Exact soliton solutions of cubic nonlinear Schrödinger equation with a momentum term

Abstract

There are various interconnections between the positive sciences, with differential equations serving as a fundamental bridge linking mathematics to other scientific disciplines. Nonlinear wave phenomena have recently gained considerable attention due to their theoretical significance and applied relevance. Nonlinear optical wave equations not only facilitate the development of advanced techniques but also play a crucial role in elucidating natural phenomena across diverse fields, including biology, nonlinear optics, and quantum physics. Among these, solitons—localized nonlinear waves—stand out as valuable tools for understanding complex nonlinear systems. Solitons are widely studied in areas such as plasma physics, nonlinear optics, and quantum mechanics. Optical solitons, in particular, have drawn significant interest due to the inherently interdisciplinary nature of soliton theory, making it a pivotal topic for advancing technologies like high-speed data transmission. The external potential strongly influences the shape and stability of optical pulses. In quantum mechanics and nonlinear optics, potentials with parity-time symmetry (PT -symmetry) are frequently utilized. Numerous studies in the literature examine the stability of nonlinear Schrödinger (NLS) equations with PT -symmetry. These equations admit various nonlinear wave solutions, including solitons, which are localized waves that propagate without distortion. Solitons demonstrate remarkable resilience during collisions, retaining their properties even after interacting with other waves. This work investigates the soliton solutions and their stability in an NLS equation incorporating a momentum term and cubic nonlinearity under an external PT -symmetric potential. The governing equation is expressed as: iu_(z) +αu_(xx) −iΓu_(x) +φ|u|^2u+V_(PT) u = 0. Here, z denotes the scaled propagation distance, u is the differentiable complex-valued slowly varying amplitude, u_(xx) represents diffraction, Γ is the momentum term taken as a constant, and V_(PT) denotes the external potential. The PT -symmetric potential is defined as: VPT = V(x) +iW(x)=V0 +V1sech(x) +V2sech^2(x)+i[W0sech(x)tanh(x) +W1tanh(x)]. Here, V(x) and W(x) represent the real and imaginary components of the potential, where V(x) is an even function and W(x) is an odd function. A detailed introduction to solitons and their interdisciplinary significance is provided in Chapter 1. The NLS equation is introduced, along with its recent developments, including the momentum term and PT -symmetry. The chapter also outlines the research objectives and the thesis hypothesis, emphasizing the importance of the momentum term in the NLS equation. Chapter 2 describes the Spectral Renormalization (SR) Method, an iterative Fourier technique used to numerically solve the NLS equation with a momentum term and a PT -symmetric potential. The method is adapted to the problem at hand, and numerical solutions are obtained. In Chapter 3, the structure of the NLS equation without potential is analyzed to investigate the effect of the momentum term. Variations in soliton structures are examined in relation to changes in the momentum term coefficient, Γ, and the propagation constant, µ. Chapter 4 explores exact solutions of the NLS equation with a momentum term and PT -symmetric potential. Using the ansatz u(x,z) = f(x)e^i(µz+g(x)), where f(x) and g(x) are real-valued functions, analytical solutions are derived. These solutions are compared with the numerical results, which shows excellent agreement. The chapter also verifies the parity-time symmetry properties of the potential, confirming that its imaginary part is odd and its real part is even. Chapter 5 focuses on the stability analysis of soliton solutions. The Split-Step Fourier method is employed to investigate nonlinear stability, while linear stability is examined through the linear spectrum. The results indicate that the solitons become unstable with even slight increases in the momentum term coefficient, Γ. Additionally, enhancing the complex component of the potential increases instability, whereas increasing the real component improves stability. The acquired results are summed up in Chapter 6. Moreover, a brief discussion on potential future research is included. All numerical results were obtained using MATLAB2023®.