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A method is proposed to construct a new extended KP hierarchy, which includes two types of KP equation with self-consistent sources and admits reductions to k-constrained KP hierarchy and to Gelfand-Dickey hierarchy with sources. It provides a general way to construct soliton equations with sources and their Lax representations.
With the square eigenfunctions symmetry constraint, we introduce a new extended matrix KP hierarchy and its Lax representation from the matrix KP hierarchy by adding a new $tau_B$ flow. The extended KP hierarchy contains two time series ${t_A}$ and $
Using the determinant representation of gauge transformation operator, we have shown that the general form of $tau$ function of the $q$-KP hierarchy is a q-deformed generalized Wronskian, which includes the q-deformed Wronskian as a special case. On
A new (gamma_n,sigma_k)-KP hierarchy with two new time series gamma_n and sigma_k, which consists of gamma_n-flow, sigma_k-flow and mixed gamma_n and sigma_k evolution equations of eigenfunctions, is proposed. Two reductions and constrained flows of
We consider solutions of the KP hierarchy which are elliptic functions of $x=t_1$. It is known that their poles as functions of $t_2$ move as particles of the elliptic Calogero-Moser model. We extend this correspondence to the level of hierarchies an
We consider solutions of the matrix KP hierarchy that are elliptic functions of the first hierarchical time $t_1=x$. It is known that poles $x_i$ and matrix residues at the poles $rho_i^{alpha beta}=a_i^{alpha}b_i^{beta}$ of such solutions as functio