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 ${\bf S}_u = {\bf S} \times {\bf S}_{ss}$, (${\bf S}^2=1$). 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