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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 16 (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.
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.
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
Journal of Nonlinear Mathematical Physics 2001, V.8, Supplement, 145148 Proceedings: NEEDS'99 Symplectic Structure of the Painleve Test
"... In this note, we present a result to show that the symplectic structures have been naturally encoded into the Painleve test. In fact, for every principal balance, there is a symplectic change of dependent variables near movable poles. ..."
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In this note, we present a result to show that the symplectic structures have been naturally encoded into the Painleve test. In fact, for every principal balance, there is a symplectic change of dependent variables near movable poles.
unknown title
, 909
"... On the GBDT version of the BäcklundDarboux transformation and its applications to the linear and nonlinear equations and Weyl theory A.L. Sakhnovich A general theorem on the GBDT version of the BäcklundDarboux transformation for systems rationally depending on the spectral parameter is treated and ..."
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On the GBDT version of the BäcklundDarboux transformation and its applications to the linear and nonlinear equations and Weyl theory A.L. Sakhnovich A general theorem on the GBDT version of the BäcklundDarboux transformation for systems rationally depending on the spectral parameter is treated and its applications to nonlinear equations are given. Explicit solutions of direct and inverse problems for Diractype systems, including systems with singularities, and for the system auxiliary to the Nwave equation are reviewed. New results on explicit construction of the wave functions for radial Dirac equation are obtained.
unknown title
, 708
"... Weyl functions, inverse problem and special solutions for the system auxiliary to the nonlinear optics equation ..."
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Weyl functions, inverse problem and special solutions for the system auxiliary to the nonlinear optics equation
Mathematical Sciences HP Laboratories Bristol
, 1998
"... threewave interaction; geometric phases; reduction; LiePoisson structure Hamiltonian LiePoisson structures of the threewave equations associated with the Lie algebras su(3) ana su(2,1) are delivered and shown to be compatible. POIsson reduction is performed using the method of invariants and geom ..."
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threewave interaction; geometric phases; reduction; LiePoisson structure Hamiltonian LiePoisson structures of the threewave equations associated with the Lie algebras su(3) ana su(2,1) are delivered 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 ~IS paper are implicit in the litet:atur~; our purpose IS to put the threewave Interaction In the modem setting of geometric mechanics and to explore some new things, such as integrability, in thIS context.