Localized waves of the coupled cubic–quintic nonlinear Schrödinger equations in nonlinear optics

Abstract

We investigate some novel localized waves on the plane wave background in the coupled cubic-quintic nonlinear Schrödinger (CCQNLS) equations through the generalized Darboux transformation (DT). A special vector solution of the Lax pair of the CCQNLS system is elaborately constructed, based on the vector solution, various types of higherorder localized wave solutions of the CCQNLS system are constructed via the generalized DT. These abundant and novel localized waves constructed in the CCQNLS system include higher-order rogue waves, higher-order rogues interacting with multi-soliton or multi-breather separately. The first-and second-order semi-rational localized waves including several free parameters are mainly discussed: (i) the semi-rational solutions degenerate to the first-and second-order vector rogue wave solutions; (ii) hybrid solutions between a first-order rogue wave and a dark or bright soliton, a second-order rogue wave and two dark or bright solitons; (iii) hybrid solutions between a first-order rogue wave and a breather, a second-order rogue wave and two breathers. Some interesting and appealing dynamic properties of these types of localized waves are demonstrated, for example, these nonlinear waves merge with each other markedly by increasing the absolute value of α. These results further uncover some striking dynamic structures in the CCQNLS system.

Mixed interactions of localized waves in the three-component coupled derivative nonlinear Schrödinger equations

“…Actually, we generalized Baronio's results in [44] to higher-order cases in the same two-component NLS system [46] and some other multicomponent coupled systems [47,48,49,50]. Compared to two-component systems [46,47,50], there can exist some novel and interesting mixed interactions of localized waves among three different components in three-component ones [48,49]. Here, we extend the two-component coupled DNLS equations in [35] and [36] to three-component case [1], and construct the corresponding Lax pair.…”

“…It is shown that Type 2-5 are four mixed interactions of localized waves among three components q 1 , q 2 and q 3 . These four mixed interactions of localized waves were also constructed in three-component Hirota equations by us in [49] and they cannot be constructed in two-component system [46,47,50] using the same method. Additionally, we can draw a conclusion that these kinds of mixed interactions of localized waves may only be obtained by DT in the nonlinear systems, whose components are more than 3 with the corresponding Lax pair including the matrices larger than 3 × 3.…”

“…It is necessary to investigate the interactional solutions between higherorder RWs and other nonlinear localized waves. Combined a special vector solution of the Lax pair and generalized DT, we successfully constructed higher-order interactions of localized waves in some multi-component coupled systems [46,47,48,49,50]. Actually, we generalized Baronio's results in [44] to higher-order cases in the same two-component NLS system [46] and some other multicomponent coupled systems [47,48,49,50].…”

“…Combined a special vector solution of the Lax pair and generalized DT, we successfully constructed higher-order interactions of localized waves in some multi-component coupled systems [46,47,48,49,50]. Actually, we generalized Baronio's results in [44] to higher-order cases in the same two-component NLS system [46] and some other multicomponent coupled systems [47,48,49,50]. Compared to two-component systems [46,47,50], there can exist some novel and interesting mixed interactions of localized waves among three different components in three-component ones [48,49].…”

Novel high‐order breathers and rogue waves in the Boussinesq equation via determinants

“…In the previous studies, high‐order rogue waves in the NLS equation have been discussed, which are also localized in both space and time and could exhibit higher main peaks in fundamental pattern. In addition to the NLS equation, families of nonlinear soliton equations have been verified possessing rogue wave solutions …”