23 research outputs found
Conformal invariant functionals of immersions of tori into R^3
We show, that higher analogs of the Willmore functional, defined on the space
of immersions M^2\rightarrow R^3, where M^2 is a two-dimensional torus, R^3 is
the 3-dimensional Euclidean space are invariant under conformal transformations
of R^3. This hypothesis was formulated recently by I.A.Taimanov
(dg-ga/9610013).
Higher analogs of the Willmore functional are defined in terms of the
Modified Novikov-Veselov hierarchy. This soliton hierarchy is associated with
the zero-energy scattering problem for the two-dimensional Dirac operator.Comment: 34 pages, LaTeX, amssym.def macros use
Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve : ergodicity, isochrony, periodicity and fractals
We study the complexification of the one-dimensional Newtonian particle in a
monomial potential. We discuss two classes of motions on the associated Riemann
surface: the rectilinear and the cyclic motions, corresponding to two different
classes of real and autonomous Newtonian dynamics in the plane. The rectilinear
motion has been studied in a number of papers, while the cyclic motion is much
less understood. For small data, the cyclic time trajectories lead to
isochronous dynamics. For bigger data the situation is quite complicated;
computer experiments show that, for sufficiently small degree of the monomial,
the motion is generically periodic with integer period, which depends in a
quite sensitive way on the initial data. If the degree of the monomial is
sufficiently high, computer experiments show essentially chaotic behaviour. We
suggest a possible theoretical explanation of these different behaviours. We
also introduce a one-parameter family of 2-dimensional mappings, describing the
motion of the center of the circle, as a convenient representation of the
cyclic dynamics; we call such mapping the center map. Computer experiments for
the center map show a typical multi-fractal behaviour with periodicity islands.
Therefore the above complexification procedure generates dynamics amenable to
analytic treatment and possessing a high degree of complexity.Comment: LaTex, 28 pages, 10 figure
Inverse scattering at fixed energy on surfaces with Euclidean ends
On a fixed Riemann surface with Euclidean ends and genus ,
we show that, under a topological condition, the scattering matrix S_V(\la)
at frequency \la > 0 for the operator determines the potential
if for all
and for some , where denotes the distance
from to a fixed point . The topological condition is given by
for and by if . In \rr^2 this
implies that the operator S_V(\la) determines any potential
such that for all .Comment: 21 page
Topological superfluid He-B: fermion zero modes on interfaces and in the vortex core
Many quantum condensed matter systems are strongly correlated and strongly
interacting fermionic systems, which cannot be treated perturbatively. However,
topology allows us to determine generic features of their fermionic spectrum,
which are robust to perturbation and interaction. We discuss the nodeless 3D
system, such as superfluid He-B, vacuum of Dirac fermions, and relativistic
singlet and triplet supercondutors which may arise in quark matter. The
systems, which have nonzero value of topological invariant, have gapless
fermions on the boundary and in the core of quantized vortices. We discuss the
index theorem which relates fermion zero modes on vortices with the topological
invariants in combined momentum and coordinate space.Comment: paper is prepared for Proceedings of the Workshop on Vortices,
Superfluid Dynamics, and Quantum Turbulence held on 11-16 April 2010, Lammi,
Finlan
The spectral curve of a quaternionic holomorphic line bundle over a 2-torus
A conformal immersion of a 2-torus into the 4-sphere is characterized by an
auxiliary Riemann surface, its spectral curve. This complex curve encodes the
monodromies of a certain Dirac type operator on a quaternionic line bundle
associated to the immersion. The paper provides a detailed description of the
geometry and asymptotic behavior of the spectral curve. If this curve has
finite genus the Dirichlet energy of a map from a 2-torus to the 2-sphere or
the Willmore energy of an immersion from a 2-torus into the 4-sphere is given
by the residue of a specific meromorphic differential on the curve. Also, the
kernel bundle of the Dirac type operator evaluated over points on the 2-torus
linearizes in the Jacobian of the spectral curve. Those results are presented
in a geometric and self contained manner.Comment: 36 page
Numerical instability of the Akhmediev breather and a finite-gap model of it
In this paper we study the numerical instabilities of the NLS Akhmediev
breather, the simplest space periodic, one-mode perturbation of the unstable
