Nonlocal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>M</mml:mi></mml:math>-component nonlinear Schrödinger equations: Bright solitons, energy-sharing collisions, and positons

The general set of nonlocal $M$-component nonlinear Schr\"odinger (nonlocal $M$-NLS) equations obeying the $\mathcal{PT}$-symmetry and featuring focusing, defocusing, and mixed (focusing-defocusing) nonlinearities that has applications in nonlinear optics settings, is considered. First, the multisoliton solutions of this set of nonlocal $M$-NLS equations in the presence and in the absence of a background, particularly a periodic line wave background, are constructed. Then, we study the intriguing soliton collision dynamics as well as the interesting positon solutions on zero background and on a periodic line wave background. In particular, we reveal the fascinating shape-changing collision behavior similar to that of in the Manakov system but with fewer soliton parameters in the present setting. The standard elastic soliton collision also occurs for particular parameter choices. More interestingly, we show the possibility of such elastic soliton collisions even for defocusing nonlinearities. Furthermore, for the nonlocal $M$-NLS equations, the dependence of the collision characteristics on the speed of the solitons is analyzed. In the presence of a periodic line wave background, we notice that the soliton amplitude can be enhanced significantly, even for infinitesimal amplitude of the periodic line waves. In addition to these solutions, by considering the long-wavelength limit of the obtained soliton solutions with proper parameter constraints, higher-order positon solutions of the nonlocal $M$-NLS equations are derived. The background of periodic line waves also influences the wave profiles and amplitudes of the positons. Specifically, the positon amplitude can not only be enhanced but also be suppressed on the periodic line wave background of infinitesimal amplitude.

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