Extended Galilean transformations for high-order systems in two integrable hierarchies

Abstract

For some integrable hierarchies, there is no Galilean transformation that keeps a high-order equation in these hierarchies invariant, although the first one or two lower-order members admit Galilean invariance. However, the case might be changed if we redefine a high-order system. The redefined system consists of the considered high-order equation itself and all the lower-order members in their hierarchy, and each member of the hierarchy is numbered by introducing temporal coordinates (t0, t1, t2, …) to replace the uniform t. For such a redefined high-order system, one may construct an extended Galilean transformation that keeps the system invariant. Two examples, the high-order Burgers system and the Korteweg–de Vries system, are employed for demonstrating our point.