这是一份liverpool利物浦大学PHYS156的成功案例

a linear invertible map $T, \mathbf{R} \rightarrow \mathbf{R}$ such that for any $\psi(x, t)$ and $\varphi(x, t) \in \mathbf{R}$. $x \in R^{n}, t \in R$ we have
$$
T(\psi \varphi)=T(\psi) T(\varphi), \quad T(1)=1 .
$$
The automorphism with the defining allows us to write down a wide class of functional-differential-difference and difference-difference equations starting from
$$
\psi_{t}(x, t)=\sum_{m=-M}^{N} U_{m} T^{m} \psi
$$
where $M$ and $N$ are integers. For example, the operator $T$ can be chosen as
$$
T \psi(x, t)=\psi(q x+\delta, t),
$$
where $q \in G L(n, \mathbb{C}), \delta \in R^{n}$. Another choice gives
$$
T \psi(x)=W \psi(x) W^{-1}, \quad W \in G L(n, \mathbb{C})
$$

PHYS156 COURSE NOTES ：

The formalism for the second DT from may be similarly constructed on the ground of the identity
$$
T^{m} \varphi=\prod_{k=0}^{m} T^{k}\left(\sigma^{-}\right) T^{-1} \varphi=B_{m}^{-}\left(\sigma^{-}\right) T^{-1} \varphi
$$
The definition of the lattice Bell polynomials of the second type $B_{m}^{-}\left(\sigma^{-}\right)$can be extracted. The evolution equation for $\sigma^{-}$is similar :
$$
\sigma_{t}^{-}=\sum_{m=-M}^{N}\left[U_{m} B_{m}^{-}\left(\sigma^{-}\right)-\sigma^{-} T^{-1}\left(U_{m}\right) B_{m-1}^{-}\left(\sigma^{-}\right)\right]
$$