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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.
Spectral Filtering Formalism and Its Application for MultiPhase DeepWater Wavetrains
"... A new formalism of spectral filtering for the description of the modulation processes is proposed. The method allows one to study the classical problem of multiphase modulations in dispersive systems. In the present paper, deepwater waves are considered. Spectral filtering results in a system of co ..."
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A new formalism of spectral filtering for the description of the modulation processes is proposed. The method allows one to study the classical problem of multiphase modulations in dispersive systems. In the present paper, deepwater waves are considered. Spectral filtering results in a system of coupled equations that describe the modulations of the carrier wave and its harmonics. The formalism may find applications in a broad range of physical situations with multiphase dynamics. 1.
BASIC ASPECTS OF SOLITON THEORY
, 2006
"... Abstract. This is a review of the main ideas of the inverse scattering method (ISM) for solving nonlinear evolution equations (NLEE), known as soliton equations. As a basic tool we use the fundamental analytic solutions χ ± (x, λ) of the Lax operator L(λ). Then the inverse scattering problem for L(λ ..."
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Abstract. This is a review of the main ideas of the inverse scattering method (ISM) for solving nonlinear evolution equations (NLEE), known as soliton equations. As a basic tool we use the fundamental analytic solutions χ ± (x, λ) of the Lax operator L(λ). Then the inverse scattering problem for L(λ) reduces to a RiemannHilbert problem. Such construction has been applied to wide class of Lax operators, related to the simple Lie algebras. We construct the kernel of the resolvent of L(λ) in terms of χ ± (x, λ) and derive the spectral decompositions of L(λ). Thus we can solve the relevant classes of NLEE which include the NLS eq. and its multicomponent generalizations, the Nwave equations etc. Applying the dressing method of Zakharov and Shabat we derive the Nsoliton solutions of these equations. Next we explain that the ISM is a natural generalization of the Fourier transform method. As appropriate generalizations of the usual exponential function we use the socalled "squared solutions " which are constructed again in terms of χ ± (x, λ) and the CartanWeyl basis of the relevant Lie algebra. One can prove the completeness relations for the "squared solutions " which in fact provide the spectral decompositions of the recursion operator Λ. These decompositions can be used to derive all fundamental properties of the corresponding NLEE in terms of Λ: i) the explicit form of the class of integrable NLEE; ii) the generating functionals of integrals of motion; iii) the hierarchies of Hamiltonian structures. We outline the importance of the classical Rmatrices for extracting the involutive integrals of motion. 1.
Reductions of Nwave interactions related to low–rank simple Lie algebras
, 2008
"... nlin.SI/0006001 The analysis and the classification of all reductions for the nonlinear evolution equations solvable by the inverse scattering method is interesting and still open problem. We show how the second order reductions of the N–wave interactions related to low–rank simple Lie algebras g ca ..."
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nlin.SI/0006001 The analysis and the classification of all reductions for the nonlinear evolution equations solvable by the inverse scattering method is interesting and still open problem. We show how the second order reductions of the N–wave interactions related to low–rank simple Lie algebras g can be embedded also in the Weyl group of g. Some of the reduced systems find applications to nonlinear optics.
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.
Interaction of modulated pulses in scalar multidimensional nonlinear lattices
, 2009
"... We investigate the macroscopic dynamics of sets of an arbitrary finite number of weakly amplitudemodulated pulses in a multidimensional lattice of particles. The latter are assumed to exhibit scalar displacement under pairwise, arbitraryrange, nonlinear interaction potentials and are embedded in a ..."
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We investigate the macroscopic dynamics of sets of an arbitrary finite number of weakly amplitudemodulated pulses in a multidimensional lattice of particles. The latter are assumed to exhibit scalar displacement under pairwise, arbitraryrange, nonlinear interaction potentials and are embedded in a nonlinear background field. By an appropriate multiscale ansatz, we derive formally the explicit evolution equations for the macroscopic amplitudes up to an arbitrarily high order of the scaling parameter, thereby deducing the resonance and nonresonance conditions on the fixed wave vectors and frequencies of the pulses, which are required for that. The derived equations are justified rigorously in time intervals of macroscopic length. Finally, for sets of up to three pulses we present a complete list of all possible interactions and discuss their ramifications for the corresponding, explicitly given macroscopic systems. Key words and phrases: nonlinear discrete lattices; interaction of modulated pulses; multiscale ansatz; derivation and justification of macroscopic dynamics. MSC 2000: 37K60; 34E13, 34C20, 70F45, 70K70, 35L45, 35L60. 1
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.
Abstract
, 1995
"... Experimental data from an experiment on drift–waves in plasma is presented. The experiment provides a space–time diagnostic and has a control parameter that permits the study of the transition from a stable plasma to a turbulent plasma. The biorthogonal decomposition is used to analyse the data. We ..."
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Experimental data from an experiment on drift–waves in plasma is presented. The experiment provides a space–time diagnostic and has a control parameter that permits the study of the transition from a stable plasma to a turbulent plasma. The biorthogonal decomposition is used to analyse the data. We introduce the notion of complex modulation for two–dimensional systems. We decompose the real physical system into complex modulated monochromatic travelling waves and give a simple model describing the speed doubling observed in the data as the control parameter increases.