Analysis of solitary waves in inhomogeneous systems

In this thesis, we aim to investigate solitary waves of three nonlinear Schrödinger (NLS)-type models, namely, the NLS equation with an asymmetric double Dirac delta potential, the NLS equation with a Dirac delta potential on star graphs, and the discrete nonlinear Schrödinger (DNLS) equation. For the first model, we obtain analytic solutions and show the difference between ground states that arise due to symmetric and asymmetric potentials. We find bifurcating asymmetric ground states at a threshold value of solution norm. In contrast to the symmetric case, pitchfork bifurcation no longer exists, and we find a saddle node one instead. For the second problem, we use coupled mode reduction method to yield conditions for symmetry breaking bifurcations. We notably obtain that the bifurcation is degenerate. There are two distinct asymmetric bifurcating solutions with the same norm. We provide an estimate of the bifurcation point. We also study non-positive definite states bifurcating from the linear solutions. Typical dynamics of unstable solutions are also presented. Finally, we study the fundamental lattice solitons of the DNLS equation and their stability via a variational method. Using a Gaussian ansatz and comparing the results with numerical computations, we report a novel observation of false instabilities. Comparing with established results and using the Vakhitov-Kolokolov criterion, we deduce that the instabilities are due to the ansatz. In the context of using the same type of ansatzs, we provide a remedy by employing multiple Gaussian functions. The results show that the higher the number of Gaussian functions used, the better the solution approximation.