The dark Korteweg-de Vries （KdV） systems are defined and classified by Kupershmidt sixteen years ago. However, there is no other classifications for other kinds of nonlinear systems. In this paper, a complete scalar...The dark Korteweg-de Vries （KdV） systems are defined and classified by Kupershmidt sixteen years ago. However, there is no other classifications for other kinds of nonlinear systems. In this paper, a complete scalar classification for dark modified KdV （MKdV） systems is obtained by requiring the existence of higher order differential polynomial symmetries. Different to the nine classes of the dark KdV case, there exist twelve independent classes of the dark MKdV equations. Fhrthermore, for the every class of dark MKdV system, there is a free parameter. Only for a fixed parameter, the dark MKdV can be related to dark KdV via suitable Miura transformation. The reeursion operators of two classes of dark MKdV systems are also given.展开更多
基金Supported by the Global Change Research Program of China under Grant No.2015Cb953904National Natural Science Foundation of China under Grant Nos.11675054,11435005,11175092,and 11205092+1 种基金Shanghai Knowledge Service Platform for Trustworthy Internet of Things(No.ZF1213)K.C.Wong Magna Fund in Ningbo University
文摘The dark Korteweg-de Vries （KdV） systems are defined and classified by Kupershmidt sixteen years ago. However, there is no other classifications for other kinds of nonlinear systems. In this paper, a complete scalar classification for dark modified KdV （MKdV） systems is obtained by requiring the existence of higher order differential polynomial symmetries. Different to the nine classes of the dark KdV case, there exist twelve independent classes of the dark MKdV equations. Fhrthermore, for the every class of dark MKdV system, there is a free parameter. Only for a fixed parameter, the dark MKdV can be related to dark KdV via suitable Miura transformation. The reeursion operators of two classes of dark MKdV systems are also given.
