A completeness study on a class of Lax pairs
Seminar Room 1, Newton Institute
Despite the existence of a Lax pair for a given equation often being used as the definition of its integrability, recently there have been few studies that seek to find or categorize nonlinear equations that using Lax pairs as their starting point. Of those studies that do begin with Lax pairs, all choose a priori a form of the Lax pair, i.e. the type of dependence on the spectral parameter, thus limiting the possible results.
We present a study that begins with Lax pairs for partial difference equations that are 2x2, where each entry of the Lax matrices contains only one separable term. Besides that the terms are general, no ansatz is made about the dependence on the spectral parameter, nor about the terms dependent on the lattice variables. We examine all possible groups of equations that can arise via the compatibility condition. Once a system of equations is extracted from the compatibility condition, it is solved in a way that preserves its generality, up to a point where nonlinear evolution equations are apparent, or the system is shown to be trivial.
In fact, of all the potential Lax pairs identified by this method, only two lead to non-trivial and unconstrained evolution equations. These are new, higher order varieties of the lattice sine-Gordon (LSG) and the lattice modified KdV (LMKDV) equations. The remaining systems are shown to be trivial, overdetermined or underdetermined, which suggests a connection between the existence of a Lax pair and the singularity confinement method of identifying integrable equations.
As we do not make any assumptions about the explicit dependence on the spectral parameter, we show that a particular nonlinear equation may have many Lax pairs, all depending on the spectral parameter in different ways. The effect that this freedom has on the process of inverse scattering is, as yet, unclear.