Equivalent classes of integrable nonlinear evolution equations and generalised Miura transformations

Abstract

The authors write down the recursion operator which defines the class of integrable nonlinear evolution equations (NEE) associated with a spectral problem (SP) for matrices of rank 2 depending quadratically on the spectral parameter. Then they show the existence of four different transformations which transform the given SP into the well known one of Zakharov and Shabat (ZS) (1972) by composing these transformations they recover, among others, the so-called elementary Backlund transformations for the ZS SP. Under some restrictions, one is able to prove the complete equivalence of the given glass of NEE to the one associated with the ZS SP.