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 ZS-AKNS 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(

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 ZS-AKNS nxn heirarchy (which includest#c non-linear Schrodingerequat#rPG modified KdV, and t#d n-waveequat#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 ZS-AKNS 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 a-act#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 sine-Gordonequat#Gor harmonic maps, Schrodinger flows onsymmet#7P spaces, Darboux ort#8794PL coordinat#LD and isomet#5G immersions of one space-form inanot#D4z 1 Research supported in part by NSF Grant DMS 9626130 2 Research supported in part by Sid Rich...

Hamiltonian Lie-Poisson structures of the three-wave 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 nonlinear-waves 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 three-wave 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.

The integrable structure of the three-wave equations is discussed in the setting of geometric mechanics. Lie-Poisson structures with quadratic Hamiltonian are associated with the three-wave 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 rigid-body 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 quasi-phase-matching 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.

This is a review of two of the fundamental tools for analysis of soliton equations: i) the algebraic ones based on Kac-Moody 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 Riemann-Hilbert 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 Zakharov-Shabat, 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 non-self-adjoint Lax operators.

Binary Bargmann symmetry constraints are applied to decompose soliton equations into finite-dimensional Liouville integrable Hamiltonian systems, generated from so-called 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 multi-wave 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 r-matrix formulation are established for the constrained flows of the multi-wave 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 multi-wave interaction equations are presented. Key words: Symmetry constraints, Binary nonlinearization, Liouville integrability PACS: 02.30.Jr- Partial differential equations.

Binary symmetry constraints of the N-wave interaction equations in 1 + 1 and 2 + 1 dimensions are proposed to reduce the N-wave interaction equations into finite-dimensional 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 N-wave interaction equations. The constraints on the potentials resulting from the symmetry constraints give rise to involutive solutions to the N-wave interaction equations, and thus the integrability by quadratures are shown for the N-wave interaction equations by the constrained flows. Running title: Symmetry Constraints of N-wave Equations 1

Abstract. The subject of this paper is the consecutive procedure of discretization and quantization of two similar classical integrable systems in three-dimensional space-time: the standard three-wave equations and less known modified three-wave 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.

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reduction of the resonant three-wave interaction to the generic sixth Painlevé equation ∗