TY - JOUR

T1 - Bäcklund transformations for nonlinear sigma models with values in Riemannian symmetric spaces

AU - Harnad, J.

AU - Saint-Aubin, Y.

AU - Shnider, S.

PY - 1984/9

Y1 - 1984/9

N2 - This work deals with Bäcklund transformations for the principal SL(n, ℂ) sigma model together with all reduced models with values in Riemannian symmetric spaces. First, the dressing method of Zakharov, Mikhailov, and Shabat is shown, for the case of a meromorphic dressing matrix, to be equivalent to a Bäcklund transformation for an associated, linearly extended system. Comparison of this multi-Bäcklund transformation with the composition of ordinary ones leads to a new proof of the permutability theorem. A new method of solution for such multi-Bäcklund transformations (MBT) is developed, by the introduction of a "soliton correlation matrix" which satisfies a Riccati system equivalent to the MBT. Using the geometric structure of this system, a linearization is achieved, leading to a nonlinear superposition formula expressing the solution explicitly in terms of solutions of a single Bäcklund transformation through purely linear algebraic relations. A systematic study of all reductions of the system by involutive automorphisms is made, thereby defining the multi-Bäcklund transformations and their solution for all Riemannian symmetric spaces.

AB - This work deals with Bäcklund transformations for the principal SL(n, ℂ) sigma model together with all reduced models with values in Riemannian symmetric spaces. First, the dressing method of Zakharov, Mikhailov, and Shabat is shown, for the case of a meromorphic dressing matrix, to be equivalent to a Bäcklund transformation for an associated, linearly extended system. Comparison of this multi-Bäcklund transformation with the composition of ordinary ones leads to a new proof of the permutability theorem. A new method of solution for such multi-Bäcklund transformations (MBT) is developed, by the introduction of a "soliton correlation matrix" which satisfies a Riccati system equivalent to the MBT. Using the geometric structure of this system, a linearization is achieved, leading to a nonlinear superposition formula expressing the solution explicitly in terms of solutions of a single Bäcklund transformation through purely linear algebraic relations. A systematic study of all reductions of the system by involutive automorphisms is made, thereby defining the multi-Bäcklund transformations and their solution for all Riemannian symmetric spaces.

UR - http://www.scopus.com/inward/record.url?scp=0010945892&partnerID=8YFLogxK

U2 - 10.1007/BF01210726

DO - 10.1007/BF01210726

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AN - SCOPUS:0010945892

SN - 0010-3616

VL - 92

SP - 329

EP - 367

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 3

ER -