765 research outputs found
Applications of Loop Group Factorization to Geometric Soliton Equations
The 1-d Schrodinger flow on 2-sphere, the Gauss-Codazzi equation for flat
Lagrangian submanifolds in C^n, and the space-time monopole equation are all
examples of geometric soliton equations. The linear systems with a spectral
parameter (Lax pair) associated to these equations satisfy the reality
condition associated to SU(n). In this article, we explain the method developed
jointly with K. Uhlenbeck, that uses various loop group factorizations to
construct inverse scattering transforms, Backlund transformations, and
solutions to Cauchy problems for these equations
Geometries and Symmetries of Soliton equations and Integrable Elliptic equations
We give a review of the systematic construction of hierarchies of soliton
flows and integrable elliptic equations associated to a complex semi-simple Lie
algebra and finite order automorphisms. For example, the non-linear
Schr\"odinger equation, the n-wave equation, and the sigma-model are soliton
flows; and the equation for harmonic maps from the plane to a compact Lie
group, for primitive maps from the plane to a -symmetric space, and constant
mean curvature surfaces and isothermic surfaces in space forms are integrable
elliptic systems. We also give a survey of
(i) construction of solutions using loop group factorizations,
(ii) PDEs in differential geometry that are soliton equations or elliptic
integrable systems,
(iii) similarities and differences of soliton equations and integrable
elliptic systems.Comment: 67 pages, to appear in Surveys on Geometry and Integrable Systems,
Advanced Studies in Pure Mathematics, Mathematical Society of Japa
Soliton Hierarchies Constructed from Involutions
We introduce two families of soliton hierarchies: the twisted hierarchies
associated to symmetric spaces. The Lax pairs of these two hierarchies are
Laurent polynomials in the spectral variable. Our constructions gives a
hierarchy of commuting flows for the generalized sine-Gordon equation (GSGE),
which is the Gauss-Codazzi equation for n-dimensional submanifolds in Euclidean
(2n-1)-space with constant sectional curvature -1. In fact, the GSGE is the
first order system associated to a twisted Grassmannian system. We also study
symmetries for the GSGE
Geometric transformations and soliton equations
We give a survey of the following six closely related topics: (i) a general
method for constructing a soliton hierarchy from a splitting of a loop algebra
into positive and negative subalgebras, together with a sequence of commuting
positive elements, (ii) a method---based on (i)---for constructing soliton
hierarchies from a symmetric space, (iii) the dressing action of the negative
loop subgroup on the space of solutions of the related soliton equation, (iv)
classical B\"acklund, Christoffel, Lie, and Ribaucour transformations for
surfaces in three-space and their relation to dressing actions, (v) methods for
constructing a Lax pair for the Gauss-Codazzi Equation of certain submanifolds
that admit Lie transforms, (vi) how soliton theory can be used to generalize
classical soliton surfaces to submanifolds of higher dimension and
co-dimension
Dispersive Geometric Curve Flows
The Hodge star mean curvature flow on a 3-dimension Riemannian or
pseudo-Riemannian manifold, the geometric Airy flow on a Riemannian manifold,
the Schrodingier flow on Hermitian manifolds, and the shape operator curve flow
on submanifolds are natural non-linear dispersive curve flows in geometric
analysis. A curve flow is integrable if the evolution equation of the local
differential invariants of a solution of the curve flow is a soliton equation.
