Peregrine Soliton Management of Breathers in Two Coupled Gross–Pitaevskii Equations with External Potential

Abstract

By solving analytically two 1D coupled Gross–Pitaevskii equations with a time-dependent harmonic trap, we find Peregrine solutions that can be effectively controlled by modulating the external potential frequency. Indeed, one observes the onset of instability in the dynamical system as the frequency is varied. This leads to the possibility of stabilizing the Peregrine solitons.

APPENDIX: FROM GPE TO MANAKOV SYSTEM

Similarity transformation and analytical setup from GPE to Manakov system: we apply the following transformation to Eqs. (1):

$${{\psi }_{1}} = {{Q}_{1}}(X,T){{e}^{{\int {h(t)dt + ia(x,t)} }}},$$

$${{\psi }_{2}} = {{Q}_{2}}(X,T){{e}^{{\int {h(t)dt + ia(x,t)} }}},$$

where

$$X = x{{e}^{{2\int {h(t)dt} }}} - 2\int {{{h}_{1}}(t)} {{e}^{{2\int {h(t)dt} }}}dt,$$

$$\begin{gathered} T = \int {{{e}^{{4\int {h(t)dt} }}}dt} , \\ a(x,t) = - \frac{1}{2}{{x}^{2}}h(t) + x{{h}_{1}}(t) + {{h}_{2}}(t), \\ \end{gathered} $$

$$\begin{gathered} {{R}_{{11}}} = {{R}_{{12}}} = {{R}_{{21}}} = {{R}_{{22}}} = 2\sigma \gamma (t), \\ \gamma (t){{e}^{{2\int {h(t)dt} }}},\,\,\,\,\sigma = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}. \\ \end{gathered} $$

Substituting the above transformation given by ${{\psi }_{1}}$ and ${{\psi }_{2}}$ in Eqs. (1) and reinforcing the following constraints:

$${{\omega }^{2}}(t) = h{{(t)}^{2}} - \frac{{h{\kern 1pt} '(t)}}{2},\,\,\,\,h(t) = \frac{{h_{1}^{'}(t)}}{{2{{h}_{1}}(t)}},\,\,\,\,h_{2}^{'}(t) = - {{h}_{1}}{{(t)}^{2}},$$

will reduce the coupled GP equations to the coupled NLS equations (2a, 2b) [20–22]

$$i\mathop {{{Q}_{1}}}\nolimits_t = \left[ { - \frac{1}{2}\mathop {{{Q}_{1}}}\nolimits_{xx} - \left( {Q_{1}^{2} + Q_{2}^{2}} \right)} \right]{{Q}_{1}},$$

$$i\mathop {{{Q}_{2}}}\nolimits_t = \left[ { - \frac{1}{2}\mathop {{{Q}_{1}}}\nolimits_{xx} - \left( {Q_{1}^{2} + Q_{2}^{2}} \right)} \right]{{Q}_{2}}.$$