Browsing by Author "TAIBI, ZEYNEB"

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    Mathematical and Numerical Analysis of the Non Linear Schrödinger Equation
    (Chaachoua Sameut Houria / P.S vinayagam, 2026) TAIBI, ZEYNEB
    This thesis presents a comprehensive analytical and numerical investigation of localized nonlinear wave structures governed by integrable Higher-Order Nonlinear Schrödinger (HNLS)-type equations. Motivated by recent advances in nonlinear wave physics and the increasing relevance of higher-order dispersive and nonlinear effects, this work demonstrates that the interplay between dispersion, nonlinearity, and higher-order terms constitutes a powerful framework for energy localization and control rather than a limitation. The study begins with a detailed overview of fundamental nonlinear wave phenomena, including solitons, breathers, and rogue waves, together with a rigorous analysis of modulation instability as the underlying mechanism for localized structure formation. Numerical tools, particularly the Split-Step Fourier Method (SSFM), are introduced and validated through the relative L 2 error norm and trajectory tracking based on the center-of-mass formalism. From an analytical perspective, the thesis employs the 3 × 3 Lax pair representation and the Darboux Transformation to construct exact solutions for integrable systems. These methods are applied to benchmark models such as the Hirota and Sasa–Satsuma equations, illustrating the iterative generation of higher-order solitonic and breather solutions and confirming the robustness of the analytical framework. The dynamics of the HNLS equation with constant coefficients are investigated in detail, revealing the role of spectral parameters in shaping wave morphology, including Akhmediev breathers, Kuznetsov–Ma solitons, and the Peregrine soliton as a limiting rational solution. The individual influence of higher-order physical parameters is isolated, highlighting the dominant role of third-order dispersion in inducing center-of-mass shifts and trajectory drift. Finally, the thesis extends the analysis to generalized HNLS equations with time-dependent coefficients, demonstrating that active modulation of dispersion and nonlinearity enables precise waveform engineering. Numerical simulations confirm the stability and physical relevance of the Darboux-derived solutions, with relative L 2 errors on the order of 10−4 , validating that the observed dynamics arise from intentional parametric control rather than numerical artifacts. Overall, this work provides a unified analytical–numerical framework for understanding and managing localized nonlinear wave phenomena, offering insights relevant to nextgeneration nonlinear physical systems in optics, hydrodynamics, and related fields.