Fundamental lattice solitons in Davey Stewartson systems

Nonlinear wave problems are of wide physical and mathematical interest and arise in a variety of scientific fields such as nonlinear optics, fluid dynamics, plasma physics, etc. The solutions of the governing nonlinear wave equations often exhibit important phenomena, such as stable localized waves (e.g., solitons) or self-similar structures and wave collapse (i.e., blow-up) where the solution tends to infinity in finite time or at finite propagation distance. Recently, wave collapse and the role of ground-state on global-existence theory are investigated for the Nonlinear Schr\"odinger equation (NLSE) with Mean Terms (NLSM or Davey Stewartson Systems). It is found that NLSM collapse can be arrested by small nonlinear saturation. Another way of arresting wave collapse is adding an external potential (lattice) to the governing equation (model). In recent years, there has been considerable interest in the study of solitons that are generated by the NLSE with various type external lattices, in particular those that can be generated in nonlinear optical materials. On the other hand, NLSM systems with additional external potentials have not been studied in current literature yet. NLSM system with an external potential is given by \begin{equation} \label{} \begin{split} iu_z + \frac{1}{2}\Delta u + \left| u \right|^2 u-\rho\phi u-V(x,y)u = 0, \\ \phi_{xx}+\nu\phi_{yy}=(\left| u \right|^2)_{xx}. \end{split} \end{equation} where $u$ corresponds to the field associated with the first-harmonic, $\phi(x,y,t)$ corresponds to the mean field, $\rho$ and $\nu$ are real constants that depend on the physical parameters, and $V(x,y)$ is an external optical potential. The NLSE can be obtained from the NLSM system by simply setting $\rho=0$. The external optical potential $V(x,y)$ can be written as the intensity of a sum of phase-modulated plane waves \begin{equation} \label{} V(x,y)=\frac{{V_0 }}{{N^2 }}\left| {\sum\limits_{n = 0}^{N - 1} {e^{i(k^n_xx+k^n_yy)} } } \right|^2 \end{equation} where $V_0>0$ is constant and corresponds to the peak depth of the potential, i.e., $V_0=max_{x,y}V(x,y)$, $(k^n_x, k^n_y)=[K cos(2\pi n/N),K sin(2\pi n/N)]$ is a wave vector. The potentials for $N=2, 3, 4, 6$ yield periodic lattices that correspond to standard $2D$ crystal structures, whereas $N=5, 7$ correspond to quasicrystals. The lattice-free medium can be obtained by setting $V_0=0$. In this dissertation, we aim to investigate the existence and stability properties of solitons in the (2+1)-dimensional NLSE and NLSM system with various types of external lattices. Chapter 2 describes the derivation of the NLSE and NLSM system by asymptotic methods. In Chapter 3, a numerical algorithm which is a modification of Spectral Renormalization (SR) method is given at the beginning. This algorithm will be used on each stage of the study to compute solutions of the models (NLSE and NLSM systems). Then, numerical methods for investigating the stability properties of solitons are explained. Chapter 4 is dedicated to the fundamental and dipole solitons in the NLSM systems. First we demonstrate the existence of solitons and, examine the stability properties of these solitons in the lattice-free medium. Then, we explain the effect of a periodic external potential as a collapse arrest mechanism in the NLSM system. The results of this part are considered as the main contribution to the thesis. Chapter 5 includes the multi-humped structures (dipoles and vortices) obtained in the NLSE with defective lattices. The NLSE is a special form of NLSM system. Therefore, understanding the dynamics of dipoles and vortices in the NLSE can be considered as a fundamental step for the NLSM systems. In Chapter 6, we present the fundamental and dipole solitons in the NLSM systems with a vacancy defect in the light of Chapter 5. Solitons in defective lattices have a significant importance in nonlinear science, this part of the study helps us to understand the effects of defects on the soliton properties in the NLSM systems. Chapter 7 deals with the existence and stability properties of fundamental solitons in the (2+1)-dimensional NLSE with a defective $\mathcal{PT}$-Symmetric external potential. Results of this dissertation are summarized in Chapter 8 where also a few ideas are outlined to further extend the research in this area.