Abstract
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.