Sato–Bäcklund transformations and string equations of the mKP hierarchy

Additional symmetry is an important kind of symmetries depending explicitly on the time and space variables, which can be expressed through Sato–Bäcklund transformations. In this paper, we construct Sato–Bäcklund transformations of the modified KP hierarchy and its constrained cases. Then the string equations of the [Formula: see text]-reduced modified KP hierarchy are established by requiring the system independent on some additional symmetry flows, which are expressed by the Lax operator [Formula: see text] and the Orlov–Shulman’s operator [Formula: see text]. At last, we obtain the negative Virasoro constraint on the two tau functions of the 2-reduced modified KP hierarchy satisfying the string equations.

Bäcklund transformation of Frobenius Painlevé equations

In this paper, in order to generalize the Painlevé equations, we give a [Formula: see text]-Painlevé IV equation which can apply Bäcklund transformations to explore. And these Bäcklund transformations can generate new solutions from seed solutions. Similarly, we also introduce a Frobenius Painlevé I equation and Frobenius Painlevé III equation. Then, we find the connection between the Frobenius KP hierarchy and Frobenius Painlevé I equation by the Virasoro constraint. Further, in order to seek different aspects of Painlevé equations, we introduce the Lax pair, Hirota bilinear equation and [Formula: see text] functions. Moreover, some Frobenius Okamoto-like equations and Frobenius Toda-like equations can also help us to explore these equations.

FLAT CURRENTS OF A GREEN–SCHWARZ SIGMA MODEL ON SUPERCOSET TARGETS WITH ℤ4m GRADING

We construct actions of Green–Schwarz sigma models on supercoset targets with ℤ4m grading whose kinetic terms only contain the target-space bosons. We consider a simple case of such kinetic term and show that there exist a one-parameter family of flat currents of the model by requiring a suitable choice of the Wess–Zumino term. Such flat currents naturally lead to a hierarchy of classical conserved nonlocal charges. We also find that the one-parameter flat currents of the model satisfy equations of motion and the Virasoro constraint. This implies that one can generate a series of classical solutions from an existing one. When m = 1, our model coincides with the well-known model given by Metsaev and Tseytlin on a supercoset PSU (2, 2|4)/[ SO (4, 1) × SO (5)] and similar models.