Solving the constrained modified KP hierarchy by gauge transformations

Abstract

In this paper, we mainly investigate two kinds of gauge transformations for the constrained modified KP hierarchy in Kupershmidt-Kiso version. The corresponding gauge transformations are required to keep not only the Lax equation but also the Lax operator. For this, by selecting the special generating eigenfunction and adjoint eigenfunction, the elementary gauge transformation operators of modified KP hierarchyx , become the ones in the constrained case. Finally, the corresponding successive applications of T D and T I on the eigenfunction Φ and the adjoint eigenfunction Ψ are discussed.

A new generalized constrained modified KP hierarchy

“…When one choose c is zero, the c-k gcmKP hierarchy will become the usual constrained mKP hierarchy. 6,19,24,27 Proposition 7. The definition of c-k gcmKP hierarchy is reasonable; that is, (1), (18), and ( 19) are consistent with each other.…”

“…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.…”

Bilinear identities for the constrained modified KP hierarchy

“…At last, we will use the gauge transformation to give the specific form of ρ(t) and σ (t). In [1], we have constructed the gauge transformation of the constrained mKP hierarchy. For the cmKP hierarchy (L ( j) ) k ≤0 = q ( j) ∂ −1 r ( j) ∂ , using the n steps of gauge transformation operator T D ,…”

The generalised q-Wronskian solutions of the q-deformed constrained modified KP hierarchy

“…On the basis of the narrow q-Wronskian solutions of the q-cmKP hierarchy, it is extended to the generalized q-Wronskian solutions of the q-cmKP hierarchy, which plays a good supplementary role in exploring the physical significance of qdeformation. We can obtain the generalized q-Wonskian solutions of the q-mKP hierarchy via the gauge transformation of the q-mKP hierarchy [26][27][28][29][30][31]. Hence, we will continue to study the generalized q-Wonskian solutions of the q-cmKP hierarchy.…”