Lax pair, binary Darboux transformation and dark solitons for the three-component Gross–Pitaevskii system in the spinor Bose–Einstein condensate

Abstract

In this paper, we investigate the dark solitons for the three-component Gross–Pitaevskii system, which describes the $F=1$ spinor Bose–Einstein condensate, with F denoting the atom’s spin. We construct a Lax pair and a binary Darboux transformation for such a system. Based on a nonzero seed solution, we obtain the kink and dark solitons corresponding to the ferromagnetic and polar states, respectively. Moreover, we obtain the W-shaped dark soliton. Interactions between the two dark solitons are discussed. We find that the interaction is inelastic if a kink soliton is included.

Appendix

Substituting Expressions (10) and (11) in Binary DT (6), we derive the dark one-soliton solutions

$$\begin{aligned}&\phi _{+1}=e^{i\theta _0}\left( \beta -2\frac{g_1}{|\Omega _{[1]}|}\right) ,\nonumber \\&\phi _{0}=e^{i\theta _0}\left( \alpha -2\frac{g_2}{|\Omega _{[1]}|}\right) ,\nonumber \\&\phi _{-1}=-e^{i\theta _0}\left( \beta +2\frac{g_4}{|\Omega _{[1]}|}\right) , \end{aligned}$$

(A.1)

where

$$\begin{aligned} g_1= & {} e^{4i\theta _1}\left( 2a_1c_{11}^*l_1{-}b_1c_{11}^{*2}{+}b_1^* l_1^2\right) \nonumber \\&{+}\,e^{{-}4i\theta _1}\left( 2a_2c_{12}^*l_3{-}b_2 c_{12}^{*2}{+}b_2^*l_3^2\right) \nonumber \\&{+}\,e^{2i\theta _1}[l_1^2m_1^*{+}c_{11}^*(2a_1l_3\nonumber \\&{-}\,2b_1 c_{12}^*{-}c_{11}^*m_1){+}2l_1\left( a_1c_{12}^*{+}b_1^*l_3{+}c_{11}^*p_1\right) ]\nonumber \\&{+}\,e^{{-}2i\theta _1}[c_{12}^*\left( 2a_2l_1{-}2b_2c_{11}^*{-}c_{12}^*m_1\right) \nonumber \\&{+}\,2l_3\left( a_2c_{11}^*{+}b_2^*l_1{+}c_{12}^*p_1\right) {+}l_3^2 m_1^*]\nonumber \\&{+}\,c_{11}^*(2a_2l_1{+}2p_1l_3{-}b_2c_{11}^*){+}c_{12}^*(2a_1l_3{+}2p_1l_1\nonumber \\&{-}\,b_1c_{12}^*){-}2m_1c_{11}^*c_{12}^*\nonumber \\&{+}\,l_1^2b_2^*{+}l_3^2b_1^*{+}2l_1l_3m_1^*, \end{aligned}$$

(A.2a)

