Equivalent Classes of Integrable Nonlinear Evolution Equations and Generalized Miura Transformations

摘要

The recursion operator that defines the class of integrable nonlinear evolution equations (NEE) associated to a spectral problem (SP) for matrices of rank two depending quadratically on the spectral parameter is presented. The existence of four different transformations which transform the given SP into that of Zakhorov and Shabat (ZS) is shown. By composing these transformations, the so called elementary Backlund transformation for the ZS SP is recovered. The complete equivalence of the given class of NEE to the one associated to the ZS SP can be proved, under given restrictions.