Lamé Potentials and the Stationary (m)kdv Hierarchy
نویسنده
چکیده
A new method of constructing elliptic finite-gap solutions of the stationary Korteweg-de Vries (KdV) hierarchy, based on a theorem due to Picard, is illustrated in the concrete case of the Lamé-Ince potentials −s(s+1)P(z), s ∈ N (P(.) the elliptic Weierstrass function). Analogous results are derived in the context of the stationary modified Korteweg-de Vries (mKdV) hierarchy for the first time.
منابع مشابه
Function of the Kdv Hierarchy
In this paper we construct a family of commuting multidimen-sional differential operators of order 3, which is closely related to the KdV hierarchy. We find a common eigenfunction of this family and an algebraic relation between these operators. Using these operators we associate a hy-perelliptic curve to any solution of the stationary KdV equation. A basic generating function of the solutions ...
متن کاملExplicit multiple singular periodic solutions and singular soliton solutions to KdV equation
Based on some stationary periodic solutions and stationary soliton solutions, one studies the general solution for the relative lax system, and a number of exact solutions to the Korteweg-de Vries (KdV) equation are first constructed by the known Darboux transformation, these solutions include double and triple singular periodic solutions as well as singular soliton solutions whose amplitude d...
متن کاملA Class of Matrix-valued Schrödinger Operators with Prescribed Finite-band Spectra
We construct a class of matrix-valued Schrödinger operators with prescribed finite-band spectra of maximum spectral multiplicity. The corresponding matrix potentials are shown to be stationary solutions of the KdV hierarchy. The methods employed in this paper rely on matrix-valued Herglotz functions, Weyl–Titchmarsh theory, pencils of matrices, and basic inverse spectral theory for matrix-value...
متن کاملSOLITARY SOLUTIONS OF COUPLED KdV AND HIROTA–SATSUMA DIFFERENTIAL EQUATIONS
By considering the set of coupled KdV differential equations as a zero curvature representation of some fourth order linear differential equation and factorizing the linear differential equation, the hierarchy of solutions of the coupled KdV differential equations have been obtained from the eigen spectrum of constant potentials.
متن کاملPicard Potentials and Hill's Equation on a Torus
An explicit characterization of all elliptic (algebro-geometric) nite-gap solutions of the KdV hierarchy is presented. More precisely, we show that an elliptic function q is an algebro-geometric nite-gap potential, i.e., a solution of some equation of the stationary KdV hierarchy, if and only if every solution of the associated diierential equation 00 + q = E is a meromorphic function of the in...
متن کامل