfy the integrable Landau-Lifshitz (LL) equation
, (). These connections
enable us to find the three surfaces swept out by the moving curves associated
with the NLS. As an example, surfaces corresponding to a stationary envelope
soliton solution of the NLS are obtained.Comment: 13 pages, 3 figure
Geometric Phase and Classical-Quantum Correspondence
We study the geometric phase factors underlying the classical and the
corresponding quantum dynamics of a driven nonlinear oscillator exhibiting
chaotic dynamics. For the classical problem, we compute the geometric phase
factors associated with the phase space trajectories using Frenet-Serret
formulation. For the corresponding quantum problem, the geometric phase
associated with the time evolution of the wave function is computed. Our
studies suggest that the classical geometric phase may be related to the the
difference in the quantum geometric phases between two neighboring eigenstates.Comment: Copy with higher resolution figures can be obtained from
http://physics.gmu.edu/~isatija by clicking on publications. to appear in the
Yukawa Institute conference proceedings, {\it Quantum Mechanics and Chaos:
From Fundamental Problems through Nano-Science} (2003
Solitons in a hard-core bosonic system: Gross-Pitaevskii type and beyond
A unified formulation that obtains solitary waves for various background
densities in the Bose-Einstein condensate of a system of hard-core bosons with
nearest neighbor attractive interactions is presented.
In general, two species of solitons appear: A nonpersistent (NP) type that
fully delocalizes at its maximum speed, and a persistent (P) type that survives
even at its maximum speed, and transforms into a periodic train of solitons
above this speed. When the background condensate density is nonzero, both
species coexist, the soliton is associated with a constant intrinsic frequency,
and its maximum speed is the speed of sound. In contrast, when the background
condensate density is zero, the system has neither a fixed frequency, nor a
speed of sound. Here, the maximum soliton speed depends on the frequency, which
can be tuned to lead to a cross-over between the NP-type and the P-type at a
certain critical frequency, determined by the energy parameters of the system.
We provide a single functional form for the soliton profile, from which diverse
characteristics for various background densities can be obtained. Using the
mapping to spin systems enables us to characterize the corresponding class of
magnetic solitons in
Heisenberg spin chains with different types of anisotropy, in a unified
fashion
Nonlinear dynamics of the classical isotropic Heisenberg antiferromagnetic chain: the sigma model sector and the kink sector
We identify two distinct low-energy sectors in the classical isotropic
antiferromagnetic Heisenberg spin-S chain. In the continuum limit, we show that
two types of rotation generators arise for the field in each sector. Using
these, the Lagrangian for sector I is shown to be that of the nonlinear sigma
model. Sector II has a null Lagrangian; Its Hamiltonian density is just the
Pontryagin term. Exact solutions are found in the form of magnons and
precessing pulses in I and moving kinks in II. The kink has `spin' S. Sector I
has a higher minimum energy than II.Comment: 4 page
Other incarnations of the Gross-Pitaevskii dark soliton
We show that the dark soliton of the Gross-Pitaevskii equation (GPE) that
describes the Bose-Einstein condensate (BEC) density of a system of weakly
repulsive bosons, also describes that of a system of strongly repulsive hard
core bosons at half filling. As a consequence of this, the GPE soliton gets
related to the magnetic soliton in an easy-plane ferromagnet, where it
describes the square of the in-plane magnetization of the system. These
relationships are shown to be useful in understanding various characteristics
of solitons in these distinct many-body systems
- …