Soliton interactions in certain square matrix nonlinear Schrödinger systems

$$\overline{\partial }$$-dressing method to PT-symmetric multi-component nonlinear Schrödinger equation

“…In 2019, in the framework of inverse scattering transform, the soliton bound state of the focusing vector NLS equation is used to analyze the asymptotic phase of modulation instability caused by noise of plane wave [35]. In 2020, Gkogkou and Prinari discussed the bright solitons and their interactions in three types of matrix NLS systems, and provided the bound states of polar solitons, ferromagnetic solitons, and polar-ferromagnetic solitons propagating at zero velocity for these three types of matrix NLS systems [36]. In addition, soliton bound states have shown great potential in fiber optic communication, material processing, and coherent pulse superposition amplification.…”

EDITORIAL: “Solitons, Integrability, Nonlinear Waves: Theory and Applications”

“…The work [7] in this FP deals with a class of square matrix nonlinear Schrödinger (MNLS) systems whose reductions include two equations that model hyperfine spin F = 1 spinor Bose-Einstein condensates in the focusing and defocusing dispersion regimes, and two novel (mixed sign) equations that were recently shown to be integrable. The main goal of the paper is to discuss the bright soliton solutions and their interactions for the focusing MNLS and for the two mixed sign systems within the framework of the IST.…”

“…The nature of the solitons and their interactions depends on whether the associated norming constants (polarization matrices) are rank-one matrices (giving rise to ferromagnetic solitons) or full rank (corresponding to polar solitons). By computing the long-time asymptotics of the 2-soliton solutions, in [7] the changes in the polarization matrix of each soliton due to the interaction are determined. Correspondingly, explicit formulas for the soliton interactions are given for all possible types of interacting solitons, and for all three inequivalent reductions of the MNLS systems that admit regular bright soliton solutions.…”

Some exact wave solutions of nonlinear partial differential equations by means of comparison with certain standard ordinary differential equations

“…This has resulted in the opening up of a number of rich areas in mathematics, most notably the subject of integrable systems 1–3 . Nonlinear PDEs turn out to be ubiquitous in modeling most natural phenomena and arise in almost all branches of the physical sciences and engineering especially in complex media 4–13 . Over the last 50 years, there have appeared a number of techniques for the analysis of such PDEs.…”

“…[1][2][3] Nonlinear PDEs turn out to be ubiquitous in modeling most natural phenomena and arise in almost all branches of the physical sciences and engineering especially in complex media. [4][5][6][7][8][9][10][11][12][13] Over the last 50 years, there have appeared a number of techniques for the analysis of such PDEs. The inverse scattering transform, Hirota bilinearization method, Darboux transformations, Lie symmetries, and Painlevé analysis [14][15][16][17][18][19][20][21][22] are extremely powerful tools for the analysis of integrable systems.…”