Study of Soliton like solutions of generalized nonlinear schrodinger equation

Abstract

This thesis involves the study of self-similar localized solutions of variants of nonlinear Schrödinger equation (NLSE), generally referred to as generalized NLSE (GNLSE). It mainly used to models the nonlinear systems such as optical fibers, Bose-Einstein condensates (BECs) and water waves. Real physical systems possess inhomogeneities due to fluctuations in environmental conditions modeled by NLSE with spatially and/or temporally varying coefficients. We study of self-similar matter wave solutions, such as bright, kink-type, rational dark and Lorentzian-type solitons/solutions of modified Gross-Pitaevskii equation with external source by employing similarity transformation technique. We study the dynamical behavior of self-similar matter waves for different types of trapping potentials such as sech- and Gaussian-type potential. We observe that the intensity of propagating waves can be increased/regulated/modified by varying the amplitude of trapping potential, external source and nonlinearities present in the system. We study the higher-order nonlinear Schrödinger equation (HNLSE) with complex potential. Under specific parametric regimes, we obtain nonlinear resonant states pertaining to wave propagation in femtosecond fiber optics. In addition, we present another systematic approach by invoking isospectral Hamiltonian technique, which provides us another efficient mechanism to amplify the intensity of propagating pulses. NLSE with an external source has application in many physical systems. We consider the HNLSE with source term and complex potential to study the dynamics of femtosecond beam propagation in twin-core inhomogeneous waveguide. The application of the fractional transformation approach has allowed the construction of periodic, cnoidal and localized soliton solutions. The complex potential should be parity-time symmetric, with the real component representing the tapering profile of the waveguide and the imaginary part representing the gain/loss profile.