Manipulating matter rogue waves in Bose–Einstein condensates trapped in time-dependent complicated potentials

Abstract

Under some constraints, we build exact and approximate first- and second-order rogue wave (RW) solutions with two control and one free parameters for a quasi-one-dimensional Gross–Pitaevskii (GP) equation with a time-varying interatomic interaction, an external trap and gain/loss term through the similarity transformation technique. Considering three different forms of the strength of the two-body interatomic interaction, we employ these rogue wave solutions to study the dynamics of matter rogue waves and superposed rogue waves in respectively one-component and coherently two-component Bose–Einstein condensates (BECs) when the gain/loss of the condensate atoms is taken into consideration. Our results show that the solution parameters (two control and one free parameters) can be used for formation and manipulating first- and second-order in BEC systems under consideration. We also show that when we change parameters appearing in the strength of the two-body interatomic interaction, first- and second-order RWs can be reduced to either one- or multiple-breather solitons or rogue wave multiplets. Our results also show that the control and free parameter appearing in the RW solutions can be used for controlling the splitting of the rogue wave components into multi-peak solutions. In the context of coherently coupled BECs, we show that linear superposition of different rogue wave solutions of the quasi-one-dimensional GP equation results into four kinds of nonlinear coherent structures namely, coexisting first–first-order (F–F) RWs, second–second-order (S–S) RWs, first–second-order (F–S) RWs, and second–first-order (S–F) RWs. These four kinds of superposed rogue waves are investigated in some detail. Also, the effects solution parameters as well as those of the intra-component strength on these four kinds of composite waves are investigated.

Appendix: (Analytical expressions for \(F\left( \xi ,\tau \right) \) and \(G\left( \xi ,\tau \right) \) appearing in Eq. (8b))

Polynomials $F\left( \xi ,\tau \right)$ and $G\left( \xi ,\tau \right)$ used in the definition of the second-order RW solution (8b) are given as

$$\begin{aligned} F\left( \xi ,\tau \right)= & {} 24\left[ \left( 3\xi -6\xi ^{2}+4\xi ^{3}-2\xi ^{4}-48\tau ^{2}\right. \right. \nonumber \\{} & {} \left. \left. +48\xi \tau ^{2}-48\xi ^{2}\tau ^{2}-160\tau ^{4}\right) \right. \nonumber \\{} & {} +i\tau \left( -12+12\xi -16\xi ^{3}+8\xi ^{4}+32\tau ^{2}\right. \nonumber \\{} & {} \left. -64\xi \tau ^{2} +64\xi ^{2}\tau ^{2}+128\tau ^{4}\right) \nonumber \\{} & {} {+}\left. 6\phi \left( 1{-}2\xi {+}\xi ^{2}{-}4i\tau {+}4i\xi \tau {-}4\tau ^{2}\right) \right. \nonumber \\{} & {} \left. +6\overline{\phi }_{1}\left( -\xi ^{2}+4i\xi \tau +4\tau ^{2}\right) \right] , \end{aligned}$$

(26a)

$$\begin{aligned} G\left( \xi ,\tau \right)= & {} 9-36\xi +72\xi ^{2}-72\xi ^{3}+72\xi ^{4}-48\xi ^{5}\nonumber \\{} & {} +16\xi ^{6} +1024\tau ^{6}+144\phi \overline{\phi } \nonumber \\{} & {} +96\tau ^{2}\left( 3+3\xi -4\xi ^{3}+2\xi ^{4}\right) \nonumber \\{} & {} +384\tau ^{4}\left( 5{-}2\xi +2\xi ^{2}\right) \nonumber \\{} & {} \quad {+}24\left( \phi {+}\overline{\phi }\right) \left( 3\xi ^{2}{-}2\xi ^{3}{-}12\tau ^{2}{+}24\xi \tau ^{2}\right) \nonumber \\{} & {} {+}48i\left( \phi {-}\overline{\phi }\right) \tau \left( 3{+}6\xi {-}6\xi ^{2}{+}8\tau ^{2}\right) , \nonumber \\ \end{aligned}$$

(26b)

where $\phi$ is an arbitrary complex number with complex conjugate $\overline{\phi }.$ Computing F and G for $\phi =-1/12,$ we find pagination

$$\begin{aligned}{} & {} F\left( \xi ,\tau \right) = 9-18\left( 2\xi -1\right) ^{2}-3\left( 2\xi -1\right) ^{4}-864\tau ^{2}\nonumber \\{} & {} \quad -3840\tau ^{4}-288\tau ^{2}\left( 2\xi -1\right) ^{2} \nonumber \\{} & {} \quad +i\tau \left[ {-}180{-}72\left( 2\xi {-}1\right) ^{2}{+}12\left( 2\xi {-}1\right) ^{4}\right. \nonumber \\{} & {} \quad \left. {+}384\tau ^{2}{+}3072\tau ^{4}{+}384\tau ^{2}\left( 2\xi {-}1\right) ^{2}\right] , \end{aligned}$$

(27)

$$\begin{aligned}{} & {} G\left( \xi ,\tau \right) =\tau ^{2}\left[ 288+1728\tau ^{2}+1024\tau ^{4}\right. \nonumber \\{} & {} \quad \left. {+}192\tau ^{2}\left( 2\xi {-}1\right) ^{2}{+}24\left( 2\xi ^{2}{-}2\xi {-}1\right) ^{2}\right] \nonumber \\{} & {} \quad {+}\frac{1}{4}\left[ 9{+}27\left( 2\xi {-}1\right) ^{2}{+}3\left( 2\xi {-}1\right) ^{4}{+}\left( 2\xi {-}1\right) ^{6}\right] . \nonumber \\ \end{aligned}$$

(28)