Rational solutions of nonlinear evolution equations, vertex operators, and bispectrality

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Abstract

We prove that if q(x, t2, ..., tm) and r(x, t2, ..., tm) are certain rational solutions of the AKNS hierarchy, then there are eigenfunctions ψ(x, k) of the AKNS/ZS operator L= ∂x -q r -∂x satisfying a differential equation in the spectral parameter k of the form B(k, ∂k)ψ = Θ(x)ψ, where B(∂k, k) is a matrix differential operator, independent of x, and Θ is a nonconstant function of x. We also discuss the relation between this result and a similar one for the rational solutions of the Schrödinger operator with potentials in the manifold of rational solutions of the KdV hierarchy.

Original languageBritish English
Pages (from-to)71-98
Number of pages28
JournalJournal of Differential Equations
Volume97
Issue number1
DOIs
StatePublished - May 1992

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