$$\begin{aligned} g_2= & {} e^{4i\theta _1}\left[ l_1(a_1c_{21}^*{+}b_1^*l_2){+}c_{11}^*(a_1l_2{-}b_1c_{21}^*)\right] \nonumber \\&{+}\,e^{{-}4i\theta _1}\left[ l_3(a_2c_{22}^*{+}b_2^*l_4){+}c_{12}^*(a_2l_4{-}b_2c_{22}^*)\right] \nonumber \\&{+}\,e^{2i\theta _1}[l_2(c_{11}^*p_1{+}l_1m_1^*{+}a_1c_{12}^*{+}b_1^*l_3)\nonumber \\&{+}\,l_1(a_1c_{22}^*{+}b_1^*l_4{+}c_{21}^*p_1){+}a_1c_{11}^*l_4{+}a_1c_{21}^*l_3\nonumber \\&{-}\,b_1c_{22}^*c_{11}^*{-}b_1c_{12}^*c_{21}^*{-}c_{21}^*c_{11}^*m_1]\nonumber \\&{+}\,e^{{-}2i\theta _1}[l_4(c_{12}^*p_1{+}l_3m_1^*{+}a_2c_{11}^*{+}b_2^*l_1)\nonumber \\&{+}\,l_3(a_2c_{21}^*{+}b_2^*l_2\nonumber \\&{+}\,c_{22}^*p_1){+}a_2c_{12}^*l_2{+}a_2c_{22}^*l_1{-}b_2c_{21}^*c_{12}^*\nonumber \\&{-}\,b_2c_{11}^*c_{22}^*{-}c_{22}^*c_{12}^*m_1]{+}(l_1l_4{+}l_2l_3)m_1^*{+}l_1l_2b_2^*\nonumber \\&{+}\,l_3l_4b_1^*{+}c_{11}^*(a_2l_2{+}p_1l_4{-}b_2c_{21}^*{-}m_1c_{22}^*)\nonumber \\&{+}\,c_{12}^*(a_1l_4{+}p_1l_2{-}b_1c_{22}^*{-}m_1c_{21}^*){+}c_{21}^*(a_2l_1\nonumber \\&{+}\,p_1l_3){+}c_{22}^*(a_1l_3{+}p_1l_1), \end{aligned}$$

(A.2b)

$$\begin{aligned} g_4= & {} e^{4i\theta _1}\left( 2a_1c_{21}^*l_2{-}b_1c_{21}^{*2}{+}b_1^* l_2^2\right) \nonumber \\&{+}\,e^{{-}4i\theta _1}\left( 2a_2c_{22}^*l_4{-}b_2 c_{22}^{*2}{+}b_2^*l_4^2\right) \nonumber \\&{+}\,e^{2i\theta _1}[l_2^2m_1^*{+}c_{21}^*(2a_1l_4\nonumber \\&{-}\,2b_1c_{22}^*{-}c_{21}^*m_1){+}2l_2\left( a_1c_{22}^*{+}b_1^*l_4{+}c_{21}^*p_1\right) ]\nonumber \\&{+}\,e^{{-}2i\theta _1}[c_{22}^*\left( 2a_2l_2{-}2b_2c_{21}^*{-}c_{22}^*m_1\right) \nonumber \\&{+}\,2l_4\left( a_2c_{21}^*{+}b_2^*l_2{+}c_{22}^*p_1\right) {+}l_4^2 m_1^*]\nonumber \\&{+}\,c_{21}^*(2a_2l_2{+}2p_1l_4{-}b_2c_{21}^*){+}c_{22}^*(2a_1l_4{+}2p_1l_2\nonumber \\&{-}\,b_1c_{22}^*){-}2m_1c_{21}^*c_{22}^*{+}l_2^2b_2^*\nonumber \\&{+}\,l_4^2b_1^*{+}2l_2l_4m_1^*, \end{aligned}$$

(A.2c)

$$\begin{aligned} |\Omega _{[1]}|= & {} (a_1^2{+}|b_1|^2)e^{4i\theta _1}{+}(a_2^2{+}|b_2|^2)e^{{-}4i\theta _1}\nonumber \\&{+}\,(b_1m_1^*{+}b_1^*m_1{+}2a_1p_1)e^{2i\theta _1}{+}(b_2m_1^*{+}b_2^*m_1\nonumber \\&{+}\,2a_2p_1)e^{{-}2i\theta _1}{+}2a_1a_2\nonumber \\&{+}\,b_1b_2^*{+}b_1^*b_2{+}p_1^2{+}|m_1|^2, \end{aligned}$$

(A.2d)

with noting $\varphi _{11} = c_{11}e^{i\theta _1}{+}c_{12}e^{{-}i\theta _1}$, $\varphi _{12} = c_{21}e^{i\theta _1}{+}c_{22}e^{{-}i\theta _1}$, $A_{11,1}=a_1e^{2i\theta _1}{+}a_{2}e^{{-}2i\theta _1}{+}p_1$ and $A_{11,3}= b_{1}e^{i\theta _1}{+}b_{2}e^{{-}i\theta _1}{+}m_1$ for simplicity.