Vector dark and bright soliton wave solutions and collisions for spin-1 Bose–Einstein condensate

Abstract

We hereby work on the generalized three-coupled Gross–Pitaevskii equations by means of the Darboux transformation and Hirota’s method. By modulating the external trap potential, atom gain or loss, and coupling coefficients, we can obtain several nonautonomous matter-wave solitons including dark–dark–dark and bright–bright–bright shapes. Propagation and interaction behaviors of the nonautonomous vector solitons are analyzed through the one- and two-soliton solutions. Then, the managements and dynamic behaviors of these solutions are investigated analytically, such as the snaking behaviors, parabolic behaviors and interaction behaviors. Interactions between the linear-type, parabolic-type and periodic-type dark and bright two solitons are elastic. The results could be of interest in such diverse fields as Bose–Einstein condensates and nonlinear fibers.

Appendices

Appendix 1

$$\begin{aligned} Q_{11}= & {} 2\,{\frac{{\mathrm{e}^{i \left( X_{{1}}-\overline{X_{{1}}} \right) }}+ \delta _{{11}}}{\mu _{{1}}-\overline{\mu _{{1}}}}},\quad Q_{12}= 2\,{\frac{{\mathrm{e}^{i \left( X_{{1}}-\overline{X_{{2}}} \right) }}+ \delta _{{21}}}{\lambda _{{1}}+\mu _{{1}}-\lambda _{{2}}- \overline{\mu _{{2}}}}},\\ Q_{21}= & {} 2\,{\frac{{\mathrm{e}^{i \left( X_{{2}}-\overline{X_{{1}}} \right) }}+ \delta _{{12}}}{\lambda _{{2}}+\mu _{{2}}-\lambda _{{1}}- \overline{\mu _{{1}}}}} ,\quad Q_{22}= 2\,{\frac{{\mathrm{e}^{i \left( X_{{2}}-\overline{X_{{2}}} \right) }}+ \delta _{{22}}}{\mu _{{2}}-\overline{\mu _{{2}}}}},\\ \beta _{km}= & {} \frac{1}{\left( \lambda _{{k}}+a_{{m}}+\mu _{{k}} \right) },\quad k=1,2,\\ X_1= & {} \mu _{{1}}\eta + \left( \lambda _{{1}}\mu _{{1}}+1/2\,{\mu _{{1}}}^{2}-1/2 \,{\lambda _{{1}}}^{2} \right) \tau ,\\ \bar{X}_1= & {} \overline{\mu _{{1}}}\eta + \left( \lambda _{{1}}\overline{\mu _{{1}}}+1/2 \, \left( \overline{\mu _{{1}}} \right) ^{2}-1/2\,{\lambda _{{1}}}^{2} \right) \tau ,\\ X_2= & {} \mu _{{2}}\eta + \left( \lambda _{{2}}\mu _{{2}}+1/2\,{\mu _{{2}}}^{2}-1/2 \,{\lambda _{{2}}}^{2} \right) \tau ,\\ \bar{X}_2= & {} \overline{\mu _{{2}}}\eta + \left( \lambda _{{2}}\overline{\mu _{{2}}}+1/2 \, \left( \overline{\mu _{{2}}} \right) ^{2}-1/2\,{\lambda _{{2}}}^{2} \right) \tau ,\\ \delta _{11}= & {} e^{2\alpha _1 \mu _{1l}},\delta _{22}=e^{2\alpha _2 \mu _{2l}},\delta _{12}=0,\delta _{21}=0. \end{aligned}$$

