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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 ..."
Abstract

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.
The Legacy of the IST
, 2002
"... Abstract. We provide a brief review of some of the major research results arising from the method of the Inverse Scattering Transform. 1. ..."
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Abstract. We provide a brief review of some of the major research results arising from the method of the Inverse Scattering Transform. 1.
Communications in Mathematical Physics manuscript No.
, 708
"... (will be inserted by the editor) How many types of soliton solutions do we know? ..."
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(will be inserted by the editor) How many types of soliton solutions do we know?
The ThreeWave Resonant Interaction: Deformation of the Plane Wave Solutions and Darboux Transformations
, 1996
"... The plane wave solutions of the threewave resonant interaction in the plane are considered. It is shown that rankone constraints over the right derivatives of invertible operators on an arbitrary linear space gives solutions of the threewave resonant interaction that can be understood as a Darbou ..."
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The plane wave solutions of the threewave resonant interaction in the plane are considered. It is shown that rankone constraints over the right derivatives of invertible operators on an arbitrary linear space gives solutions of the threewave resonant interaction that can be understood as a Darboux transformation of the plane wave solutions. The method is extended further to obtain general Darboux transformations: for any solution of the threewave interaction problem and vector solutions of the corresponding Lax pair large families of new solutions, expressed in terms of Grammian type determinants of these vector solutions, are given.