在這篇論文，於第一章中我們陳述孤立子的歷史和一些非線性微分方程的解為孤立子。在第二章中我們推導出空間和時間中孤立子所應滿足的非線性薛定諤方程，並簡要說明線性穩定性分析和變分原理。在第三章中我們解已修正非線性薛定諤方程在非局部擴散介質中無位能之情形下的解，而獲得孤立子族的解。我們使用線性穩定性分析來測試孤立子在非局部擴散介質中是否穩定。結果所有的孤立子在非局部擴散介質中都穩定。
在第四章中我們分析局域具有自聚焦和非局域具有自散焦的週期性材料存在著間隙孤立子。由於材料局域具有自聚焦和非局域具有自散焦的特性，在局域和非局域的競爭下，間隙孤立子在第一個帶隙會分岔和形狀變換。不穩定的間隙孤立子會改變形狀由單峰模式變平頂模式而再變為雙峰模式。這項工作的結果可作為藉由局域和非局域的競爭處理非線性光學信號的可能性。
在第五章中，我們分析在兩個不同位能V1和V2之下存在著二維表面波，位能V1為圓環而其中心帶有一個圓，而電位V2單純為圓環。在位能V1和V2所形成的表面波為弦月波，而兩者形成的機制是不同的。 在位能V1下弦月波的產生是沒有功率閾值，然而在位能V下要有一個功率閾值才能形成弦月波。
在第六章中我們引入幾何對稱性破缺，我們將位能由圓環改為橢圓。位能為橢圓產生的弦月波僅存在橢圓的長短軸的地方。 短軸上弦月波的功率大於長軸上弦月波的功率。短軸上弦月波存在功率閾值，而當橢圓的長短軸比超過1.8的時候，在長軸的弦月波不再有功率閾值。
最後，我們總結論文已完成的成果和未來工作的展望。

In this thesis we state that the history of solitons and several nonlinear differ- ential equations which the solutions are solitons in chapter one. In chapter two we deduce the nonlinear Schr ̈odinger equation for spatial and temporal solitons, and briefly state that linear stability analysis and variational principle. In chapter three we solve modified nonlinear Schro ̈dinger equation to obtain soliton families in nonlocal diffusive medium under no trapping potential. We find the properties of solitons and use linear stability analysis to test solitons in nonlocal diffusive medium are stable or not. All the solitons are stable in nonlocal diffusive medium.

In chapter four we analyze the existence, bifurcations, and shape transforma- tions of one-dimensional gap solitons (GSs) in the first finite bandgap induced by a periodic potential built into materials with local self-focusing and nonlocal self-defocusing nonlinearities. Originally stable on-site GS modes become unsta- ble near the upper edge of the bandgap with the introduction of the nonlocal self-defocusing nonlinearity with a small nonlocality radius . Unstable off-site GSs bifurcate into a new branch featuring single-humped, double-humped, and flat- top modes due to the competition between local and nonlocal nonlinearities. The mechanism underlying the complex bifurcation pattern and cutoff effects (termina- tion of some bifurcation branches) is illustrated in terms of the shape transforma- tion under the action of the varying degree of the nonlocality. The results of this work suggest a possibility of optical-signal processing by means of the competing nonlocal and local nonlinearities.

In chapter five we analyze the existance, and stability of two dimensional sur- face soliton families with two different potential V1 and V2. Potential V1 is a circular ring with a centric circle attendant, and potential V2 is the only a circilar ring. To form surface solitons with potential V1 and V2 are two different mechanisms. There exists a power threshold for the crescent wave under trapped potential V2, but the crescent wave under trapped potential V2 is power thresholdless. In chapter six by introducing the symmetry-breaking in geometry, we reveal the existence of thresholdless crescent waves, i.e., nonlinear surface modes pinged to the boundary of a curvature, in an elliptical ring. An effective nonlinear
Schro ̈dinger equation along the azimuthal direction is derived by taking the trans- formation in the curvilinear coordinate of elliptical symmetry, which illustrates the formation of trapping potentials (barriers) along the semi-major (minor) axis. Our results demonstrate an alternative but efficient approach to access optical surface modes with a variety of micro-structures.

Finally we summarize the works which have be done in the thesis and future works will be studied in conclusion.

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