Multiple double-pole bright-bright and bright-dark solitons and energy-exchanging collision in the M-component nonlinear Schrödinger equations.

Multiple double-pole bright-bright and bright-dark soliton solutions for the multicomponent nonlinear Schrödinger (MCNLS) system comprising three types of nonlinearities, namely, focusing, defocusing, and mixed (focusing-defocusing) nonlinearities, arising in different physical settings are constructed. An interesting type of energy-exchanging phenomenon during collision of these double-pole solitons is unraveled. To explore the objectives, we consider the general solutions of a set of generalized MCNLS equations and by taking the long-wavelength limit with proper parameter choices of single-pole bright-bright and bright-dark soliton pairs, the multiple double-pole bright-bright and bright-dark soliton solutions are constructed in terms of determinants. The regular double-pole bright-bright solitons exist in the focusing and focusing-defocusing MCNLS equations and undergo a particular type of energy-sharing collision for M≥2 in addition to the usual elastic collisions. A striking feature observed in the process of energy-sharing collisions is that the double-pole two-soliton possessing unequal intensities before collision indeed exactly exchange their intensities after collision. Further, the existence of double-pole bright-dark solitons in the MCNLS equations with focusing, defocusing, and mixed (focusing-defocusing) nonlinearities is analyzed by constructing explicit determinant form solutions, where the double-pole bright solitons exhibit elastic and energy-exchanging collisions while the double-pole dark solitons undergo mere elastic collision. The double-pole bright-dark solitons possess much richer localized coherent patterns than their counterpart double-pole bright-bright solitons. For particular choices of parameters, we demonstrate that the solitons would degenerate into the background, resulting in a lower number of solitons. Another important observation is the formation of doubly localized rogue waves with extreme amplitude, in the case of double-pole bright-dark four-solitons. Our results should stimulate interest in such special multipole localized structures and are expected to have ramifications in nonlinear optics.