52 research outputs found
Quasiperiodic Solutions of the Fibre Optics Coupled Nonlinear Schr{\"o}dinger Equations
We consider travelling periodical and quasiperiodical waves in single mode
fibres, with weak birefringence and under the action of cross-phase modulation.
The problem is reduced to the ``1:2:1" integrable case of the two-particle
quartic potential. A general approach for finding elliptic solutions is given.
New solutions which are associated with two-gap Treibich-Verdier potentials are
found. General quasiperiodic solutions are given in terms of two dimensional
theta functions with explicit expressions for frequencies in terms of theta
constants. The reduction of quasiperiodic solutions to elliptic functions is
discussed.Comment: 24 page
The Lax pairs for the Holt system
By using non-canonical transformation between the Holt system and the
Henon-Heiles system the Lax pairs for all the integrable cases of the Holt
system are constructed from the known Lax representations for the Henon-Heiles
system.Comment: 7 pages, LaTeX2e, a4.st
Identities for hyperelliptic P-functions of genus one, two and three in covariant form
We give a covariant treatment of the quadratic differential identities
satisfied by the P-functions on the Jacobian of smooth hyperelliptic curves of
genera 1, 2 and 3
Separation of variables for the generalized Henon-Heiles system and system with quartic potential
We consider two well-known integrable systems on the plane using the concept
of natural Poisson bivectors on Riemaninan manifolds. Geometric approach to
construction of variables of separation and separated relations for the
generalized Henon-Heiles system and the generalized system with quartic
potential is discussed in detail.Comment: 9 pages, LaTeX with AMS font
Wannier functions of elliptic one-gap potentials
Wannier functions of the one dimensional Schroedinger equation with elliptic
one gap potentials are explicitly constructed. Properties of these functions
are analytically and numerically investigated. In particular we derive an
expression for the amplitude of the Wannier function in the origin, a power
series expansion valid in the vicinity of the origin and an asymptotic
expansion characterizing the decay of the Wannier function at large distances.
Using these results we construct an approximate analytical expression of the
Wannier function which is valid in the whole spatial domain and is in good
agreement with numerical results.Comment: 24 pages, 5 figure
Elliptic Solitons and Groebner Bases
We consider the solution of spectral problems with elliptic coefficients in
the framework of the Hermite ansatz. We show that the search for exactly
solvable potentials and their spectral characteristics is reduced to a system
of polynomial equations solvable by the Gr\"obner bases method and others. New
integrable potentials and corresponding solutions of the Sawada-Kotera,
Kaup-Kupershmidt, Boussinesq equations and others are found.Comment: 18 pages, no figures, LaTeX'2
Nonequilibrium Cooper pairing in the nonadiabatic regime
We obtain a complete solution for the mean-field dynamics of the BCS paired
state with a large, but finite number of Cooper pairs in the non-adiabatic
regime. We show that the problem reduces to a classical integrable Hamiltonian
system and derive a complete set of its integrals of motion. The condensate
exhibits irregular multi-frequency oscillations ergodically exploring the part
of the phase-space allowed by the conservation laws. In the thermodynamic limit
however the system can asymptotically reach a steady state.Comment: 4 pages, no figure
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