A general scheme for analysing reductions of dispersionless integrable hierarchies is presented. It is based on a method for determining the S-function by means of a system of first-order differential equations. Compatibility systems of nonlinear partial differential equations of Bourlet type characterizing both reductions and hodograph solutions of the dKP hierarchy are obtained. Wide classes of illustrative explicit examples are exhibited.

The authors present a unified treatment of a modified singular manifold expansion method as an improved variant of the Painleve analysis for partial differential equations with two branches in the Painleve expansion. They illustrate the method by fully applying it to the Boussinesq classical system and the Mikhailov-Shabat system.

We give an exhaustive characterization of the complex saddle point configurations of the Gross-Witten-Wadia matrix model in the large-$N$ limit. In particular, we characterize the cases in which the saddles accumulate in one, two, or three arcs, in terms of the values of the coupling constant and of the fraction of the total unit density that is supported in one of the arcs, and derive an explicit condition for gap closing associated with nonvacuum saddles. By applying the idea of large-$N$ instanton we also give direct analytic derivations of the weak-coupling and strong-coupling instanton actions.

Wide classes of explicit solutions of the Manin‐Radul and Jacobian supersymmetric KP hierarchies are constructed by using line bundles over complex supercurves based on the Riemann sphere. Their construction extends several ideas of the standard KP theory, such as wave functions, ∂̄‐equations and τ‐functions. Thus, supersymmetric generalizations of N‐soliton solutions, including a new purely odd ‘‘solitino’’ solution, as well as rational solutions, are found and characterized.