Progress of Theoretical Physics, Yol. 75, No.5, May 1986 The WKIS System: Backlund Transformations, Generalized Fourier Transforms and All That*) M. BOlT!, V. S. GERDJIKOY**) and F. PEMPINELLI Department of Physics, University of Leece, 73100 Leece (Received August 29, 1985) 1111 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. § 1. Introduction Several approaches to the nonlinear evolution equations. (NLEEs) solvable by the inverse scattering method (ISM) (see the monographs in Refs. 1) 3)) were started by the famous paper of Ablowitz, Kaup, Newell and Segur (AKNS).4) In particular, AKNS proposed a recursion approach to the description of the integrable NLEEs related to a given auxiliary linear problem L. They started from the Lax representation in the form L t- Mx+ [L, M]=O, which is the compatibility condition for the linear problems </JAx, t,A)=L(x, t,A)</J(X, t,A), </Jt(x, t,A)=M(x, t,A)</J(X, t,A)-</J(X, t,A)C(A). (I-I) (1- 2) (1-3) Here L, M and Care 2x2 matrix-valued polynomials in A and A-I. Land M depend on x and t through the set of potentials in L, while C is x and t independent. C does not appear in (1-1) and is introduced for later convenience. Choosing L = [== i ( - A63 + Q) and requiring that (1-1) is fulfilled identically with respect to A, the authors of Ref. 4) obtained recurrent relations, which allowed them to determine the coefficients of M = 1\71 through Q(x, t). They showed that in solving these recurrent relations a fundamental role is played by some integrQ-differential operators L± which, later, have been called recursion (or generating) operators. Studying the generalized Wronskian identities Calogero and Degasperis proposed in Ref. 5) a generalization A± of the recursion operators L±. Using A± they were able to describe the class of Backlund transformations (BT) for the corresponding NLEEs. Although the authors of Refs. 4) and 5) formulated their approaches on the example *) This work has been partially supported by M. P. I. and I. N. F. N. **) Permanent address: Institute for Nuclear Research and Nuclear Energy, 1184 Sofia, Bulgaria. Downloaded from https://academic.oup.com/ptp/article-abstract/75/5/1111/1849917 by guest on 07 June 2020