發展具頻散關係及辛結構保持之時域有限差分法求解非線性薛丁格方程

Development of a Dispersion-relation-equation (DRE) error reducing and symplectic FDTD scheme for the cubic nonlinear Schrodinger equation (NLS)

林樂(Le Lin)

指導教授 ： 許文翰

國立臺灣大學/工學院/工程科學及海洋工程學研究所/碩士(2014年)

https://doi.org/10.6342/NTU.2014.02912

摘要

非線性薛丁格方程在許多物理領域的研究中扮演相當重要的角色。尤其一維的非線性薛丁格方程屬於可積系統，故隱含著相當豐富理論特性及守恆性。除此之外，此方程也同時屬於漢彌爾頓系統，故具有辛結構之性質。因為這些豐富的理論特性及廣大的工程運用，非線性薛丁格方程一直是吾人感興趣的主題之一，且凸顯出其數值研究的重要性。
本論文提出具有較好的頻散關係及辛結構守恆的時域有限差分法來求解非線性薛丁格方程。在本文所提出的方法中，先將該方程拆解成線性及非線性之偏微分方程。在求解線性方程時，對於時間微分項，本文所提出的方法採取四階準確度且具辛結構守恆之特性的離散方法；而對於空間微分項，本文所提出的方法則對於其數值的頻散關係做了最佳化。另一方面，在求解此方程非線性部分時，由於其對於時間具有不變性，故可以求得其「實解」。為了驗證本文中所提出的方法之可行性，本論文測試了許多具實解及典型的測試問題。由結果可知，本論文所提出之方法，在所有的測試問題中均能保有相當好的準確度。
本文接著將所提出之數值方法應用於探討瘋狗浪及非線性薛丁格方程的解之漸進行為。憑藉著本文所提出的方法，瘋狗浪的形成機制及其結構、特性可以被更多的了解及探討。而對於薛丁格方程的解的漸進行為，此種複雜的解的特性及不同解之間的過渡行為也可以透過本文所提出的方法得以了解。

關鍵字

非線性薛丁格方程 ； 分步方法 ； 顯式具辛結構法 ； 頻散關係 ； 瘋狗浪 ； 解之漸進行為

並列摘要

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.

並列關鍵字

cubic Schrodinger equation ； method of fractional steps ； explicit symplectic scheme ； dispersion relation equation ； rogue wave ； asymptotics ； Painleve equations.

參考文獻

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