Inverse Scattering Problem for Vector Fields and the Cauchy Problem for the Heavenly Equation

We solve the inverse scattering problem for multidimensional vector fields and we use this result to construct the formal solution of the Cauchy problem for the second heavenly equation of Plebanski, a scalar nonlinear partial differential equation in four dimensions relevant in General Relativity, which arises from the commutation of multidimensional Hamiltonian vector fields.Comment: 15 pages, submitted to Phisics Letters A. This paper replaces nlin.SI/051204

Hyper-elliptic Nambu flow associated with integrable maps

We study hyper-elliptic Nambu flows associated with some $n$ dimensional maps and show that discrete integrable systems can be reproduced as flows of this class.Comment: 13 page

Linearly Coupled Bose-Einstein Condensates: From Rabi Oscillations and Quasi-Periodic Solutions to Oscillating Domain Walls and Spiral Waves

In this paper, an exact unitary transformation is examined that allows for the construction of solutions of coupled nonlinear Schr{\"o}dinger equations with additional linear field coupling, from solutions of the problem where this linear coupling is absent. The most general case where the transformation is applicable is identified. We then focus on the most important special case, namely the well-known Manakov system, which is known to be relevant for applications in Bose-Einstein condensates consisting of different hyperfine states of $^{87}$Rb. In essence, the transformation constitutes a distributed, nonlinear as well as multi-component generalization of the Rabi oscillations between two-level atomic systems. It is used here to derive a host of periodic and quasi-periodic solutions including temporally oscillating domain walls and spiral waves.Comment: 6 pages, 4 figures, Phys. Rev. A (in press

Functional representation of the Volterra hierarchy

In this paper I study the functional representation of the Volterra hierarchy (VH). Using the Miwa's shifts I rewrite the infinite set of Volterra equations as one functional equation. This result is used to derive a formal solution of the associated linear problem, a generating function for the conservation laws and to obtain a new form of the Miura and Backlund transformations. I also discuss some relations between the VH and KP hierarchy.Comment: 17 pages, submitted to Journal of Nonlinear Mathematical Physic

Storing and processing optical information with ultra-slow light in Bose-Einstein condensates

We theoretically explore coherent information transfer between ultra-slow light pulses and Bose-Einstein condensates (BECs) and find that storing light pulses in BECs, by switching off the coupling field, allows the coherent condensate dynamics to process optical information. We develop a formalism, applicable in both the weak and strong probe regimes, to analyze such experiments and establish several new results. Investigating examples relevant to Rb-87 experimental parameters we see a variety of novel two-component BEC dynamics occur during the storage, including interference fringes, gentle breathing excitations, and two-component solitons. We find the dynamics when the levels |F=1, M_F=-1> and |F=2, M_F=+1> are well suited to designing controlled processing of the information. By switching the coupling field back on, the processed information is rewritten onto probe pulses which then propagate out as slow light pulses. We calculate the fidelity of information transfer between the atomic and light fields upon the switch-on and subsequent output. The fidelity is affected both by absorption of small length scale features and absorption of regions of the pulse stored near the condensate edge. In the strong probe case, we find that when the oscillator strengths for the two transitions are equal the fidelity is not strongly sensitive to the probe strength, while when they are unequal the fidelity is worse for stronger probes. Applications to distant communication between BECs, squeezed light generation and quantum information are anticipated.Comment: 19 pages, 12 figures, submitted to Phys. Rev.

On the Equivalence of Different Lax Pairs for the Kac-van Moerbeke Hierarchy

We give a simple algebraic proof that the two different Lax pairs for the Kac-van Moerbeke hierarchy, constructed from Jacobi respectively super-symmetric Dirac-type difference operators, give rise to the same hierarchy of evolution equations. As a byproduct we obtain some new recursions for computing these equations.Comment: 8 page

Iso-spectral deformations of general matrix and their reductions on Lie algebras

We study an iso-spectral deformation of general matrix which is a natural generalization of the Toda lattice equation. We prove the integrability of the deformation, and give an explicit formula for the solution to the initial value problem. The formula is obtained by generalizing the orthogonalization procedure of Szeg\"{o}. Based on the root spaces for simple Lie algebras, we consider several reductions of the hierarchy. These include not only the integrable systems studied by Bogoyavlensky and Kostant, but also their generalizations which were not known to be integrable before. The behaviors of the solutions are also studied. Generically, there are two types of solutions, having either sorting property or blowing up to infinity in finite time.Comment: 25 pages, AMSLaTe

A few things I learnt from Jurgen Moser

A few remarks on integrable dynamical systems inspired by discussions with Jurgen Moser and by his work.Comment: An article for the special issue of "Regular and Chaotic Dynamics" dedicated to 80-th anniversary of Jurgen Mose

Instabilities in the two-dimensional cubic nonlinear Schrodinger equation

The two-dimensional cubic nonlinear Schrodinger equation (NLS) can be used as a model of phenomena in physical systems ranging from waves on deep water to pulses in optical fibers. In this paper, we establish that every one-dimensional traveling wave solution of NLS with trivial phase is unstable with respect to some infinitesimal perturbation with two-dimensional structure. If the coefficients of the linear dispersion terms have the same sign then the only unstable perturbations have transverse wavelength longer than a well-defined cut-off. If the coefficients of the linear dispersion terms have opposite signs, then there is no such cut-off and as the wavelength decreases, the maximum growth rate approaches a well-defined limit.Comment: 4 pages, 4 figure

Interaction of pulses in nonlinear Schroedinger model

The interaction of two rectangular pulses in nonlinear Schroedinger model is studied by solving the appropriate Zakharov-Shabat system. It is shown that two real pulses may result in appearance of moving solitons. Different limiting cases, such as a single pulse with a phase jump, a single chirped pulse, in-phase and out-of-phase pulses, and pulses with frequency separation, are analyzed. The thresholds of creation of new solitons and multi-soliton states are found.Comment: 9 pages, 7 figures. Accepted to Phys. Rev. E, 200