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Poisson Actions and Scattering Theory for Integrable Systems
 J. Differential Geometry
, 1998
"... Conservat#se laws, heirarchies,scat#G556D t#cat# and Backlundt#ndPD9575PL54G are known t# bet#I building blocks of int#GG849P part#49 di#erent##e equat#PDDG We ident#en t#enP asfacet# of at#II46 of Poisson groupact#pPGG and applyt#p t#plyP t# t#p ZSAKNS nxn heirarchy (which includest#c nonlinear S ..."
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Cited by 28 (10 self)
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Conservat#se laws, heirarchies,scat#G556D t#cat# and Backlundt#ndPD9575PL54G are known t# bet#I building blocks of int#GG849P part#49 di#erent##e equat#PDDG We ident#en t#enP asfacet# of at#II46 of Poisson groupact#pPGG and applyt#p t#plyP t# t#p ZSAKNS nxn heirarchy (which includest#c nonlinear Schrodingerequat#rPG modified KdV, and t#d nwaveequat#7I4P Wefirst find a simple model Poisson group act#pP t#t# cont#nPD flows forsyst#G4 wit# a Lax pair whose t#ose all decay on R.Backlundt#ndPG9zD4PL47G and flows arise from subgroups of t#fP single Poisson group. Fort#G ZSAKNS nxn heirarchy defined by aconst#G t a # u(n), t#P simple model is no longercorrect# The adet#86PL44 t woseparat# Poissonst#sonP44G9 The flows come fromt#o Poissonact#so of t#P cent#nPDzID H a of a in t#P dual Poisson group (due t# t#u behavior of e a#xat infinit y). When a hasdist#DzG eigenvalues, H a is abelian and it act# symplect#L4I65 . The phase space oft#PG9 flows is t#P space S a ofleft coset# of t#P cent#PIII58 of a in D , where D is acert#87 loop group. The group D cont#nP4 bot# a Poisson subgroup correspondingt# t#d cont#n uousscat#GGG8P dat#t and arat#5z7P loop group correspondingt# t#d discret# scat#t#587 dat#t The H aact#95 is t#P right dressingact#si on S a .Backlundt#ndPI59I8PLD46 arise fromt#o act#PG of t#P simplerat#lePD loops on S a by right mult#7PLDDG84Pt Variousgeomet#I9 equat##I9 arise from appropriat# choice of a andrest#G94PLDD oft#I phase space and flows. Inpart#DzPLD we discussapplicat#z4G t# t#p sineGordonequat#Gor harmonic maps, Schrodinger flows onsymmet#7P spaces, Darboux ort#8794PL coordinat#LD and isomet#5G immersions of one spaceform inanot#D4z 1 Research supported in part by NSF Grant DMS 9626130 2 Research supported in part by Sid Rich...
Geometries and symmetries of soliton equations and integrable elliptic systems
 IN SURVEYS ON GEOMETRY AND INTEGRABLE SYSTEMS, ADVANCED STUDIES IN PURE MATHEMATICS, MATHEMATICAL SOCIETY OF JAPAN NORTHEASTERN UNIVERSITY AND UC IRVINE EMAIL ADDRESS: TERNG@NEU.EDU MSRI, BERKELEY, CA 94720 EMAIL ADDRESS: EWANG@MRSI.ORG
, 2002
"... We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semisimple Lie algebra and finite order automorphisms. For example, the nonlinear Schrödinger equation, the nwave equation, and the sigmamodel are soliton flow ..."
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Cited by 26 (4 self)
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We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semisimple Lie algebra and finite order automorphisms. For example, the nonlinear Schrödinger equation, the nwave equation, and the sigmamodel 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 ksymmetric space, and constant mean curvature surfaces and isothermic surfaces in space forms are integrable elliptic systems. We also give a survey of • construction of solutions using loop group factorizations, • PDEs in differential geometry that are soliton equations or elliptic integrable systems, • similarities and differences of soliton equations and integrable elliptic
Bäcklund transformation, Ward solitons, and unitons
"... Abstract. The Ward equation, also called the modified 2 + 1 chiral model, is obtained by a dimension reduction and a gauge fixing from the selfdual YangMills field equation on R 2,2. It has a Lax pair and is an integrable system. Ward constructed solitons whose extended solutions have distinct sim ..."
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Abstract. The Ward equation, also called the modified 2 + 1 chiral model, is obtained by a dimension reduction and a gauge fixing from the selfdual YangMills 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 2solitons 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 Bäcklund 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)
The Submanifold Geometries associated to Grassmannian Systems
 735, viii + 95
"... There is a hierarchy of commuting soliton equations associated to each symmetric space U/K. When U/K has rank n, the first n flows in the hierarchy give rise to a natural first order nonlinear system of partial di#erential equations in n variables, the so called U/Ksystem. Let G m,n denote the Gr ..."
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Cited by 12 (1 self)
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There is a hierarchy of commuting soliton equations associated to each symmetric space U/K. When U/K has rank n, the first n flows in the hierarchy give rise to a natural first order nonlinear system of partial di#erential equations in n variables, the so called U/Ksystem. Let G m,n denote the Grassmannian of ndimensionallinear subspaces in R m+n , and G 1 m,n the Grassmannian of spacelike mdimensionallinear subspaces in the Lorentzian space R m+n,1 . In this paper, we use techniques from soliton theory to study submanifolds in space forms whose GaussCodazzi equations are gauge equivalent to the G m,n system or the G 1 m,n system. These include submanifolds with constant sectional curvatures, isothermic surfaces, and submanifolds admitting principal curvature coordinates. The dressing actions of simple elements on the space of solutions of the G m,n and G 1 m,n systems correspond toB acklund, Darboux and Ribaucour transformations for submanifolds. Tabl# of Contents 1.
