Lax-Sato Formulation of the KP Hierarchy

In this chapter, we briefly review the Lax formulation of the KP hierarchy, which consists of an infinite set of linear equations whose compatibility conditions give rise to the flows corresponding to the KP hierarchy. The main purpose of this section is to highlight the basic framework of integrability underlying the KP theory. Then we show that the multi-component Burgers hierarchy discussed in the previous chapter appears as a finite reduction in the Sato theory. In particular, we emphasize the importance of the \(\tau \)-function and explain the central role of the \(\tau \)-function in the KP hierarchy. The materials discussed in this chapter can also be found in [27, 35, 37, 91, 100, 111–113, 132].