A generalized dressing approach for solving the extended KP and the extended mKP hierarchy

Abstract

A new multicomponent CKP hierarchy and solutionsA combination of dressing method and variation of constants as well as a formula for constructing the eigenfunction is used to solve the extended KP hierarchy, which is a hierarchy with one more series of time flow and based on the symmetry constraint of KP hierarchy. Similarly, extended mKP hierarchy is formulated and its zero-curvature form, Lax representation, and reductions are presented. Via gauge transformation, it is easy to transform dressing solutions of extended KP hierarchy to the solutions of extended mKP hierarchy. Wronskian solutions of extended KP and extended mKP hierarchies are constructed explicitly.

The KP hierarchy with self–consistent sources: construction, Wronskian solutions and bilinear identities

“…Then by modifying the dressing approach for the original KP hierarchy, we can get a dressing approach for the KP hierarchy with self-consistent sources as the following [14]:…”

“…In [14], a gauge transformation between the KP hierarchy with sources and the mKP hierarchy with sources is constructed, and a Wronskian solution (including soliton solutions) for the mKP hierarchy with sources is also obtained with the help of dressing approach (Proposition 2) for the KP hierarchy with sources.…”

“…This is a systematic method to generate the SESCSs, and can be used to study the case of BKP, CKP [22], q-deformed KP [10,11], and some other cases. A generalized dressing method was also derived for these soliton hierarchy with sources, and their Wronskian solutions (including soliton solutions) were obtained [14]. Recently, a bilinear identity for the KP hierarchy with self-consistent sources (KPHwS) and their Hirota's bilinear equations (in a simpler form) were obtained [12].…”

Новая Расширенная Матричная Иерархия Кадомцева - Петвиашвили И Ее Решения

“…Эта расширенная КП-иерархия и ее t n -и τ k -редукции предоставляют единообразный способ нахождения двух типов КП-уравнения с само-согласованными источниками и ряда (1 + 1)-мерных солитонных уравнений с само-согласованными источниками. Решения расширенной КП-иерархии и расширенной мкКП-иерархии можно получить в рамках обобщенного метода одевания [14].…”

Discrete KP equation with self-consistent sources

“…This systematic method has been used to generate extensions with self-consistent sources of the CKP [57], multicomponent KP [22], and some other hierarchies. A generalized dressing method has been also derived for these soliton hierarchies with sources, and some soliton solutions were obtained [36]. Recently, a bilinear identity for the KP hierarchy with sources and their Hirota bilinear equations were obtained [34].…”