51 research outputs found
On certain classes of solutions of the Weierstrass-Enneper system inducing constant mean curvature surfaces
Analysis of the generalized Weierstrass-Enneper system includes the
estimation of the degree of indeterminancy of the general analytic solution and
the discussion of the boundary value problem. Several different procedures for
constructing certain classes of solutions to this system, including potential,
harmonic and separable types of solutions, are proposed. A technique for
reduction of the Weierstrass-Enneper system to decoupled linear equations, by
subjecting it to certain differential constraints, is presented as well. New
elementary and doubly periodic solutions are found, among them kinks, bumps and
multi-soliton solutions
Links between different analytic descriptions of constant mean curvature surfaces
Transformations between different analytic descriptions of constant mean
curvature (CMC) surfaces are established. In particular, it is demonstrated
that the system descriptive of CMC surfaces within the
framework of the generalized Weierstrass representation, decouples into a
direct sum of the elliptic Sh-Gordon and Laplace equations. Connections of this
system with the sigma model equations are established. It is pointed out, that
the instanton solutions correspond to different Weierstrass parametrizations of
the standard sphere
Generalized Weierstrass representation for surfaces in multidimensional Riemann spaces
Generalizations of the Weierstrass formulae to generic surface immersed into
, and into multidimensional Riemann spaces are proposed. Integrable
deformations of surfaces in these spaces via the modified Veselov-Novikov
equation are discussed.Comment: LaTeX, 20 pages, minor misprints correcte
Quantum effects for extrinsic geometry of strings via the generalized Weierstrass representation
The generalized Weierstrass representation for surfaces in is
used to study quantum effects for strings governed by Polyakov-Nambu-Goto
action. Correlators of primary fields are calculated exactly in one-loop
approximation for the pure extrinsic Polyakov action. Geometrical meaning of
infrared singularity is discussed. The Nambu-Goto and spontaneous curvature
actions are treated perturbatively.Comment: Latex, 13 page
On the dbar-dressing method applicable to heavenly equation
The \dbar-dressing scheme based on local nonlinear vector \dbar-problem
is developed. It is applicable to multidimensional nonlinear equations for
vector fields, and, after Hamiltonian reduction, to heavenly equation.
Hamiltonian reduction is described explicitely in terms of the \dbar-data. An
analogue of Hirota bilinear identity for heavenly equation hierarchy is
introduced, -function for the hierarchy is defined. Addition formulae
(generating equations) for the -function are found. It is demonstrated
that -function for heavenly equation hierarchy is given by the action for
\dbar-problem evaluated on the solution of this problem.Comment: 11 page
Nonlinear Beltrami equation and tau-function for dispersionless hierarchies
It is proved that the action for nonlinear Beltrami equation (quasiclassical
dbar-problem) evaluated on its solution gives a tau-function for dispersionless
KP hierarchy. Infinitesimal transformations of tau-function corresponding to
variations of dbar-data are found. Determinant equations for the function
generating these transformations are derived. They represent a dispersionless
analogue of singular manifold (Schwarzian) KP equations. Dispersionless 2DTL
hierarchy is also considered.Comment: 12 page
Coisotropic deformations of associative algebras and dispersionless integrable hierarchies
The paper is an inquiry of the algebraic foundations of the theory of
dispersionless integrable hierarchies, like the dispersionless KP and modified
KP hierarchies and the universal Whitham's hierarchy of genus zero. It stands
out for the idea of interpreting these hierarchies as equations of coisotropic
deformations for the structure constants of certain associative algebras. It
discusses the link between the structure constants and the Hirota's tau
function, and shows that the dispersionless Hirota's bilinear equations are,
within this approach, a way of writing the associativity conditions for the
structure constants in terms of the tau function. It also suggests a simple
interpretation of the algebro-geometric construction of the universal Whitham's
equations of genus zero due to Krichever.Comment: minor misprints correcte
Explode-decay dromions in the non-isospectral Davey-Stewartson I (DSI) equation
In this letter, we report the existence of a novel type of explode-decay
dromions, which are exponentially localized coherent structures whose amplitude
varies with time, through Hirota method for a nonisospectral Davey-Stewartson
equation I discussed recently by Jiang. Using suitable transformations, we also
point out such solutions also exist for the isospectral Davey-Stewartson I
equation itself for a careful choice of the potentials
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
Generalized Weierstrass Relations and Frobenius reciprocity
This article investigates local properties of the further generalized
Weierstrass relations for a spin manifold immersed in a higher dimensional
spin manifold from viewpoint of study of submanifold quantum mechanics. We
show that kernel of a certain Dirac operator defined over , which we call
submanifold Dirac operator, gives the data of the immersion. In the derivation,
the simple Frobenius reciprocity of Clifford algebras and plays
important roles.Comment: 17pages. to be published in Mathematical Physics, Analysis and
Geometr
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