The gauge transformations generated by the wave functions in the constrained modified KP hierarchy

Abstract

There are two ways to choose the generating functions of the gauge transformations [Formula: see text] and [Formula: see text], when dicussing the gauge transformations for the constrained modified KP hierarchy. The first is to select the (adjoint) eigenfunctions, while the second is the (adjoint) wave functions. In this paper, we will mainly discuss the gauge transformations obtained by the second method. The corresponding successive applications are considered. Also, we investigate the results of the gauge transformation derived through the union of these two methods.

A new generalized constrained modified KP hierarchy

“…The famous AKNS hierarchy and Yajima-Oikawa hierarchy can be found in the frame of the constrained KP hierarchies. 14 By now, much work has been done in the aspects of the constrained KP and mKP hierarchies, for example, additional Viraroso symmetries, 6,21,22 Hamiltonian structures, 12,15,23 Darboux transformations, [24][25][26] and bilinear formulations. 16,17,27 Here in this paper, we generalize the constrained mKP hierarchy from (L k ) ≤0 = q𝜕 −1 r into a more general case by adding a multiple of the inverse of Lax operator in the nonpositive part, that is, (L k ) ≤0 = q𝜕 −1 r + cL −1 , which is called the generalized constrained mKP (gcmKP for short) hierarchy.…”

“…The famous AKNS hierarchy and Yajima–Oikawa hierarchy can be found in the frame of the constrained KP hierarchies 14 . By now, much work has been done in the aspects of the constrained KP and mKP hierarchies, for example, additional Viraroso symmetries, 6,21,22 Hamiltonian structures, 12,15,23 Darboux transformations, 24–26 and bilinear formulations 16,17,27 . Here in this paper, we generalize the constrained mKP hierarchy from $$$ {\left({L}^k\right)}_{\le 0}=q{\partial}^{-1}r $$$ into a more general case by adding a multiple of the inverse of Lax operator in the nonpositive part, that is, $$$ {\left({L}^k\right)}_{\le 0}=q{\partial}^{-1}r+c{L}^{-1} $$$, which is called the generalized constrained mKP (gcmKP for short) hierarchy.…”