Exact and Multiple Solitons with Conservation Laws for Nonlinear Schrodinger Equations

Abstract

Asoliton being atraveling wave solution of a nonlinear partial differential equation (NLPDE) has remarkable properties. These wave solutions are stable and have significant role in the mathematical physics particularly in fiber technology. A soliton (dromion) maintains its speed, amplitude as well as shape over large distance in a nonlinear (NL) and non-local media [75]. In the domain of NL optical fibers, the importance of optical dromions is vi tal. They possess a significant potential of becoming information carriers in the field of telecommunication because of their tremendous feature of long propagation without any change in shape and attenuation [46]. These dromions being light pulses are described and governed by nonlinear Schr¨ odinger equation (NLSE). Optical solitons are the solutions of NLSEs and they can be evaluated by utilizing different kinds of nonlinearities. A NLSE is a classical field equation whose major applications involve the transmission of light in NL optical fibers and planar waveguides [5]. In the areas of engineering and applied sciences, numerous NLSEs exhibit conservation of energy, mass and momentum as well as electric charges. The presence of an infinite num ber of conservation laws (CLs) of a NLSE assures the complete integrability of the NLSE [52]. In this thesis, we obtain CLs for various NLSEs like modulated compressional disper sive Alfv´en (MCDA) model [57], Heisenberg ferromagnetic spin chains equation (HFSCE) [56], Biswas-Arshed model (BAM) [58], perturbed Radhakrishnan-Kundu-Lakshmanan (PRKL) model [19], Chen-Lee-Liu equation (CLLE) [24] and some other NLSEs. For the investigation of solvability of NLSEs, we have a very impressive technique known as the Painlev´e test (P-test) [16]. This tremendous test was firstly initiated for the analysis of the singularities structure for ordinary differential equations (ODEs). In this dissertation, we investigate the integrability by means of P-test for above mentioned NLSEs. At the end, we study different kinds of soliton solutions of numerous NLSEs by means of x Bernoulli’s equation approach (BEA) [74], sine-cosine method (SCM) [74], unified method (UM) [1], Hirota bilinear method (HBM) [105] and sub-ODE approach [114]. We analyse optical soliton solutions for above mentioned models and some other NLSEs. We also investigate optical dromions for Parity-Time symmetry (PTS) lattices with quadratic and anti cubic nonlinearities of the propagation equation [35]. Over and above board, we obtain multiple solitons for PRKL model and discuss one soliton transformation, two and three soliton interactions using HBM.