孤立子理論是一門美麗地、跨學科間的主題，它已經對數學和物理科學產生巨大衝擊。在本篇論文中，我們將考慮偶核耦合非線性薛丁格系統，其中擾動係數屬於實數，而且系統可分別對應於聚光雷射或散光雷射在非線性雙折射偶核光纖中之量子波傳導方程式。在適當條件下，此系統會簡化成為（可積的）擾動的非線性薛丁格方程式。值得注意的一件事實是，在擾動的非線性薛丁格方程式中，干擾項不是造成光纖中非線性波傳播衰退的理由。
在此論文，我們將給出上述偶核耦合非線性薛丁格系統的 Lax 表現，證明其為一無窮維希耳伯特空間中之可積系統。利用其無窮維對稱性的幾何特性，我們也將給出系統的 Bäcklund-gauge 轉換。藉由 Bäcklund-gauge 轉換，我們造出系統的2-孤立子解，此處數字“2”表示離散的光譜值個數，而且也會觀察到“黑暗孤立子”解的存在性證據。最後，我們會造出由不穩定單平面波所引起之同相奇異波 (homoclinic wave)，同時證明當時間趨近正負無窮大時，同相奇異波會趨近到平面波解。

Soliton theory is a beautifully interdisciplinary subject which has had enormous impact, both on mathematics and the physical sciences. In this thesis, we consider the dual-core CNLS system with the perturbed constants are real numbers and the system can corresponding to the quasi-monochromatic electromagnetic waves in the dual-core focusing or the dual-core defocusing birefringent optical fiber, respectively. This coupled system can reduces to the (integrable) perturbed nonlinear Schrödinger equation. The perturbed term in perturbed NLS equation is not the reason that why the nonlinear wave propagation decay in optical fiber.
We give a Lax representation of the above dual-core CNLS system to verify it is an infinite dimensional integrable system on Hilbert space. Using the geometric property of symmetry, we will construct a Bäcklund-gauge transformation for the system. By means of Bäcklund-gauge transformation, we construct 2-soliton solutions of system, where 2 is the number of discrete spectral values, and observe the existence of dark soliton solutions. Finally, we construct the homoclinic waves arising from the unstable simple plane waves and show that the homoclinic solutions are truly homoclinic to the plane waves in the sense that they approach the plane wave solutions in different orientations as time t increasing infinitely.

I. Introduction …………………………………………………………1
II. Lax Representation of the Dual-Core Coupled Nonlinear Schrödinger system………………………………………………………3
III. Bäcklund-Gauge Transformation of the Dual-Core CNLS System………………………………………………………………………7
IV. N-Solitons and Dark Solitons of the Dual-Core CNLS System………………………………………………………………………14
V. N-Whiskered Homoclinic Waves of the Dual-Core Focusing CNLS System or the Dual-Core Defocusing CNLS System………………………………………………………………………19
VI. Discussions …………………………………………………………34
VII. Figure Captions……………………………………………………………………36
References…………………………………………………………………38
Figures ……………………………………………………………………39

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