Squared eigenfunction symmetries of the constrained integrable hierarchies

Abstract

In this paper, we mainly investigate the constrained Kadomtsev–Petviashvili (KP) and modified KP (mKP) hierarchies. Squared eigenfunction (SE) symmetry is an important symmetry defined by eigenfunctions and adjoint eigenfunctions, which can be viewed as the generator of the additional symmetries. Due to special constraints on the Lax operators, it is found that there are three different cases for the generating eigenfunctions and adjoint eigenfunctions of the SE symmetries. Then we discuss the changes of the SE symmetries under the Miura links between the KP and mKP hierarchies. Next, the exponential of the SE symmetries is shown to be the binary Darboux transformations. At last based upon the results above, new additional symmetries for the constrained KP and mKP hierarchies are constructed, different from the additional Virasoro symmetries defined by Aratyn.