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The problem of integrable discretization: Hamiltonian approach
 Progress in Mathematics, Volume 219. Birkhäuser
"... this paper. Hence, the nature of this discretization as a member of the Toda hierarchy was not understood properly. A complete account was given in Suris (1995). The work Rutishauser (1954) contains not only the denition of the qd algorithm from the viewpoint of the numerical analist, but also the e ..."
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Cited by 71 (2 self)
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this paper. Hence, the nature of this discretization as a member of the Toda hierarchy was not understood properly. A complete account was given in Suris (1995). The work Rutishauser (1954) contains not only the denition of the qd algorithm from the viewpoint of the numerical analist, but also the equations of motion of the Toda lattice (under the name of a \continuous analogue of the qd algorithm")! The relation of the qd algorithm to integrable systems might have important implications for the numerical analysis, cf. Deift et al. (1991), Nagai and Satsuma (1995).
Bäcklund Transformations and Loop Group Actions
 Comm. Pure. Appl. Math
"... We constru#V a local action of thegrou# of rational maps from S 2 to GL(n, C) on localsolu#175fi of flows of the ZSAKNS sl(n, C)hierarchy. We show that the actions of simple elements (linear fractional transformations) give local Backlu#8 transformations, and we derive a permu#mB1586 y formu#r f ..."
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Cited by 68 (17 self)
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We constru#V a local action of thegrou# of rational maps from S 2 to GL(n, C) on localsolu#175fi of flows of the ZSAKNS sl(n, C)hierarchy. We show that the actions of simple elements (linear fractional transformations) give local Backlu#8 transformations, and we derive a permu#mB1586 y formu#r from di#erent factorizations of a qu#8198BV element. We prove that the action of simple elements on the vacu#1 may give either global smoothsolu#596fi or solu#1519 with singu#fi1fiBV474 However, the action of thesu#B3496 of the rational maps that satisfy theU (n)reality condition g(
Multidimensional quadrilateral lattices are integrable
, 1996
"... The notion of multidimensional quadrilateral lattice is introduced. It is shown that such a lattice is characterized by a system of integrable discrete nonlinear equations. Different useful formulations of the system are given. The geometric construction of the lattice is also discussed and, in part ..."
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Cited by 55 (22 self)
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The notion of multidimensional quadrilateral lattice is introduced. It is shown that such a lattice is characterized by a system of integrable discrete nonlinear equations. Different useful formulations of the system are given. The geometric construction of the lattice is also discussed and, in particular, it is clarified the number of initial–boundary data which define the lattice uniquely.
Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems
, 2000
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The symmetries of solitons
 Bull. Amer. Math. Soc
, 1997
"... Abstract. In this article we will retrace one of the great mathematical adventures of this century—the discovery of the soliton and the gradual explanation of its remarkable properties in terms of hidden symmetries. We will take an historical approach, starting with a famous numerical experiment car ..."
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Cited by 37 (1 self)
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Abstract. In this article we will retrace one of the great mathematical adventures of this century—the discovery of the soliton and the gradual explanation of its remarkable properties in terms of hidden symmetries. We will take an historical approach, starting with a famous numerical experiment carried out by Fermi, Pasta, and Ulam on one of the first electronic computers, and with Zabusky and Kruskal’s insightful explanation of the surprising results of that experiment (and of a followup experiment of their own) in terms of a new concept they called “solitons”. Solitons however raised even more questions than they answered. In particular, the evolution equations that govern solitons were found to be Hamiltonian and have infinitely many conserved quantities, pointing to the existence of many nonobvious symmetries. We will cover next the elegant approach to solitons in terms of the Inverse Scattering Transform and Lax Pairs, and finally explain how those ideas led stepbystep to the discovery that Loop Groups, acting by “Dressing Transformations”, give
Projectively flat Finsler 2spheres of constant curvature
 Selecta Math
"... Abstract. After recalling the structure equations of Finsler structures on surfaces, I define a notion of ‘generalized Finsler structure ’ as a way of microlocalizing the problem of describing Finsler structures subject to curvature conditions. I then recall the basic notions of path geometry on a ..."
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Cited by 32 (0 self)
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Abstract. After recalling the structure equations of Finsler structures on surfaces, I define a notion of ‘generalized Finsler structure ’ as a way of microlocalizing the problem of describing Finsler structures subject to curvature conditions. I then recall the basic notions of path geometry on a surface and define a notion of ‘generalized path geometry ’ analogous to that of ‘generalized Finsler structure’. I use these ideas to study the geometry of Finsler structures on the 2sphere that have constant FinslerGauss curvature K and whose geodesic path geometry is projectively flat, i.e., locally equivalent to that of straight lines in the plane. I show that modulo diffeomorphism there is a 2parameter family of projectively flat Finsler structures on the sphere whose FinslerGauss curvature K is identically 1. Hilbert’s Fourth Problem was entitled “Problem of the straight line as the shortest distance between two points”. It concerned, in its most general formulation, the problem of characterizing the notnecessarilysymmetric distance functions d that could be defined on (convex) subsets U ⊂ R 2 so that the lines were geodesics,
Proof of the projective Lichnerowicz–Obata conjecture
 J. Differential Geom
"... We solve two classical conjectures by showing that if an action of a connected Lie group on a complete Riemannian manifold preserves the geodesics (considered as unparameterized curves), then the metric has constant positive sectional curvature, or the group acts by affine transformations. ..."
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We solve two classical conjectures by showing that if an action of a connected Lie group on a complete Riemannian manifold preserves the geodesics (considered as unparameterized curves), then the metric has constant positive sectional curvature, or the group acts by affine transformations.
Transformations of quadrilateral lattices
 J. Math. Phys
"... Abstract. We investigate the τfunction of the quadrilateral lattice using the nonlocal ¯ ∂dressing method, and we show that it can be identified with the Fredholm determinant of the integral equation which naturally appears within that approach. ..."
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Cited by 30 (16 self)
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Abstract. We investigate the τfunction of the quadrilateral lattice using the nonlocal ¯ ∂dressing method, and we show that it can be identified with the Fredholm determinant of the integral equation which naturally appears within that approach.
Darboux transformation and perturbation of linear functionals
 Linear Algebra and Appl
"... Let L be a quasidefinite linear functional defined on the linear space of polynomials with real coefficients. In the literature, three canonical transformations of this functional are studied: xL, L + Cδ(x) and 1L + Cδ(x) where δ(x) denotes the linear x functional (δ(x))(xk) = δk,0, and δk,0 is th ..."
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Cited by 28 (9 self)
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Let L be a quasidefinite linear functional defined on the linear space of polynomials with real coefficients. In the literature, three canonical transformations of this functional are studied: xL, L + Cδ(x) and 1L + Cδ(x) where δ(x) denotes the linear x functional (δ(x))(xk) = δk,0, and δk,0 is the Kronecker symbol. Let us consider the sequence of monic polynomials orthogonal with respect to L. This sequence satisfies a threeterm recurrence relation whose coefficients are the entries of the socalled monic Jacobi matrix. In this paper we show how to find the monic Jacobi matrix associated with the three canonical perturbations in terms of the monic Jacobi matrix associated with L. The main tools are Darboux transformations. In the case that the LU factorization of the monic Jacobi matrix associated with xL does not exist and Darboux transformation does not work, we show how to obtain the monic Jacobi matrix associated with x2L as a limit case. We also study perturbations of the functional L that are obtained by combining the canonical cases. Finally, we present explicit algebraic relations between the polynomials orthogonal with respect to L and orthogonal with respect to the perturbed functionals.
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...