Modulational instability analysis and localized solutions of generalized nonlinear Schrodinger equation

Abstract

The thesis is primarily devoted towards the extensive study of existence and stability of localized solutions of generalized higher-order nonlinear Schrödinger equation that describes wave propagation in inhomogeneous nonlinear physical system. Some of the inhomogeneities are inherently present in these nonlinear systems while others appear due to external factors such as environmental fluctuations. Initially proposed in quantum mechanics, the novel idea of PT symmetry is implemented in optics by identifying the mathematical similarity in the structure of Schrödinger equation of quantum mechanics and the nonlinear Schrödinger equation (NLSE) of optics. We study MI of constant amplitude waves in PT-symmetric tapered graded-index waveguide with gain/loss regions. The MI analysis is performed at different points along longitudinal direction for self-focusing and defocusing nonlinearity. The effect of external source on MI is also investigated. Next, we report stationary Hermite-Gaussian soliton solutions in graded-index waveguide that under specific parametric constraints exhibit PT symmetry. Using isospectral Hamiltonian approach that results in Riccati parameter, the control on individual humps has been elucidated. Further, stability of Hermite-Gaussian soliton and Riccati generalized Hermite-Gaussian solitons is investigated. By employing similarity transformation, generalized higher-order NLSE reduces to constant coefficient higher-order NLSE. Further, we illustrated that for judicious choice of tapering parameter, intensity of self-similar waves can be controlled. The system is modelled by higher-order NLSE. The soliton solutions are identified in different regimes of the parameter space. The dark soliton for specific parametric constraints appears as wide localized solution whose width can be varied through a free parameter keeping the amplitude same. Further, the effect of wave parameters on the amplitude of surface gravity waves is also studied. newline