Coupled KP and BKP hierarchies and the corresponding symmetric functions

We introduce a series of two-component symmetric functions, investigate the coupled Kadomtsev–Petviashvili (KP) and B-type Kadomtsev–Petviashvili (BKP) hierarchy through their relationship with symmetric functions. Then the Plücker equations derived from their bilinear identity for these two hierarchy are presented in the form of composite Schur functions by using some results from the classical theory of symmetric functions. Finally, we give a combinatorial proof of the facts that two-component Schur polynomials solve the coupled KP hierarchy and two-component Schur Q-polynomials solve the coupled BKP hierarchy.