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 $k$-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 $R^3$ and on $R^{2,1}$, the geometric Airy flow on $R^n$, 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 $\lambda$-family ($\lambda \in \R \setminus \{0\}$) 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 $\R^n$ when $\lambda \to 0$. 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 $R^{2,2}$. 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 $k$ 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 $x, y$ and $t$. (iv) Since stationary Ward solitons are unitons, our method also gives an explicit construction of all $k$-unitons from finitely many rational maps from $C$ to $C^n$.Comment: 38 page

N-dimension Central Affine Curve Flows

We construct a sequence of commuting central affine curve flows on $R^n\backslash 0$ invariant under the action of $SL(n,R)$ 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$_n$) hierarchy on the space of n-th order differential operators. (b) We use the solution of the Cauchy problems of the GD$_n$ 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