N. A. Kostov

N-wave interactions related to simple Lie algebras. ℤ2-reductions and soliton solutions

The reductions of the integrable N-wave type equations solvable by the inverse scattering method with the generalized Zakharov-Shabat systems L and related to some simple Lie algebra are analysed. The Zakharov-Shabat dressing method is extended to the case when is an orthogonal algebra. Several types of one-soliton solutions of the corresponding N-wave equations and their reductions are studied. We show that one can relate a (semi-)simple subalgebra of to each soliton solution. We illustrate our results by four-wave equations related to so(5) which find applications in Stokes-anti-Stokes wave generation.

Reductions of N -wave interactions related to low-rank simple Lie algebras: I. Z2-reductions

The analysis and the classification of all reductions for the nonlinear evolution equations solvable by the inverse scattering method is an interesting and still open problem. We show how the second-order reductions of the N -wave interactions related to low-rank simple Lie algebras g can be embedded also in the Weyl group of g. This allows us to display along with the well known ones a number of new types of integrable N -wave systems. Some of the reduced systems find applications to nonlinear optics.

Solutions of multi-component NLS models and Spinor Bose–Einstein condensates

Abstract Three- and five-component nonlinear Schrodinger-type models, which describe spinor Bose–Einstein condensates (BEC’s) with hyperfine structures F = 1 and F = 2 , respectively, are studied. These models for particular values of the coupling constants are integrable by the inverse scattering method. They are related to symmetric spaces of BD.I-type ≃ SO(2r + 1) / SO(2) × SO(2r −1) for r = 2 and r = 3 . Using conveniently modified Zakharov–Shabat dressing procedure we obtain different types of soliton solutions.

On classification of soliton solutions of multicomponent nonlinear evolution equations

We consider several ways of how one could classify the various types of soliton solutions related to multicomponent nonlinear evolution equations which are solvable by the inverse scattering method for the generalized Zakharov–Shabat system related to a simple Lie algebra g. In doing so we make use of the fundamental analytic solutions, the Zakharov–Shabat dressing procedure, the reduction technique and other tools characteristic for that method. The multicomponent solitons are characterized by several important factors: the subalgebras of g and the way these subalgebras are embedded in g, the dimension of the corresponding eigensubspaces of the Lax operator L, as well as by additional constraints imposed by reductions.

On the soliton solutions of the spinor Bose-Einstein condensate

A three-component nonlinear Schrodinger-type model which describes spinor Bose-Einstein condensates is studied. These types of χ3-interactions are integrable by the inverse scattering method. We analyze its Hamiltonian properties and outline an algebraic procedure to derive their three types of soliton solutions based on Zakharov-Shabat dressing method. Their applications to spinor model of Bose-Einstein condensates are discussed.

Inverse scattering transform analysis of Stokes-anti-Stokes stimulated Raman scattering.

Zakharov-Shabat--Ablowitz-Kaup-Newel-Segur representation for Stokes-anti-Stokes stimulated Raman scattering is proposed. Periodical waves, solitons and self-similarity solutions are derived. Transient and bright threshold solitons are discussed.

Exact Solutions for Equations of Bose-Fermi Mixtures in One-Dimensional Optical Lattice

We present two new families of stationary solutions for equations of Bose-Fermi mixtures with an elliptic function potential with modulus k. We also discuss particular cases when the quasiperiodic solutions become periodic ones. In the limit of a sinusoidal potential (k ! 0) our solutions model a quasi-one dimensional quantum degenerate Bose- Fermi mixture trapped in optical lattice. In the limit k ! 1 the solutions are expressed by hyperbolic function solutions (vector solitons). Thus we are able to obtain in an unified way quasi-periodic and periodic waves, and solitons. The precise conditions for existence of every class of solutions are derived. There are indications that such waves and localized objects may be observed in experiments with cold quantum degenerate gases.

Bose-Einstein condensates with F=1 and F=2: reductions and soliton interactions of multi-component NLS models

We analyze a class of multicomponent nonlinear Schrödinger equations (MNLS) related to the symmetric BD.I-type symmetric spaces and their reductions. We briefly outline the direct and the inverse scattering method for the relevant Lax operators and the soliton solutions. We use the Zakharov-Shabat dressing method to obtain the two-soliton solution and analyze the soliton interactions of the MNLS equations and some of their reductions.

Fordy-Kulish model and spinor Bose-Einstein condensate

Abstract A three-component nonlinear Schrodinger-type model which describes spinor Bose-Einstein condensate ( BEC) is considered. This model is integrable by the inverse scattering method and using Zakharov-Shabat dressing method we obtain three types of soliton solutions. The multi-component nonlinear Schrödinger type models related to symmetric spaces C.I ≃ Sp(4)/U(2) is studied.

N-Wave Equations with Orthogonal Algebras: Z 2 and Z 2 ◊ Z 2 Reductions and Soliton Solutions ?

A method and apparatus for the fabrication of optical lenses by injection/compression molding of thermoplastic includes a plurality of sleeves, each having a bore therethrough, and a plurality of mold inserts dimensioned to be received in the bore with minimal clearance for sliding fit. The mold inserts each include a precision optical surface adapted to form a front or back surface of a lens. A selected pair of mold inserts are placed in the bore of a sleeve with front and back surface forming optical surfaces in confronting relationship to define a mold cavity, and the assembly is heated to a temperature above the glass transition temperature of the thermoplastic to be molded. An injection port extends through the sleeve to the bore, and is positioned to inject thermoplastic that has been heated to a fluid state into the cavity. After injection of the thermoplastic, the mold inserts are compressed together, and excess thermoplastic is forced out of the mold cavity. The mold inserts are then translated together relative to the sleeve to uncouple the injection port from the cavity. Compressive pressure is then maintained on the mold inserts while the mold assembly is cooled below the glass transition temperature. The mold inserts are pulled from the sleeve, and the finished lens is removed.We consider