Reduction technique for matrix nonlinear evolution equations solvable by the spectral transform

Publisher Website
Google Scholar
Abstract
The main purpose of this paper is to describe a technique of reduction, whereby from the class of evolution equations for matrices of order N solvable via the spectral transform associated to the (matrix) linear Schrödinger eigenvalue problem, one derives subclasses of nonlinear evolution equations involving less than N2 fields. To illustrate the method, from the equations for matrices of order 2 two subclasses of equations for 2 fields (rather than 4) are obtained. The first class coincides, or rather includes, that solvable via the spectral transform associated to the generalized Zakharov–Shabat spectral problem; further reduction to nonlinear evolution equations for a single field reproduces a number of well-known equations, but also yields a novel one (highly nonlinear). The second class also yields highly nonlinear equations; some examples are given, including another novel evolution equation for a single field.