Non-differentiable fractional odd-soliton solutions of local fractional generalized Broer-Kaup system by extending Darboux transformation

Abstract

In this paper, a local fractional generalized Broer-Kaup (gBK) system is first de?rived from the linear matrix problem equipped with local space and time fractional partial derivatives, i.e, fractional Lax pair. Based on the derived fractional Lax pair, the second kind of fractional Darboux transformation (DT) mapping the old potentials of the local fractional gBK system into new ones is then established. Finally, non-differentiable frcational odd-soliton solutions of the local fractional gBK system are obtained by using two basic solutions of the derived fractional Lax pair and the established fractional DT. This paper shows that the DT can be extended to construct non-differentiable fractional soliton solutions of some local fractional non-linear evolution equations in mathematical physics.

Fractional Soliton and Semirational Solutions of a Fractional Two-Component Generalized Hirota Equation

“…Based on the derived fractional N-fold DT, as shown in Equations ( 21)-( 23), the first-step fractional GDTs, as shown in Equations ( 26)-( 28), second-step fractional GDTs, as shown in Equations ( 31)-( 33), and third-step fractional GDTs, as shown in Equations ( 36)-(38), of the fractional TCGH Equation (1) were constructed, respectively. Then, the fractional soliton solutions, as shown in Equations ( 40) and (41), and semirational solutions, as shown in Equations ( 46) and (47), were obtained by using Equations ( 27), ( 28), (32), and (33). This indicates that one advantage of the fractional DT and GDT presented in this article is that they can be used to handle fractional integrable systems.…”

“…Using the fractional single-soliton solution, as shown in Equations ( 41) and ( 42), and the second-step fractional GDT, as shown in Equations ( 32) and (33), we obtain the fractional semirational solutions of the fractional TCGH Equation (1):…”

“…These aspects are different from those of the traditional DT [29] and GDT [30]. Although we have some preliminary studies [33,34] on fractional DT and GDT, the starting point of this work is to reveal the novel dynamic characteristics dominated by fractional order of fractional soliton and semirational solutions from a new fractional integrable system.…”