ON THE MODULI OF CONSTANT MEAN CURVATURE CYLINDERS OF FINITE TYPE IN THE 3SPHERE
, 2008
"... We show that onesided Alexandrov embedded constant mean curvature cylinders of finite type in the 3sphere are surfaces of revolution. This confirms a conjecture by Pinkall and Sterling that the only embedded constant mean curvature tori in the 3sphere are rotational. ..."
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Cited by 10 (1 self)
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We show that onesided Alexandrov embedded constant mean curvature cylinders of finite type in the 3sphere are surfaces of revolution. This confirms a conjecture by Pinkall and Sterling that the only embedded constant mean curvature tori in the 3sphere are rotational.
Curved flats, exterior differential systems, and conservation laws, Complex, contact and symmetric manifolds
 235–254, Progr. Math., 234, Birkhauser
, 2005
"... Abstract. Let σ be an involution of a real semisimple Lie group U, U0 the subgroup fixed by σ, and U/U0 the corresponding symmetric space. Ferus and Pedit called a submanifold M of a rank r symmetric space U/U0 a curved flat if TpM is tangent to an rdimensional flat of U/U0 at p for each p ∈ M. Th ..."
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Cited by 8 (3 self)
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Abstract. Let σ be an involution of a real semisimple Lie group U, U0 the subgroup fixed by σ, and U/U0 the corresponding symmetric space. Ferus and Pedit called a submanifold M of a rank r symmetric space U/U0 a curved flat if TpM is tangent to an rdimensional flat of U/U0 at p for each p ∈ M. They noted that the equation for curved flats is an integrable system. Bryant used the involution σ to construct an involutive exterior differential system Iσ such that integral submanifolds of Iσ are curved flats. Terng used r first flows in the U/U0hierarchy of commuting soliton equations to construct the U/U0system. She showed that the U/U0system and the curved flat system are gauge equivalent, used the inverse scattering theory to solve the Cauchy problem globally with smooth rapidly decaying initial data, used loop group factorization to construct infinitely many families of explicit solutions, and noted that many these systems occur as the GaussCodazzi equations for submanifolds in space forms. The main goals of this paper are: (i) give a review of these known results, (ii) use techniques from soliton theory to construct infinitely many integral submanifolds and conservation laws for the exterior differential system Iσ. 1.
Initial value problems of the sineGordon equation and geometric solutions
 Ann. Global Anal. Geom
, 2005
"... Recent results using inverse scattering techniques interpret every solution ϕ(x,y) of the sineGordon equation as a nonlinear superposition of solutions along the axes x = 0 and y = 0. This has a wellknown geometric interpretation, namely that every weakly regular surface of Gauss curvature K = −1 ..."
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Cited by 8 (0 self)
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Recent results using inverse scattering techniques interpret every solution ϕ(x,y) of the sineGordon equation as a nonlinear superposition of solutions along the axes x = 0 and y = 0. This has a wellknown geometric interpretation, namely that every weakly regular surface of Gauss curvature K = −1, in arc length asymptotic line parametrization, is uniquely determined by the values ϕ(x,0) and ϕ(0, y) of its coordinate angle along the axes. We introduce a generalized Weierstrass representation of pseudospherical surfaces that depends only on these values, and we explicitely construct the associated family of pseudospherical immersions corresponding to it.
Equivariant harmonic cylinders
"... Abstract. We prove that a primitive harmonic map is equivariant if and only if it admits a holomorphic potential of degree one. We investigate when the equivariant harmonic map is periodic, and as an application discuss constant mean curvature cylinders with screw motion symmetries. ..."
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Cited by 7 (3 self)
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Abstract. We prove that a primitive harmonic map is equivariant if and only if it admits a holomorphic potential of degree one. We investigate when the equivariant harmonic map is periodic, and as an application discuss constant mean curvature cylinders with screw motion symmetries.
Transformations of flat Lagrangian immersions and Egoroff nets, accepted
 Asian Journal of Mathematics
"... Abstract. We associate a natural λfamily (λ ∈ R \ {0}) of flat Lagrangian immersions in C n with nondegenerate 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 U( ..."
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Abstract. We associate a natural λfamily (λ ∈ R \ {0}) of flat Lagrangian immersions in C n with nondegenerate 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 U(n)⋉Cn O(n)⋉Rn. An interesting observation is that the family degenerates to an Egoroff net on R n when λ → 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 CP n−1 and ∂invariant Egoroff nets follow nicely via a spherical restriction and Hopf fibration. 1.
The BianchiDarboux transform of Lisothermic surfaces
 Sch] [WP] W.K. Schief. Isothermic surfaces in spaces of
"... Abstract. We study an analogue of the classical Bäcklund transformation for Lisothermic surfaces in Laguerre geometry, the Bianchi–Darboux transformation. We show how to construct the Bianchi–Darboux transforms of an Lisothermic surface by solving an integrable linear differential system. We then ..."
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Abstract. We study an analogue of the classical Bäcklund transformation for Lisothermic surfaces in Laguerre geometry, the Bianchi–Darboux transformation. We show how to construct the Bianchi–Darboux transforms of an Lisothermic surface by solving an integrable linear differential system. We then establish a permutability theorem for iterated Bianchi–Darboux transforms. 1. Introduction. Certain types of integrable nonlinear PDEs (soliton equations) arise in differential geometry as compatibility conditions for the linear equations obeyed by frames adapted to surfaces in higher dimensional manifolds. In a number of situations, the construction of new solutions of the arising PDE relies on the existence of Bäcklund type