Noncommutative coupled complex modified Korteweg–de Vries equation: Darboux and binary Darboux transformations

Abstract

Higher-order nonlinear evolution equations are important for describing the wave propagation of second- and higher-order number of fields in optical fiber systems with higher-order effects. One of these equations is the coupled complex modified Korteweg–de Vries (ccmKdV) equation. In this paper, we study noncommutative (nc) generalization of ccmKdV equation. We present Darboux and binary Darboux transformations (DTs) for the nc-ccmKdV equation and then construct its Quasi-Grammian solutions. Further, single and double-hump soliton solutions of first- and second-order are given in commutative settings.

Darboux Transformation for the Hirota Equation

“…Furthermore, our solutions for the HE are written in terms of quasideterminants [8,9] rather than determinants. It has been proved that quasideterminants are very useful for constructing exact solutions of integrable equations [10,11,21,29,31,35,41], enabling these solutions to be expressed in a simple and compact form. Finally, we prove that the DT we present here can be used to obtain various solutions of the HE (1.3) such as multi-solitons, breathers and rogue waves.…”

Darboux Transformation for the Hirota equation

“…Furthermore, our solutions for the HE are written in terms of quasideterminants [7,8] rather than determinants. It has been proved that quasideterminants are very useful for constructing exact solutions of integrable equations [9,10,19,27,28,32,38], enabling these solutions to be expressed in a simple and compact form.…”

Soliton solutions for a two-component generalized Sasa-Satsuma equation

“…which is also called a Sasa-Satsuma equation. Equation (4) has been extensively studied by different methods, such as inverse scattering transform [7,8], Darboux transformation [9][10][11][12][13][14] and Hirota bilinear method [15,16].…”