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

*(English)*Zbl 0532.58036This 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.

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

##### MSC:

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 |

##### Keywords:

soliton; dressing method; multi-Bäcklund transformation; matrix Riccati equations; Grassmann manifolds
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\textit{J. Harnad} et al., Commun. Math. Phys. 92, 329--367 (1984; Zbl 0532.58036)

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##### References:

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