On equations solvable by the inverse scattering method for the Dirac operator
The paper aims to explore the characteristic property of Lax representations of integrable evolution equations. For the so-called Dirac operator, which leads to the AKNS spectral problem, the author proves that the \(S\)-matrix (or the scattering data) of the Dirac operator under an evolution equation satisfies a regular first-order ordinary differential equation, with an independent variable being time, if and only if the evolution equation possesses a Lax representation associated with the Dirac operator. Such Lax representations are equivalent to zero curvature representations and need the isospectral condition. The technique developed is constructive, which shows the relation between Lax representations and the governing equations for the scattering data. It would be also possible to apply the author's technique of using the \(S\)-matrix to characterize integrable evolution equations associated with other spectral problems and even under nonisospectral conditions.
scientific article; zbMATH DE number 1892195