Appendix 2

$$\begin{aligned} \eta _l= & {} k_l(\eta +ik_l \tau ),(l=1,2),\\ e^{R_u}= & {} \frac{\kappa _{u}}{k_u+k_u^*},\\ e^{\delta _0}= & {} \frac{\kappa _{12}}{k_1+k_2^*},\\ e^{\delta _0^*}= & {} \frac{\kappa _{21}}{k_2+k_1^*},\\ e^{\delta _{uv}^{(j)}}= & {} \frac{\gamma \alpha _v ^{(j)*}\sum _{l=1}^3(\alpha _u^{(l)})^2}{2(k_u+k_v^*)^2},\\ e^{\delta _{u}^{(j)}}= & {} \frac{\gamma \alpha _u^{(j)*}\sum _{l=1}^3 \left( \alpha _1^{(l)}\alpha _2^{(l)}\right) +(k_1-k_2)(\alpha _1^{(j)}\kappa _{2u} -\alpha _2^{(j)}\kappa _{1u}}{(k_1+k_u^*)(k_2+k_u^*)},\\ e^{\epsilon _{uv}}= & {} \frac{\gamma ^2\sum _{j=1}^3\left( \alpha _u^{(j)}\right) ^2 \sum _{j=1}^3\left( \alpha _v^{(j)*}\right) ^2}{4(k_u+k_v^*)^4},\\ e^{\tau _{u}}= & {} \frac{\gamma ^2\sum _{j=1}^3\left( \alpha _1^{(j)*}\alpha _2^{(j)*}\right) \sum _{j=1}^3\left( \alpha _u^{(j)}\right) ^2}{2(k_u+k_1^*)^2(k_u+k_2^*)^2},\\ e^{\mu _{uv}^{(j)}}= & {} \frac{\gamma ^2(k_1-k_2)^2 \alpha _{3-u}^{(j)}\sum _{l=1}^3\left( \alpha _u^{(l)}\right) ^2 \sum _{l=1}^3\left( \alpha _v^{(l)*}\right) ^2}{4(k_u+k_v^*)^4(k_{3-u}+k_v^*)^2},\\ e^{\theta _{uv}}= & {} \frac{\gamma ^3(k_1-k_2)^2 (k_1^*-k_2^*)^2}{4 \tilde{D}(k_u+k_v^*)^2 } \sum _{j=1}^3\left( \alpha _u^{(j)}\right) ^2 \sum _{j=1}^3\left( \alpha _v^{(j)*}\right) ^2 \\&\times \sum _{j=1}^3\left( \alpha _{3-u}^{(j)*} \alpha _{3-v}^{(j)*}\right) ,\\ e^{\phi _{u}^{(j)}}= & {} \frac{\gamma ^3(k_1-k_2)^4 (k_1^*-k_2^*)^2 }{8 \tilde{D}(k_1+k_u^*)^2 (k_2+k_u^*)^2 } \alpha _{3-u}^{(j)*} \sum _{l=1}^3\left( \alpha _1^{(l)}\right) ^2\\&\times \sum _{l=1}^3\left( \alpha _2^{(l)*}\right) ^2 \sum _{l=1}^3\left( \alpha _{u}^{(l)*}\right) ^2,\\ e^{R_3}= & {} \frac{ |k_1-k_2|^2 (\kappa _{11}\kappa _{22}-\kappa _{12}\kappa _{21}) +\gamma ^2 |\sum _{j=1}^3\left( \alpha _1^{(j)}\alpha _2^{(j)}\right) |^2}{ (k_1+k_1^*)(k_2+k_2^*)|k_1+k_2^*|^2 },\\ e^{R_{4}}= & {} \frac{\gamma ^4(k_1-k_2)^8 }{16 \tilde{D}^2 } \sum _{j=1}^3\left( \alpha _1^{(j)}\right) ^2 \sum _{j=1}^3\left( \alpha _1^{(j)*}\right) ^2\\&\times \sum _{j=1}^3( \alpha _{2}^{(j)})^2\sum _{j=1}^3( \alpha _{2}^{(j)*})^2,\\ e^{ \mu _u^{(j)}}= & {} \frac{\gamma ^2(k_1-k_2)^2 }{2 \tilde{D} } \sum _{j=1}^3\left( \alpha _u^{(l)}\right) ^2 \left( \left[ (k_{3-u}+k_1^*)^2\right. \right. \\&\left. +\,(k_{2}^*-k_1^*)(k_{3-u}+k_2^*)\right] \alpha _{3-u}^{(j)}\alpha _{1}^{(j)*}\alpha _{2}^{(j)*}\\&+\,(k_{3-u}^*+k_1^*)(k_{1}^*-k_2^*)\alpha _{1}^{(j)*} \sum _{l=1,l\ne j}^3\left( \alpha _{3-u}^{(l)}\alpha _{2}^{(l)*}\right) \\&-(k_{1}^*-k_2^*)(k_{3-u}^*+k_2^*)\alpha _{2}^{(j)*} \sum _{l=1,l\ne j}^3\left( \alpha _{3-u}^{(l)}\alpha _{1}^{(l)*}\right) \\&+\,(k_{3-u}+k_1^*)(k_{3-u}+k_2^*)\alpha _{3-u}^{(j)*} \left. \sum _{l=1,l\ne j}^3\left( \alpha _{1}^{(l)}\alpha _{2}^{(l)*}\right) \right) , \end{aligned}$$

with

$$\begin{aligned}&\tilde{D}=(k_1+k_1^*)^2 (k_1^*+k_2)^2(k_1+k_2^*)^2(k_2+k_2^*)^2,\\&\kappa _{uv}=\frac{\gamma \sum _{j=1}^3\left( \alpha _u^{(j)} \alpha _v^{(j)*}\right) }{(k_u+k_v^*)},\\&\qquad \qquad u,v=1,2, \quad j,l=1,2,3. \end{aligned}$$