Darboux transformation for differential-difference principal chiral equation and its continuous limits (Q2738073)

The continuous chiral equation \((R_x R^{-1})_t+ (R_t R^{-1})_x= 0\) has the well-known Lax pair NEWLINE\[NEWLINE\varphi_x= \lambda R_x R^{-1}\varphi, \qquad \varphi_t= \frac{\lambda} {2\lambda-1} R_t R^{-1} \varphi.NEWLINE\]NEWLINE The author investigates differential-difference chiral equation NEWLINE\[NEWLINE\frac{d}{dt} (R_x R^{-1})+ \Delta \Biggl( \frac{dR}{dt} R^{-1}\Biggr)= 0,NEWLINE\]NEWLINE the relevant generalized Lax pair NEWLINE\[NEWLINE\psi_+= (\lambda+I+R_+ R^{-1})\psi, \qquad \frac{d\psi}{dt}= -\frac{dR}{dt} R^{-1} \frac{\psi}{\lambda}NEWLINE\]NEWLINE (abbreviations \(P= P(n,t)\), \(\psi= \psi(n,t)\), \(A_+= A(n+1,t)\), \(\Delta= A_+-A\)) and its (rather useful) adaptation by substitution \(\lambda h-1\) for \(\lambda\). Darboux transformations (for \(Gl(n)\) and \(U(n)\) cases) of these Lax pairs are thoroughly discussed. If the size \(h\) of the lattice tends to zero, the limits provide two different series of solutions of the continuous chiral equation.