Integration of non-Abelian Langmuir type lattices by the inverse spectral problem method
The author deals with Cauchy problems of the type \[ \dot C_n= \sum^q_{i= 1} C_{n+ i}C_n- C_n \sum^q_{i= 1} C_{n- i}\quad (q\in\mathbb{N}),\tag{1} \] \[ C_n= C_n(t)\in{\mathcal L}(B),\quad t\in [0,T],\quad 0<T\leq\infty,\quad \cdot={d\over dt}, \] in semi-infinite \((n\in\mathbb{Z}_+)\) and infinite \((n\in\mathbb{Z})\) cases; here \(B\) is a Banach space. In the scale case for \(q= 1\) and positive \(C_n\) this system is the well-known Langmuir chain, for \(q> 1\) it is an discrete analogue of the Korteweg-de Vries equation. The author describes an effective procedure for constructing bounded solutions to (1) in terms of the corresponding Lax representation \(\dot L= [L,A]\) of system (1) and its Weil matrix \({\mathfrak M}(\lambda)\). The main results are necessary and sufficient conditions (of algebraic type) under which this procedure really defines a unique solution to (1).
scientific article; zbMATH DE number 1179571