Nonautonomous solitons, breathers and rogue waves for the Gross–Pitaevskii equation in the Bose–Einstein condensate

• Lax pair, conservation laws and Darboux transformation have been constructed.
• Nonautonomous solitons and breathers have been derived with the help of the Darboux transformation.
• First- and second-order nonautonomous rogue waves have been constructed via the generalized Darboux transformation.
• Influences of the nonlinear parameter, linear external potential parameter, harmonic external potential parameter and the spectral parameter on the propagation and interaction of the nonautonomous solitons, breathers and rogue waves have been discussed.

Under investigation in this paper is the Gross–Pitaevskii equation which describes the dynamics of the Bose–Einstein condensate. Lax pair, conservation laws and Darboux transformation (DT) are constructed. Nonautonomous solitons and breathers are derived based on the DT obtained. A kind of modulation instability process is generated. Nonautonomous rogue waves are obtained via the generalized DT. Influence of the nonlinearity, linear external potential, harmonic external potential, and spectral parameter on the propagation and interaction of the nonautonomous solitons, breathers and rogue waves is also discussed.Amplitude of the first-order nonautonomous soliton is proportional to the imaginary part of the spectral parameter and inversely proportional to the nonlinearity parameter. Linear external potential parameter affects the location of the first-order nonautonomous soliton. Head-on interaction, overtaking interaction and bound-state-like nonautonomous solitons can be formed based on the signs of the real parts of the spectral parameters. Quasi-periodic behaviors are exhibited for the nonautonomous breathers. If the harmonic external potential parameter is negative, quasi-period decreases along the positive time axis, with an increase in the amplitude and a compression in the width. Quasi-period decreases with the increase of the nonlinearity parameter. The second-order nonautonomous rogue wave can split into three first-order ones. Nonlinearity parameter has an effect on the amplitude of the rogue wave. Linear external potential parameter influences the location of the rogue wave, while harmonic external potential parameter affects the curved direction of the background.