Toda lattice hierarchy. I
A sequence of mutually commuting higher evolutions (i.e. a hierarchy) for the two dimensional Toda lattice equation (TL) \(\partial_{x_ 1}\partial_{y_ 1}u(s)=e^{u(s)-u(s-1)}-e^{u(s+1)-u(s)},\) with \(u(s)=u(s;x_ 1,y_ 1),\) \(s\in Z\), is presented. Let \(x=(x_ 1,x_ 2,...)\) and \(y=(y_ 1,y_ 2,...)\) be independent variables with infinitely many components. One introduces the matrices of size \(Z\times Z:\) \(L=\sum_{-\infty <n\leq 1}diag[b_ n(s;x,y)]\Lambda^ n,\quad b_ 1=0;\quad M=\sum_{-1\leq n<\infty}diag[c_ n(s;x,y)]\Lambda^ n,\quad c_{-1}\neq 0+\quad B_ n=(L^ n)_+=\sum^{n}_{m=0}diag[b_{n,m}(s;x,y)]\Lambda^{n- m};\quad C_ n=(M^ n)_-=\sum^{n- 1}_{m=0}diag[c_{n,m}(s;x,y)]\Lambda^{n-m}.\) The quantities \(b_ j(j\leq 1)\) and \(c_ j(j\geq -1)\) serve as unknown functions in the hierarchy. The TL hierarchy considered by the author is defined by the system of Lax type: \[ \partial_{x_ n}L=[B_ n,L],\quad \partial_{y_ n}L=[C_ n,L],\quad \partial_{x_ n}M=[B_ n,M],\quad \partial_{y_ n}M=[C_ n,M], \] n\(=1,2,... \). The linearization of the TL hierarchy is achieved by introducing the wave matrices \(W^{(\infty)}\) and \(W^{(0)}\) of size \(Z\times Z\) with the properties: \(LW^{(\infty)}=W^{(\infty)}\Lambda,\) \(MW^{(0)}=W^{(0)}\Lambda^{-1},\) \(\partial_{x_ n}W=B_ nW,\) \(\partial_{y_ n}W=C_ nW,\) \(W=W^{(\infty)},W^{(0)}.\) The bilinearization of the TL hierarchy is also obtained using the \(\sigma\) function defined by \(u(s)=\log [\sigma (s+1)/\sigma (s)].\)
scientific article; zbMATH DE number 3879429
scientific article