Differential equations in the spectral parameter for matrix differential operations of AKNS type.

摘要

Let {dollar}L{dollar} be an AKNS operator, i.e., an operator of the form {dollar}L = Jsp{lcub}-1{rcub}(partialsb{lcub}x{rcub} - Q(x)){dollar}, where {dollar}Q(x){dollar} is off-diagonal and {dollar}J = diag(1,omega, ..., omegasp{lcub}N-1{rcub}){dollar} with {dollar}omega = esp{lcub}2pi i/N{rcub}{dollar}. In this work we shall be concerned with the connection between bispectral operators and the following two objects: matrix Darboux transformations and completely integrable hierarchies of nonlinear evolution equations generated by Lax Pair/Zakharov-Shabat/AKNS constructions from the operator {dollar}L{dollar} and their manifolds of rational solutions. We say that an operator {dollar}L{dollar} has the bispectral property if there exists a family of eigenfunctions {dollar}varphi(x,k){dollar} of {dollar}L{dollar} satisfying a differential equation in the spectral parameter of the form {dollar}B(k, partialsb{lcub}k{rcub})varphi = Theta(x)varphi{dollar}, where {dollar}B{dollar} is a linear differential operator of positive order independent of {dollar}x,{dollar} and {dollar}Theta(x){dollar} is a matrix valued function. Also, by a matrix Darboux transformation, we mean a gauge transformation of the form {dollar}varphi mapsto (k - A(x))varphi{dollar} that maps solutions of {dollar}Lvarphi = kvarphi{dollar} into solutions of {dollar}tilde L tildevarphi = ktildevarphi{dollar}.; We show that for {dollar}N = 2{dollar} and{dollar}{dollar}Q=leftlbrackmatrix{lcub}0&qcr r&0cr{rcub}rightrbrack{dollar}{dollar}we can construct families of bispectral operators by successive applications of matrix Darboux transformations to {dollar}Q = 0{dollar}. This implies that the operators {dollar}L{dollar} associated to certain families of rational solutions of the AKNS hierarchy possess the bispectral property. Such families include all the rational solutions, decaying at infinity, of the mKdV hierarchy, as well as more general rational solutions of the AKNS hierarchy such that {dollar}q{dollar} and {dollar}r{dollar} are functionally independent. This answers the question of whether there are potentials {dollar}Q(x){dollar} other than those in the mKdV hierarchy that have the bispectral property.; For {dollar}N geq 2{dollar} and {dollar}Q{dollar} satisfying certain symmetry conditions, we use matrix Darboux transformations to obtain nontrivial bispectral potentials by starting with potentials of the form {dollar}{lcub}1over x{rcub}Qsb0,{dollar} where {dollar}Qsb0{dollar} is constant.; We characterize the bispectral AKNS operators that have two linearly independent eigenfunctions, {dollar}varphisb1{dollar} and {dollar}varphisb2{dollar}, satisfying a differential equation in {dollar}k{dollar} of the form {dollar}Bsb0(k)partialsb{lcub}k{rcub}varphisb{lcub}i{rcub} + Bsb1(k)varphisb{lcub}i{rcub} = Theta(x)varphisb{lcub}i{rcub},{dollar} where {dollar}Bsb0{dollar} is a scalar valued function, and {dollar}Bsb1{dollar} and {dollar}Theta{dollar} are matrix valued functions. The answer is connected with some special solutions of nonlinear evolution equations in the AKNS hierarchy. (Abstract shortened with permission of author.)