Exact soliton solutions of coupled nonlinear Schrödinger equations: Shape-changing collisions, logic gates, and partially coherent solitons

Abstract

The different dynamical features underlying soliton interactions in coupled nonlinear Schrödinger equations, which model multimode wave propagation under varied physical situations in nonlinear optics, are studied. In this paper, by explicitly constructing multisoliton solutions (up to four-soliton solutions) for two-coupled and arbitrary N-coupled nonlinear Schrödinger equations using the Hirota bilinearization method, we bring out clearly the various features underlying the fascinating shape changing (intensity redistribution) collisions of solitons, including changes in amplitudes, phases and relative separation distances, and the very many possibilities of energy redistributions among the modes of solitons. However, in this multisoliton collision process the pairwise collision nature is shown to be preserved in spite of the changes in the amplitudes and phases of the solitons. Detailed asymptotic analysis also shows that when solitons undergo multiple collisions, there exists the exciting possibility of shape restoration of at least one soliton during interactions of more than two solitons represented by three- and higher-order soliton solutions. From an application point of view, we have shown from the asymptotic expressions how the amplitude (intensity) redistribution can be written as a generalized linear fractional transformation for the N-component case. Also we indicate how the multisolitons can be reinterpreted as various logic gates for suitable choices of the soliton parameters, leading to possible multistate logic. In addition, we point out that the various recently studied partially coherent solitons are just special cases of the bright soliton solutions exhibiting shape-changing collisions, thereby explaining their variable profile and shape variation in collision process.

Novel energy sharing collisions of multicomponent solitons

“…Later, Radhakrishnan et al have shown that the solitons in the Manakov system (1) exhibit certain novel inelastic (energy sharing) collisions [11] in contrast to single component NLS system. Kanna et al, have obtained the multisoliton solutions for the multicomponent Manakov system (2) using the Hirota's method [15]. Thus the system has been well studied in the literature [8,11,[13][14][15][16] and the existence of N -soliton (for arbitrary N ) solution and also its proof has been obtained.…”

“…Kanna et al, have obtained the multisoliton solutions for the multicomponent Manakov system (2) using the Hirota's method [15]. Thus the system has been well studied in the literature [8,11,[13][14][15][16] and the existence of N -soliton (for arbitrary N ) solution and also its proof has been obtained. In this section, we restrict our review to the interaction of two solitons in the Manakov system.…”

“…This exchange phenomenon also satisfies the energy conservation of both the solitons before and after collision and also the conservation of energy in individual components which has been discussed in detail in Refs. [11,[13][14][15]. For illustrative purpose, we show the energy sharing collision characterized by intensity redistribution, amplitude dependent phase-shift and change in relative separation distances in the Manakov system in Fig.…”

“…The above system admits bright soliton solutions for the Manakov (focusing) case and it supports both bright and bright-dark soliton solutions for the mixed-ICNLS case [11][12][13][14][15][16][17][18]. Particularly, the bright multi-soliton solutions of the multicomponent generalization of Manakov system have been obtained by Kanna et al using the Hirota bilinearization method and a detailed investigation on the soliton collisions, such as energy sharing collision and elastic type interactions, have been explored [14][15][16][17]. In this paper, we will review the results of two component systems only.…”

“…These Manakov and mixed-ICNLS systems find important applications in optical communication and in artificial metamaterials. They have been intensively studied in literature [2,[8][9][10][11][12][13][14][15][16][17][18][19]. Also, the integrable multicomponent generalization of ICNLS system (1) can be written as…”