WAVE PROPAGATION IN THE LUGIATO-LEFEVER EQUATION

• 主要作者: Anggito Pramudhani R, Bramasta
• 格式: Final Project
• 語言: Indonesia
• 在線閱讀: https://digilib.itb.ac.id/gdl/view/77561
• 機構: Institut Teknologi Bandung

This final project will study the Lugiato-Lefever equation, which, in physical terms, is an experiment involving the shooting of laser light onto a plate. The Lugiato- Lefever equation is a nonlinear Schr¨odinger equation that describes the behavior of light in a nonlinear optical resonator, such as Kerr frequency waves in an optical microresonator. The nonlinear Schr¨odinger equation is a mathematical equation that explains the time evolution of a physical system. It takes the form of a differential equation with wave-like characteristics and is used as a mathematical model for wave propagation. The Lugiato-Lefever equation contains a secondorder derivative operator, so methods are employed to approximate the second derivative, namely pseudo-spectral and methods. In this final project, we will seek to determine the bifurcation diagram generated, assess the stability of this diagram, and if there are unstable points, investigate where the solutions will escape to. A bifurcation diagram is a chart that depicts the relationship between variables and parameters concerning equilibrium points. It represents the solution diagram of the problem at hand. The methods used include the Newton-Raphson method to find equilibrium values, numerical continuation to identify turning points, and Runge- Kutta for examining solutions in the unstable regions. Scaling methods will also be employed to compare previous results, but the second derivative approximation used will differ, utilizing fast Fourier transform. The Lugiato-Lefever equation also contains diffusion terms, so the author also attempts to find the patterns generated in the 2D domain using fast Fourier transform, Newton-Raphson, and Runge-Kutta. Simulations are conducted using MATLAB, and for stability determination, the author utilizes Maple 18.