An algebraic characterization of the bilinear relations of the matrix hierarchy associated with a commutative algebra of \(k\times k\)-matrices
A purely algebraic characterization for the equations of the matrix hierarchy that can be associated to a maximal commutative subalgebra of the \(k\times k\) matrices is presented. One starts with a description of the algebraic background of the corresponding system of equations and it is shown how they result from linearization. Then, the eigenfunctions occurring in this linear system are obtained. It is shown that these eigenfunctions, named wave functions, completely determine the solutions of the non-linear equations. One associates with them an eigenfunction -- the dual wave function -- and a characterization of the duality property is given. The algebraic framework used in this paper is adequate for the description of the linearization and the Lax form of the hierarchy. It is suggested that the results obtained allow an algebraic interpretation of the bilinear form of the respective system of nonlinear equations.
scientific article; zbMATH DE number 5700578
scientific article