A new extended matrix KP hierarchy and its solutions

Starting from the matrix KP hierarchy and adding a new $\tau_{B}$ flow, we obtain a new extended matrix KP hierarchy and its Lax representation with the symmetry constraint on squared eigenfunctions taken into account. The new hierarchy contains two sets of times $t_{A}$ and $\tau_{B}$ and also eigenfunctions and adjoint eigenfunctions as components. We propose a generalized dressing method for solving the extended matrix KP hierarchy and present some solutions. We study the soliton solutions of two types of $(2+1)$-dimensional AKNS equations with self-consistent sources and two types of Davey–Stewartson equations with self-consistent sources.