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25
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 38 (16 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(
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 21 (9 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...
Geometric phases, reduction and LiePoisson structure for the resonant threewave interaction
 Physica D
, 1998
"... Hamiltonian LiePoisson structures of the threewave equations associated with the Lie algebras su(3) and su(2, 1) are derived and shown to be compatible. Poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. These resul ..."
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Cited by 15 (5 self)
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Hamiltonian LiePoisson structures of the threewave equations associated with the Lie algebras su(3) and su(2, 1) are derived and shown to be compatible. Poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. These results can be applied to applications of nonlinearwaves in, for instance, nonlinear optics. Some of the general structures presented in the latter part of this paper are implicit in the literature; our purpose is to put the threewave interaction in the modern setting of geometric mechanics and to explore some new things, such as explicit geometric phase formulas, as well as some old things, such as integrability, in this context.
Zamolodchikov’s tetrahedron equation and hidden structure of quantum groups
 J. Phys. A
"... The tetrahedron equation is a threedimensional generalization of the YangBaxter equation. Its solutions define integrable threedimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution o ..."
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Cited by 13 (10 self)
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The tetrahedron equation is a threedimensional generalization of the YangBaxter equation. Its solutions define integrable threedimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices. Obviously, any such threedimensional model can be viewed as a twodimensional integrable model on a square lattice, where the additional third dimension is treated as an internal degree of freedom. Therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this “hidden third dimension”. In this paper we construct a new solution of the tetrahedron equation, which provides in this way the twodimensional solvable models related to finitedimensional highest weight representations for all quantum affine algebra Uq ( ̂ sl(n)), where the rank n coincides with the size of the hidden dimension. These models are related with an anisotropic deformation of the sl(n)invariant Heisenberg magnets. They were extensively studied for a long time, but the hidden 3d structure was hitherto unknown. Our results lead to a remarkable exact “ranksize ” duality relation for the nested Bethe Ansatz solution for these models. Note also, that the above solution of the tetrahedron equation arises in the quantization of the “resonant threewave scattering ” model, which is a wellknown integrable classical system in 2 + 1 dimensions.
Geometry and control of threewave interactions
 in The Arnoldfest
, 1997
"... The integrable structure of the threewave equations is discussed in the setting of geometric mechanics. LiePoisson structures with quadratic Hamiltonian are associated with the threewave equations through the Lie algebras su(3) and su(2, 1). A second structure having cubic Hamiltonian is shown to ..."
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Cited by 4 (1 self)
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The integrable structure of the threewave equations is discussed in the setting of geometric mechanics. LiePoisson structures with quadratic Hamiltonian are associated with the threewave equations through the Lie algebras su(3) and su(2, 1). A second structure having cubic Hamiltonian is shown to be compatible. The analogy between this system and the rigidbody or Euler equations is discussed. Poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. We show that using piecewise continuous controls, the transfer of energy among three 1 waves can be controlled. The so called quasiphasematching control strategy, which is used in a host of nonlinear optical devices to convert laser light from one frequency to another, is described in this context. Finally, we discuss the connection between piecewise constant controls and billiards.
Algebraic and Analytic Aspects of Soliton Type Equations
, 2002
"... This is a review of two of the fundamental tools for analysis of soliton equations: i) the algebraic ones based on KacMoody algebras, their central extensions and their dual algebras which underlie the Hamiltonian structures of the NLEE; ii) the construction of the fundamental analytic solutions ( ..."
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Cited by 4 (4 self)
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This is a review of two of the fundamental tools for analysis of soliton equations: i) the algebraic ones based on KacMoody algebras, their central extensions and their dual algebras which underlie the Hamiltonian structures of the NLEE; ii) the construction of the fundamental analytic solutions (FAS) of the Lax operator and the RiemannHilbert problem (RHP) which they satisfy. The fact that the inverse scattering problem for the Lax operator can be viewed as a RHP gave rise to the dressing ZakharovShabat, one of the most effective ones for constructing soliton solutions. These two methods when combined may allow one to prove rigorously the results obtained by the abstract algebraic methods. They also allow to derive spectral decompositions for nonselfadjoint Lax operators.
QUANTIZATION OF THREEWAVE EQUATIONS
, 2007
"... Abstract. The subject of this paper is the consecutive procedure of discretization and quantization of two similar classical integrable systems in threedimensional spacetime: the standard threewave equations and less known modified threewave equations. The quantized systems in discrete spacetime ..."
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Cited by 4 (3 self)
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Abstract. The subject of this paper is the consecutive procedure of discretization and quantization of two similar classical integrable systems in threedimensional spacetime: the standard threewave equations and less known modified threewave equations. The quantized systems in discrete spacetime may be understood as the regularized integrable quantum field theories. Integrability of the theories, and in particular the quantum tetrahedron equations for vertex operators, follow from the quantum auxiliary linear problems. Principal object of the lattice field theories is the Heisenberg discrete time evolution operator constructed with the help of vertex operators.
Binary Bargmann symmetry constraints of soliton equations. Nonlinear Analysis 47
 in Proceedings of the Third World Congress of Nonlinear Analysts, Nonlinear Analysis 47
, 2001
"... Binary Bargmann symmetry constraints are applied to decompose soliton equations into finitedimensional Liouville integrable Hamiltonian systems, generated from socalled constrained flows. The resulting constraints on the potentials of soliton equations give rise to involutive solutions to soliton ..."
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Cited by 3 (2 self)
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Binary Bargmann symmetry constraints are applied to decompose soliton equations into finitedimensional Liouville integrable Hamiltonian systems, generated from socalled constrained flows. The resulting constraints on the potentials of soliton equations give rise to involutive solutions to soliton equations, and thus the integrability by quadratures are shown for soliton equations by the constrained flows. The multiwave interaction equations associated with the 3 × 3 matrix AKNS spectral problem are chosen as an illustrative example to carry out binary Bargmann symmetry constraints. The Lax representations and the corresponding rmatrix formulation are established for the constrained flows of the multiwave interaction equations, and the integrals of motion generated from the Lax representations are utilized to show the Liouville integrability for the resulting constrained flows. Finally, involutive solutions to the multiwave interaction equations are presented. Key words: Symmetry constraints, Binary nonlinearization, Liouville integrability PACS: 02.30.Jr Partial differential equations.
Binary symmetry constraints of Nwave interaction equations in 1+1 and 2+1 dimensions
 J. Math. Phys
"... Binary symmetry constraints of the Nwave interaction equations in 1 + 1 and 2 + 1 dimensions are proposed to reduce the Nwave interaction equations into finitedimensional Liouville integrable systems. A new involutive and functionally independent system of polynomial functions is generated from a ..."
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Cited by 3 (1 self)
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Binary symmetry constraints of the Nwave interaction equations in 1 + 1 and 2 + 1 dimensions are proposed to reduce the Nwave interaction equations into finitedimensional Liouville integrable systems. A new involutive and functionally independent system of polynomial functions is generated from an arbitrary order square matrix Lax operator and used to show the Liouville integrability of the constrained flows of the Nwave interaction equations. The constraints on the potentials resulting from the symmetry constraints give rise to involutive solutions to the Nwave interaction equations, and thus the integrability by quadratures are shown for the Nwave interaction equations by the constrained flows. Running title: Symmetry Constraints of Nwave Equations 1