本論文主要的目的在探討孤立波在非局域非線性介質中的巨觀特性。 這些
性質預期可以在非線性光學系統及原子光學系統中發現並探演重要的角色,改變
該系統特性並增加系統的可控制性. 非局域非線性反應存在熱-光學物質,光折變
晶體,液晶，電漿以及玻色凝態中,已知具有降低非線性維度，穩定非線性孤立波
的作用. 本研究在不失一般性並兼具不同系統之獨特性的前提下深入非局域非
線性介質孤立波的分叉點性質及各項穩定度. 在非局域非線性效應下,向量式耦
合亮-暗光孤子對中的亮光孤子形成所需要的能量會因非局域強度變大而減少,
對於在週期性光晶格下的純量波動,非局域非線性則提供了對能帶的可操控性及
波動在晶格中的可移動性. 此外非局域非線性對所有對稱破壞不穩定度,包含調
製不穩定度, 振盪不穩定度, 橫向調製不穩定度具有明顯的抑制作用. 這樣效應
是來自於非局域反應使介質中的折射率對波動強度的穩態反應緩慢,造成系統中
的微小擾動不容易成長,尤其對於較高頻的微擾,壓制的程度更甚. 因此在這個類
似”低通瀘波器”效果的系統中所有的噪音,微擾在各個方向上(空間及時間上)的
成長受到限制,導致等效非線性維度下降,有助於孤立波在介質中的穩定存在 另
外,在本研究中也建構了一個一般性的數學模型來分析具有圓對稱性的光孤子以
光渦漩. 利用這個模型,本研究發現了一系列新形態的楕圓光孤子,楕圓光渦漩及楕圓光
孤子環.並且研究這些新的光孤子的穩定性及演化特性.

The focus of this work is to explore the macroscopic property of solitary waves in nonlocal nonlinear materials which extend to nonlinear optical systems and atomic optical systems. Nonlocal nonlinearity are known to improve the stability of solitary waves by reducing the nonlinear power. This work reveals insights to bifurcate behavior and instability nature of solitary waves in nonlocal nonlinearity in general aspects and in specific details. Nonlocal nonlinear response reduces
the formation power a guided bright soliton in a dark-bright vector soliton pair, providing controllability over the band structure of scalar waves and enhence the mobility of wave packets in periodic potential; also the symmetry-breaking in-
stabilities are remarkably suppressed ascribable to the fact that nonlocal response exhibits slow steady-state spatial response to the noise spectrum. Such a ”low pass filter” ’effect restricts the noise to grow in all the transverse dimensions and results in the reduction of the effective order of nonlinearity in the system. Furthermore, elliptical solitons in 2D nonlinear Schr\"{o}dinger equations are proposed and studied numerically with a more generalized formulation showing the existence and
symmetry-breaking instabilties of new families of solitons, vortices, and soliton rings with elliptical symmetry, which paves the way for future insvestigation of 2D nonlocal nonlinear system.

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