The WKIS System: Backlund Transformations, Generalized Fourier Transforms and All That

Abstract

The spectral problem of Wadati·Konno·Ichikawa-Shimizu (WKIS) and the nonlinear evolution equa­ tions (NLEEs) related to it, which are solvable by the use of the inverse scattering method, are considered. The generating operators L± and A± of these NLEEs and of their Backlund transformations (BTs) together with the completeness relations of the eigenfunctions of both operators are derived. The action-angle variables and the hierarchies of Hamiltonian structures for these NLEE are constructed. ' The interrelations between the hierarchies of the WKIS system and its gauge equivalents (the Zakharov-Shabat system, etc.) are established. A convenient gauge transformation of the WKIS spectral problem is used to get in an alternative way the general BT. The so-called elementary BTs are obtained and it is shown that they can be cast into a form similar to that found by Darboux for the Schrodinger spectral problem. The nonlinear superposition formulae are also explicitly written.