Multi-field Volterra-like Systems

Abstract

We have seen in Chapter 12 that factorizing the Lax matrix $T\left( {b,{a^{\left( 1 \right)}},...,{a^{\left( m \right)}},\lambda } \right)$ of the multi-field Toda lattice TL_m+1 (or, more precisely, the matrix $I + \in T$ ) into just two factors gives rise to the modified multi-field Toda lattice. One of these factors is two-diagonal, and the other is m,-diagonal in the (m + 1)-field situation. For the two-field systems this exhausts the possibilities of factorization, so that the Volterra lattice (coming from factorizing T itself) is more or less equivalent to the modified Toda lattice. This is no more true in the multi-field situation, and the present chapter is devoted to the case of the ultimate factorization, when all the factors are bi-diagonal. The corresponding generalization of the Volterra lattice is a system with m fields $v^{\left( j \right)} ,1 \leqslant j \leqslant m,$ Satisfying

$$\dot{v}_{k}^{{\left( j \right)}} = v_{k}^{{\left( j \right)}}\left( {\sum\limits_{{i = 1}}^{{j - 1}} {\left( {v_{{k + 1}}^{{\left( i \right)}} - v_{k}^{{\left( i \right)}}} \right) + \sum\limits_{{i = j + 1}}^{m} {\left( {v_{k}^{{\left( i \right)}} - v_{{k - 1}}^{{\left( i \right)}}} \right)} } } \right). $$

(15.1.1)