| Summary: | 碩士 === 國立臺灣大學 === 工程科學及海洋工程學研究所 === 102 === Nonlinear Schrodinger (NLS) equation appears in many studies of theoretical physics and possesses many fascinating properties. This equation in one space dimension is an example of integrable model, therefore, permitting an infinite number of conserved quantities such as the momentum and energy. This classical field equation can be rewritten as a system of equations involving Hamiltonian functions. This equation also possesses multisymplectic geometric structure and can be therefore constructed in a multi-symplectic form. Because of both these remarkable properties and wide physical and engineering applications, the nonlinear Schrodinger (NLS) equation has been the subjects of intensive study. Therefore, numerical study on the Schrodinger equation with cubic nonlinearity is essential.
In this dissertation, developed scheme with a better dispersion-relation-equation error reducing and symplecticity for the cubic nonlinear Schrodinger equation is proposed. Over one time step from tn to tn+1, the linear part of Schrodinger equation is solved firstly through four time integration steps. In this part of simulation, the explicit symplectic scheme of fourth order accuracy is adopted to approximate the temporal derivative
term. The second-order spatial derivative term in the linear Schrodinger equation is approximated by centered scheme. The resulting symplectic and space centered difference scheme renders an optimized numerical dispersion relation equation. In the second part of simulation, the solution of the nonlinear equation can be computed exactly thanks to the embedded invariant nature within each time increment. The proposed semidiscretized symplectic scheme underlying the modified equation analysis of second kind and the method of dispersion error minimization has been assessed in terms of the spatial modified wavenumber or the temporal angular frequency resolution.
Furthermore, several applications of the proposed new finite difference scheme for the calculation of Schrodinger equations are included such as the rogue waves in deep-water and the asymptotic problems accompanied with many remarkable quantities of Painleve equations. One of the objectives of this dissertation is to increase the knowledge about the rogue waves by following two steps. The first one is to explore the solution nature
in localized region near the point of gradient catastrophe. The second one is to enlighten the solution in the transitional region bounded by the breaking curves that separate two completely different smooth and oscillatory regions, which are manifested by the modulated plane waves and the two-phase nonlinear waves, respectively. Another objective of this study is to understand how the solution of cubic nonlinear Schrodinger (NSL) equation behaves at large time. The study on long-time asymptotics for cubic nonlinear Schrodinger (NSL) equation, in particularly in the transition zones separating the solution into the different regions, is carried out. Through the proposed scheme, the intricate phenomena in the transitional zone can be clearly visualized, which facilitates one to realize how the solution translate between different solution regions. The connection between the Painleve equations of types II and IV and the nonlinear Schrodinger (NSL) equation will be particularly addressed as well.
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