Classical \(r\)-matrix approach to the system associated with the three-wave interaction equations (Q2717974)

The authors consider the following three-wave interaction equations: NEWLINE\[NEWLINEu_{ij+1}= \frac{\beta_i-\beta_j} {\alpha_i-\alpha_j} u_{ijx}+ \sum _{\substack{ k=1\\ i,j\neq k}}^3 \Biggl( \frac{\beta_k-\beta_i} {\alpha_k-\alpha_i}- \frac{\beta_k-\beta_j} {\alpha_k-\alpha_j} \Biggr) u_{ik}u_{kj}, \quad i\neq j,\;1\leq i,j\leq 3.\tag{1}NEWLINE\]NEWLINE Here they present a Lax representation for this system and an \(r\)-matrix is found from the Lax operator. From the \(r\)-matrix, they obtain \(3N\) functionally independent and involutive integrals of motion, showing complete integrability in the Liouville sense of the system.