8 research outputs found
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
Small-amplitude excitations in a deformable discrete nonlinear Schroedinger equation
A detailed analysis of the small-amplitude solutions of a deformed discrete
nonlinear Schr\"{o}dinger equation is performed. For generic deformations the
system possesses "singular" points which split the infinite chain in a number
of independent segments. We show that small-amplitude dark solitons in the
vicinity of the singular points are described by the Toda-lattice equation
while away from the singular points are described by the Korteweg-de Vries
equation. Depending on the value of the deformation parameter and of the
background level several kinds of solutions are possible. In particular we
delimit the regions in the parameter space in which dark solitons are stable in
contrast with regions in which bright pulses on nonzero background are
possible. On the boundaries of these regions we find that shock waves and
rapidly spreading solutions may exist.Comment: 18 pages (RevTex), 13 figures available upon reques
Doubly periodic waves of a discrete nonlinear Schr\"odinger system with saturable nonlinearity
A system of two discrete nonlinear Schr\"odinger equations of the
Ablowitz-Ladik type with a saturable nonlinearity is shown to admit a doubly
periodic wave, whose long wave limit is also derived. As a by-product, several
new solutions of the elliptic type are provided for NLS-type discrete and
continuous systems.Comment: 12 pages, to appear, Journal of nonlinear mathematical physic
Rotational Surfaces in and Solutions in the Nonlinear Sigma Model
The Gauss map of non-degenerate surfaces in the three-dimensional Minkowski
space are viewed as dynamical fields of the two-dimensional O(2,1) Nonlinear
Sigma Model. In this setting, the moduli space of solutions with rotational
symmetry is completely determined. Essentially, the solutions are warped
products of orbits of the 1-dimensional groups of isometries and elastic curves
in either a de Sitter plane, a hyperbolic plane or an anti de Sitter plane. The
main tools are the equivalence of the two-dimensional O(2,1) Nonlinear Sigma
Model and the Willmore problem, and the description of the surfaces with
rotational symmetry. A complete classification of such surfaces is obtained in
this paper. Indeed, a huge new family of Lorentzian rotational surfaces with a
space-like axis is presented. The description of this new class of surfaces is
based on a technique of surgery and a gluing process, which is illustrated by
an algorithm.Comment: PACS: 11.10.Lm; 11.10.Ef; 11.15.-q; 11.30.-j; 02.30.-f; 02.40.-k. 45
pages, 11 figure
'Universality' of the Ablowitz-Ladik hierarchy
Consiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7, Rome / CNR - Consiglio Nazionale delle RichercheSIGLEITItal