For example, the Hodge star mean curvature flow on and on , the
geometric Airy flow on , the Schrodingier flow on compact Hermitian
symmetric spaces, and the shape operator curve flow on an Adjoint orbit of a
compact Lie group are integrable. In this paper, we give a survey of these
results, describe a systematic method to construct integrable curve flows from
Lax pairs of soliton equations, and discuss the Hamiltonian aspect and the
Cauchy problem of these curve flows.Comment: 51 page
Isothermic hypersurfaces in R^{n+1}
A diagonal metric sum_{i=1}^n g_{ii} dx_i^2 is termed Guichard_k if
sum_{i=1}^{n-k}g_{ii}-sum_{i=n-k+1}^n g_{ii}=0. A hypersurface in R^{n+1} is
isothermic_k if it admits line of curvature co-ordinates such that its induced
metric is Guichard_k. Isothermic_1 surfaces in R^3 are the classical isothermic
surfaces in R^3. Both isothermic_k hypersurfaces in R^{n+1} and Guichard_k
orthogonal co-ordinate systems on R^n are invariant under conformal
transformations. A sequence of n isothermic_k hypersurfaces in R^{n+1}
(Guichard_k orthogonal co-ordinate systems on R^n resp.) is called a Combescure
sequence if the consecutive hypersurfaces (orthogonal co-ordinate systems
resp.) are related by Combescure transformations. We give a correspondence
between Combescure sequences of Guichard_k orthogonal co-ordinate systems on
R^n and solutions of the O(2n-k,k)/O(n)xO(n-k,k)-system, and a correspondence
between Combescure sequences of isothermic_k hypersurfaces in R^{n+1} and
solutions of the O(2n+1-k,k)/O(n+1)xO(n-k,k)-system, both being integrable
systems. Methods from soliton theory can therefore be used to construct
Christoffel, Ribaucour, and Lie transforms, and to describe the moduli spaces
of these geometric objects and their loop group symmetries.Comment: 21 pages, reworked definition to correct error in theore
Central Affine Curve Flow on the Plane
We give the following results for Pinkall's central affine curve flow on the
plane: (i) a systematic and simple way to construct the known higher commuting
curve flows, conservation laws, and a bi-Hamiltonian structure, (ii) Baecklund
transformations and a permutability formula, (iii) infinitely many families of
explicit solutions. We also solve the Cauchy problem for periodic initial data.Comment: to appear in Journal of Fixed Point Theory and Applications
(Choquet-Bruhat Festschrift
Transformations of flat Lagrangian immersions and Egoroff nets
We associate a natural -family ()
of flat Lagrangian immersions in \C^n with non-degenerate normal bundle to
any given one. We prove that the structure equations for such immersions admit
the same Lax pair as the first order integrable system associated to the
symmetric space \frac{\U(n) \ltimes \C^n}{\OO(n) \ltimes \R^n}. An
interesting observation is that the family degenerates to an Egoroff net on
when . We construct an action of a rational loop group on
such immersions by identifying its generators and computing their dressing
actions. The action of the generator with one simple pole gives the geometric
Ribaucour transformation and we provide the permutability formula for such
transformations. The action of the generator with two poles and the action of a
rational loop in the translation subgroup produce new transformations. The
corresponding results for flat Lagrangian submanifolds in \C P^{n-1} and
\p-invariant Egoroff nets follow nicely via a spherical restriction and Hopf
fibration.Comment: 23 pages, submitte
Backlund transformations, Ward solitons, and unitons
The Ward equation, also called the modified 2+1 chiral model, is obtained by
a dimension reduction and a gauge fixing from the self-dual Yang-Mills field
equation on . It has a Lax pair and is an integrable system. Ward
constructed solitons whose extended solutions have distinct simple poles. He
also used a limiting method to construct 2-solitons whose extended solutions
have a double pole. Ioannidou and Zakrzewski, and Anand constructed more
soliton solutions whose extended solutions have a double or triple pole. Some
of the main results of this paper are: (i) We construct algebraic B\"acklund
transformations (BTs) that generate new solutions of the Ward equation from a
given one by an algebraic method. (ii) We use an order limiting method and
algebraic BTs to construct explicit Ward solitons, whose extended solutions
have arbitrary poles and multiplicities. (iii) We prove that our construction
gives all solitons of the Ward equation explicitly and the entries of Ward
solitons must be rational functions in and . (iv) Since stationary
Ward solitons are unitons, our method also gives an explicit construction of
all -unitons from finitely many rational maps from to .Comment: 38 page
N-dimension Central Affine Curve Flows
We construct a sequence of commuting central affine curve flows on
invariant under the action of and prove the
following results:
(a) The central affine curvatures of a solution of the j-th central affine
curve flow is a solution of the j-th flow of Gelfand-Dickey (GD) hierarchy
on the space of n-th order differential operators. (b) We use the solution of
the Cauchy problems of the GD flow to solve the Cauchy problems for the
central affine curve flows with periodic initial data and also with initial
data whose central affine curvatures are rapidly decaying. (c) We obtain a
bi-Hamiltonian structure for the central affine curve flow hierarchy and prove
that it arises naturally from the Poisson structures of certain co-adjoint
orbits. (d) We construct Backlund transformations, infinitely many families of
explicit solutions and give a permutability formula for these curve flows.Comment: 32 pages (this version adds a section on Backlund transformations and
makes some changes of the first version
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