Lax pairs

Lax pairs are a mathematical tool used to analyze certain complex systems, such as nonlinear equations, by rewriting them as a compatibility condition between two related operators or matrices. Think of them like two interconnected parts of a machine—if their relationship stays consistent over time, the system's behavior can be solved more easily. This approach helps identify conserved quantities and simplifies understanding the system's dynamics, especially in integrable systems, where the solution remains stable and predictable over time.