Whitham deformations of top-like integrable dynamical systems
type: Item
instance of: scholarly article
title: Whitham deformations of top-like integrable dynamical systems
zbMATH Open document ID: 0809.34011
DOI: 10.1007/BF01085982
publication date: 1994-11-08T00:00:00Z
review text: The note is devoted to the finite-gap integration method for the equations \[ dL(\lambda)/dt+ [L(\lambda),A(\lambda)]= 0 \] in the Lax form employing the Baker-Akhiezer function \(\Psi\), a solution of the system \(L(\lambda)\Psi= \mu\Psi\), \(d\Psi/dt= A(\lambda)\Psi\). This \(\Psi\) can be found in terms of theta functions of the Riemann surface \(\Gamma\) determined by \(\text{det}(L(\lambda)= \mu I)= 0\). It is possible to introduce deformations that do not destroy neither the Liouville torus \(\text{Jac }\Gamma\) nor the quasi-periodic flow on them. These deformations are described by using the branch points \(\lambda_ k\) of \(\Gamma\) and are determined by the Whitham equation \[ (\tau d\Omega(p,\tau)+ d\Omega_ n(p,\tau))_{\lambda= \lambda_ j(\tau)}\partial_ \tau \lambda_ j(\tau)= 0, \] where \(\Omega\), \(\Omega_ n\) are appropriate Abelian integrals.
reviewed by: Jan Chrastina
Mathematics Subject Classification ID: 34M99
Mathematics Subject Classification ID: 34A25
Mathematics Subject Classification ID: 37C55
zbMATH DE Number: 652156
zbMATH Keywords: finite-gap integration method
zbMATH Keywords: Baker-Akhiezer function
zbMATH Keywords: Riemann surface
zbMATH Keywords: deformations
zbMATH Keywords: quasi-periodic flow
zbMATH Keywords: Whitham equation
zbMATH Keywords: Abelian integrals
author: V. Yu. Novokshenov
MaRDI profile type: Publication
cites work: Q4090743
cites work: Q4275431
cites work: The Kowalewski top 99 years later: a Lax pair, generalizations and explicit solutions
cites work: Multiphase averaging and the inverse spectral solution of the Korteweg—de Vries equation
cites work: Theta functions on Riemann surfaces
cites work: The hyperbolic nature of the zero dispersion Kdv limit
cites work: Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory
cites work: Q3743738
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 652156