zbMATH — the first resource for mathematics

Bäcklund transformations for nonlinear sigma models with values in Riemannian symmetric spaces. (English) Zbl 0532.58036
This paper deals with the Bäcklund transformations (BTs) for the principal SL(n,C) sigma model for which the governing equation reads (1) \(A_{\eta}+B_{\xi}=0\), where \(A=g_{\xi}g^{-1}\) and \(B=g_{\eta}g^{-1}\) with g taking values in the Lie group SL(n,C). The linearized equations, whose integrability condition is the Eq. (1), are (2) \(\psi_{\xi}=(A/(1+\lambda))\psi\), \(\psi_{\eta}=(B/(1- \lambda))\psi\). A simple BT for Eq. (1) was found first by Ogielski et al. in 1980. By repeated differentiation of the above Eqs. (2), the authors construct from the original Eq. (1) an extended system \[ g_{j,\xi}+g_{j-1,\xi}=Ag_ j,\quad g_{j,\eta}-g_{j-1,\eta}=Bg_ j,\quad j=1,...,K-1, \] where \(g_ 0=g=\psi(0)\), \(g_ j=(1/j!)(d^ j/d\lambda^ j)(\psi(\lambda))|_{\lambda =0}.\)
Then they apply the dressing method of Zakharov and Shabat to derive a BT for this extended system. This BT turns out to be a K-multi-Bäcklund transformation (K-MBT), whose effect is the same as iterated sequence of K simple BTs. Since the composition of two simple BTs, characterized by two sets of parameters, is equal to a 2-MBT which can be shown to be symmetric with respect to the two sets of parameters, the authors give therefore a beautiful proof of the permutability of BTs. It seems that this idea could be applied to other soliton equations as well. In order to solve the MBT, the authors introduce a new quantity, they call it the soliton correlation matrix M, which is constructed from the residues of the product \(\chi^{-1}(\mu)\chi(\lambda)\), where \(\chi\) (\(\lambda)\) is the dressing matrix. This matrix M is shown to satisfy a system of matrix Riccati equations, which in its turn can be solved by a linear fractional transformation. Since the matrix Riccati equations are in fact equivalent to the MBT, the authors thus develop a method for its solution. They also give a geometric interpretation of this Riccati equation in terms of group actions on Grassmann manifolds. By virtue of this interpretation the authors reduce the Riccati equations to a system of linear matrix equations in n or 2n dimensions. In the last part of the paper the authors study all reductions of the system by involutive automorphisms. The sigma fields with values in various irreducible Riemannian symmetric spaces and the corresponding MBTs are discussed in some detail.
Reviewer: G.Tu

58J90 Applications of PDEs on manifolds
35Q99 Partial differential equations of mathematical physics and other areas of application
53C80 Applications of global differential geometry to the sciences
53C35 Differential geometry of symmetric spaces
81T60 Supersymmetric field theories in quantum mechanics
Full Text: DOI
[1] Harnad, J., Saint-Aubin, Y., Shnider, S.: Superposition of solutions to Bäcklund transformations for the SU(n) principal sigma model. Preprint CRMA-1074. J. Math. Phys. (to appear) (1983)
[2] Harnad, J., Saint-Aubin, Y., Shnider, S.: Quadratic Pseudopotentials for GL(n, ?) principal sigma models. Preprint CRMA-1075. Physica D (to appear) (1983) · Zbl 0584.58016
[3] Ogielski, A.T., Prasad, M.K., Sinha, A., Wang, L.L.C.: Bäcklund transformations and local conservation laws for principal chiral fields. Phys. Lett.91 B, 387 (1980)
[4] Zakharov, V.E., Mikhailov, A.V.: Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method. Zh. Eksp. Teor. Fiz.74, 1953 (1973) [Sov. Phys. JETP47, 1017 (1979)]
[5] Zakharov, V.E., Shabat, A.B.: Funk. Anal. Pr.13, 13 (1979) [Funct. Anal. Appl.13, 166 (1979)]
[6] Zakharov, V.E., Mikhailov, A.V.: On the integrability of classical spinor models in two-dimensional space-time. Commun. Math. Phys.74, 21 (1980)
[7] Saint-Aubin, Y.: Bäcklund transformations and soliton-type solutions for ? models with values in real Grassman spaces. Lett. Math. Phys.6, 441 (1983) · Zbl 0522.53057
[8] Eichenherr, H., Forger, M.: On the dual symmetry of the non-linear sigma models. Nucl. Phys. B155, 381 (1979); More about non-linear sigma models on symmetric spaces. Nucl. Phys. B164, 528 (1980)
[9] Harnad, J., Winternitz, P., Anderson, A.L.: Superposition principles for matrix Riccati equations. J. Math. Phys.24, 1062 (1983) · Zbl 0556.34015
[10] Helgason, S.: Differential geometry, Lie groups and symmetric spaces. London, New York: Academic Press 1978 · Zbl 0451.53038
[11] Mikhailov, A.V.: Reduction in integrable systems. The reduction group. JETP Lett.32, 174 (1980)
[12] Pohlmeyer, K.: Integrable Hamiltonian systems and interactions through quadratic constraints. Commun. Math. Phys.46, 207 (1976) · Zbl 0996.37504
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.