280 research outputs found
The model equation of soliton theory
We consider an hierarchy of integrable 1+2-dimensional equations related to
Lie algebra of the vector fields on the line. The solutions in quadratures are
constructed depending on arbitrary functions of one argument. The most
interesting result is the simple equation for the generating function of the
hierarchy which defines the dynamics for the negative times and also has
applications to the second order spectral problems. A rather general theory of
integrable 1+1-dimensional equations can be developed by study of polynomial
solutions of this equation under condition of regularity of the corresponding
potentials.Comment: 17
Elementary Darboux transformations and factorization
A general theorem on factorization of matrices with polynomial entries is
proven and it is used to reduce polynomial Darboux matrices to linear ones.
Some new examples of linear Darboux matrices are discussed.Comment: 10 page
The Canonical Symmetry for Integrable Systems
The properties of discrete nonlinear symmetries of integrable equations are
investigated. These symmetries are shown to be canonical transformations. On
the basis of the considered examples, it is concluded, that the densities of
the conservation laws are changed under these transformations by spatial
divergencies.Comment: 17 pages, LaTeX, IHEP 92-14
Discrete symmetry's chains and links between integrable equations
The discrete symmetry's dressing chains of the nonlinear Schrodinger equation
(NLS) and Davey-Stewartson equations (DS) are consider. The modified NLS (mNLS)
equation and the modified DS (mDS) equations are obtained. The explicitly
reversible Backlund auto-transformations for the mNLS and mDS equations are
constructed. We demonstrate discrete symmetry's conjugate chains of the KP and
DS models. The two-dimensional generalization of the P4 equation are obtained.Comment: 20 page
Dressing chain for the acoustic spectral problem
The iterations are studied of the Darboux transformation for the generalized
Schroedinger operator. The applications to the Dym and Camassa-Holm equations
are considered.Comment: 16 pages, 6 eps figure
Discrete Darboux transformation for discrete polynomials of hypergeometric type
Darboux Transformation, well known in second order differential operator
theory, is applied here to the difference equation satisfied by the discrete
hypergeometric polynomials(Charlier, Meixner-Krawchuk, Hahn)
Towards the theory of integrable hyperbolic equations of third order
The examples are considered of integrable hyperbolic equations of third order
with two independent variables. In particular, an equation is found which
admits as evolutionary symmetries the Krichever--Novikov equation and the
modified Landau--Lifshitz system. The problem of choice of dynamical variables
for the hyperbolic equations is discussed.Comment: 22
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