Search results for "Integrable systems"
showing 10 items of 256 documents
Stationary problems for equation of the KdV type and dynamical r-matrices
1995
We study a quite general family of dynamical $r$-matrices for an auxiliary loop algebra ${\cal L}({su(2)})$ related to restricted flows for equations of the KdV type. This underlying $r$-matrix structure allows to reconstruct Lax representations and to find variables of separation for a wide set of the integrable natural Hamiltonian systems. As an example, we discuss the Henon-Heiles system and a quartic system of two degrees of freedom in detail.
Modular Schrödinger equation and dynamical duality.
2008
We discuss quite surprising properties of the one-parameter family of modular (Auberson and Sabatier (1994)) nonlinear Schr\"{o}dinger equations. We develop a unified theoretical framework for this family. Special attention is paid to the emergent \it dual \rm time evolution scenarios which, albeit running in the \it real time \rm parameter of the pertinent nonlinear equation, in each considered case, may be mapped among each other by means of an "imaginary time" transformation (more seriously, an analytic continuation in time procedure).
Superfield commutators for D = 4 chiral multiplets and their apppications
1987
The superfield commutators and their corresponding equal-time limits are derived in a covariant way for the D=4 free massive chiral multiplet. For interesting chiral multiplets, the general KAllen-Lehmann representation is also introduced. As applications of the free superfield commutators, the general solution of the Cauchy problem for chiral superfields is given, and an analysis of the closure of the bilinear products of superfields which desrcibe the extension of the internal currents for free supersymmetric chiral matter is performed.
The Poisson Bracket Structure of the SL(2, R)/U(1) Gauged WZNW Model with Periodic Boundary Conditions
2000
The gauged SL(2, R)/U(1) Wess-Zumino-Novikov-Witten (WZNW) model is classically an integrable conformal field theory. A second-order differential equation of the Gelfand-Dikii type defines the Poisson bracket structure of the theory. For periodic boundary conditions zero modes imply non-local Poisson brackets which, nevertheless, can be represented by canonical free fields.
Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schrodinger equations
2006
International audience; A different kind of shape changing (intensity redistribution) collision with potential application to signal amplification is identified in the integrable N-coupled nonlinear Schrodinger (CNLS) equations with mixed signs of focusing- and defocusing-type nonlinearity coefficients. The corresponding soliton solutions for the N=2 case are obtained by using Hirota's bilinearization method. The distinguishing feature of the mixed sign CNLS equations is that the soliton solutions can both be singular and regular. Although the general soliton solution admits singularities we present parametric conditions for which nonsingular soliton propagation can occur. The multisoliton …
Numerical study of a multiscale expansion of the Korteweg de Vries equation and Painlev\'e-II equation
2007
The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\e^2$, $\e\ll 1$, is characterized by the appearance of a zone of rapid modulated oscillations. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. Whereas the difference between the KdV and the asymptotic solution decreases as $\epsilon$ in the interior of the Whitham oscillatory zone, it is known to be only of order $\epsilon^{1/3}$ near the leading edge of this zone. To obtain a more accurate description near the leading edge of the oscillatory zone we present a…
Trivial S-Matrices, Wigner-Von Neumann Resonances and Positon Solutions of the Integrable Nonlinear Evolution Equations
1996
It is well known that the scattering matrix is different from the unit matrix in the case of 1-dimensional Schrodinger operator with smooth rapidly decreasing nonzero potential. This no more true in the case of the slowly decreasing and oscillating potentials for which the absence of scattering is accompanied by the occurrence of the Wigner-von Neumann resonances embedded in the positive absolutely continuous spectrum. Taken as initial conditions in the KdV like integrable partial differential equations these potentials generate interesting family of explicit solutions. Below we will call them positon or multipositon solutions. The interaction of an arbitrary finite number of positons and s…
Quantized separations of phase-locked soliton pairs in fiber lasers
2003
Quantized separations of phase-locked soliton pairs in fiber lasers were presented. The relation between the Kelly sidebands and the quantized separations between solitons was confirmed. Simulation results showed that the solitons can see each other at relatively larger distances than they would in the absence of radiation.
Propagation, Stability and Interactions of Novel Three-Wave Parametric Solitons
2006
International audience; We found a new class of analytic soliton solutions that describe the parametric wave mixing of optical pulses in quadratic nonlinear crystals. We analyze the stability properties, interactions and collisions of these solitons.
q-deformed solitons and quantum solitons of the Maxwell-Bloch lattice
2000
We report for the first time exact solutions of a completely integrable nonlinear lattice system for which the dynamical variables satisfy a q-deformed Lie algebra - the Lie-Poisson algebra su_q(2). The system considered is a q-deformed lattice for which in continuum limit the equations of motion become the envelope Maxwell-Bloch (or SIT) equations describing the resonant interaction of light with a nonlinear dielectric. Thus the N-soliton solutions we here report are the natural q-deformations, necessary for a lattice, of the well-known multi-soliton and breather solutions of self-induced transparency (SIT). The method we use to find these solutions is a generalization of the Darboux-Backl…