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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.
Dubrovin equations and integrable systems on hyperelliptic curves, in preparation
"... Abstract. We introduce the most general version of Dubrovintype equations for divisors on a hyperelliptic curve Kg of arbitrary genus g ∈ N, and provide a new argument for linearizing the corresponding completely integrable flows. Detailed applications to completely integrable systems, including th ..."
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Cited by 8 (7 self)
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Abstract. We introduce the most general version of Dubrovintype equations for divisors on a hyperelliptic curve Kg of arbitrary genus g ∈ N, and provide a new argument for linearizing the corresponding completely integrable flows. Detailed applications to completely integrable systems, including the KdV, AKNS, Toda, and the combined sineGordon and mKdV hierarchies, are made. These investigations uncover a new principle for 1 + 1dimensional integrable soliton equations in the sense that the Dubrovin equations, combined with appropriate trace formulas, encode all hierarchies of soliton equations associated with hyperelliptic curves. In other words, completely integable hierarchies of soliton equations determine Dubrovin equations and associated trace formulas and, vice versa, Dubrovintype equations combined with trace formulas permit the construction of hierarchies of soliton equations. 1.
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.
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 ..."
Abstract
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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.