Existence Of Solutions In Nonlinear Photonic Lattice Model

In the field of nonlinear photonic lattice, Physical scientists use optical induction,the interference of two or more plane waves in a photosensitive material , to create a 2D photonic lattice in which the solitons form. On this two-dimensional discrete soliton research, We have only the experimental observations. In this paper, we use the principle of variational method, Mountain Pass Lemma, Fixed point method to show existence of steady-state solutions in two-dimensional soliton model. Our result provides a theoretical basis for a variety of experiments and research in photonic lattices and crystals. Finally, Our approach also can be applied to research the proposed lattice solitons in Bose-Einstein condensates.Besides, In this paper, the third chapter uses Mountain Pass Lemma to proof existence of solutions in two-dimensional geometric constraint magnetic wall model . In the case of periodic boundary, it can describe a magnetic crystal structure.The whole thesis consists of three chapters. In the first chapter, we introduce the relative background knowledge and the main result. In chapter 2, under the various parameters, we discuss in detail the existence of solutions in the photonic lattice model . In chapter 3, we discuss the existence of solutions in two-dimensional geometric constraints of magnetic wall model .