background, limiting our considerations to the simplest case of one unstable
mode. In agreement with recent theoretical findings of the authors, in the
situation in which the round-off errors are negligible with respect to the
perturbations due to the discrete scheme used in the numerical experiments, the
split-step Fourier method (SSFM), the numerical output is well-described by a
suitable genus 2 finite-gap solution of NLS. This solution can be written in
terms of different elementary functions in different time regions and,
ultimately, it shows an exact recurrence of rogue waves described, at each
appearance, by the Akhmediev breather. We discover a remarkable empirical
formula connecting the recurrence time with the number of time steps used in
the SSFM and, via our recent theoretical findings, we establish that the SSFM
opens up a vertical unstable gap whose length can be computed with high
accuracy, and is proportional to the inverse of the square of the number of
time steps used in the SSFM. This neat picture essentially changes when the
round-off error is sufficiently large. Indeed experiments in standard double
precision show serious instabilities in both the periods and phases of the
recurrence. In contrast with it, as predicted by the theory, replacing the
exact Akhmediev Cauchy datum by its first harmonic approximation, we only
slightly modify the numerical output. Let us also remark, that the first rogue
wave appearance is completely stable in all experiments and is in perfect
agreement with the Akhmediev formula and with the theoretical prediction in
terms of the Cauchy data.Comment: 27 pages, 8 figures, Formula (30) at page 11 was corrected, arXiv
admin note: text overlap with arXiv:1707.0565
The type numbers of closed geodesics
A short survey on the type numbers of closed geodesics, on applications of
the Morse theory to proving the existence of closed geodesics and on the recent
progress in applying variational methods to the periodic problem for Finsler
and magnetic geodesicsComment: 29 pages, an appendix to the Russian translation of "The calculus of
variations in the large" by M. Mors
Quantum phase transitions from topology in momentum space
Many quantum condensed matter systems are strongly correlated and strongly
interacting fermionic systems, which cannot be treated perturbatively. However,
physics which emerges in the low-energy corner does not depend on the
complicated details of the system and is relatively simple. It is determined by
the nodes in the fermionic spectrum, which are protected by topology in
momentum space (in some cases, in combination with the vacuum symmetry). Close
to the nodes the behavior of the system becomes universal; and the universality
classes are determined by the toplogical invariants in momentum space. When one
changes the parameters of the system, the transitions are expected to occur
between the vacua with the same symmetry but which belong to different
universality classes. Different types of quantum phase transitions governed by
topology in momentum space are discussed in this Chapter. They involve Fermi
surfaces, Fermi points, Fermi lines, and also the topological transitions
between the fully gapped states. The consideration based on the momentum space
topology of the Green's function is general and is applicable to the vacua of
relativistic quantum fields. This is illustrated by the possible quantum phase
transition governed by topology of nodes in the spectrum of elementary
particles of Standard Model.Comment: 45 pages, 17 figures, 83 references, Chapter for the book "Quantum
Simulations via Analogues: From Phase Transitions to Black Holes", to appear
in Springer lecture notes in physics (LNP
Conformal invariant functionals of immersions of tori into R³
We show, that higher analogs of the Willmore functional, defined on the space of immersions M 2 ! R 3 , where M 2 is a two-dimensional torus, R 3 is the 3-dimensional Euclidean space are invariant under conformal transformations of R 3 . This hypothesis was formulated recently by I. A. Taimanov. Higher analogs of the Willmore functional are defined in terms of the Modified Novikov-Veselov hierarchy. This soliton hierarchy is associated with the zero-energy scattering problem for the two-dimensional Dirac operator. 1 Introduction To start with, we would like to recall the following interesting fact from the theory of 2-dimensional surfaces in R 3 (see [20], p. 110 and references therein). Let X : M 2 ! R 3 be a smooth immersion of a compact orientable surface M 2 into the Euclidean space R 3 (i.e. a smooth map from M 2 to R 3 1 This work was fulfilled during the author's visit to the Freie Universitat, Berlin, Germany, which was supported by the Humboldt-Foundat..