Three interaction scenarios of two orthogonally polarised optical pulses modelled by the variable coefficient coupled nonlinear Schrödinger equations

• Published in: Nonlinear dynamics Vol. 112; no. 24; pp. 22355 - 22378
• Main Author: Das, Prakash Kumar
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.12.2024; Springer Nature B.V
• Subjects: Approximation; Automotive Engineering; Bose-Einstein condensates; Classical Mechanics; Control; Differential equations; Dynamical Systems; Engineering; Fiber lasers; Matter waves; Mechanical Engineering; Nonlinear dynamics; Nonlinear systems; Optical pulses; Ordinary differential equations; Polynomials; Schrodinger equation; Solitary waves; Theorems; Topology; Vibration; Rapidly convergent approximation method; Fibre optical transmission; Exact solutions; Boundedness; W-shaped; Boomerang-like solution; Parabolic-shaped; The coupled variable coefficient nonlinear Schrödinger equation; Two-hump M-shaped; Snake-like
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-024-10217-6

This paper investigates the coupled variable coefficient nonlinear Schrödinger equation, which depicts the evolution of two orthogonally polarised optical pulses and vector soliton dynamics in inhomogeneous fibres. The dynamics of a matter wave in a quasi-one-dimensional two-component Bose–Einstein condensate can also be effectively described by this models. The model is examined in three different scenarios of solitons interaction. They are converted to a stationary form, or nonlinear system of ordinary differential equations, using the travelling transformation. The rapidly convergent approximation method is then used to solve four separate integrable instances of these systems, three in coupled form and one in decoupled form. Three theorems are offered, each independently proved and demonstrated, for the boundedness requirements of these derived solutions. To validate the theorems, a few specific values of the parameters that correspond to the theorem requirements are taken, plotted, and unique soliton profiles of the system are analysed. The solution has M-shaped, W-shaped, parabolic-like, snake-like, and boomerang-like two-hump asymmetric bright–bright, bright–dark, and dark–dark soliton solutions for different values of the dispersion coefficient with periodic and polynomial effects. We can adjust the topologies of the previously stated solitons by changing the variable coefficient. In the last integrable case, the system is reduced to a scalar equation and solved using rapidly convergent approximation method. After translating it to different hyperbolic and trigonometric forms, we compared the results to the current solutions in the literature. To the best of the authors’ knowledge, the first three integrable case solutions are distinct and haven’t been reported before. The findings of the study could aid in the development of polarised optical pulses and soliton fibre lasers.