SNAKING OF DISCRETE NONLINEAR SCHROÂ¨DINGER EQUATION WITH ONE-DIMENSIONAL DIAMOND LATTICE - Mubarok, Nahrul
The nonlinear Schr¨odinger equation is a widely applied equation in various scientific fields, both for continuous cases and discrete cases. Both cases can result in a snaking case with the main factor in the discrete case is the Laplacian par. In this thesis, we will examine snaking of the discrete Schr¨odinger nonlinear equation with a one-dimensional diamond lattice. This equation can be transformed into the time-independent discrete Allen-Cahn equation. Some of the main numerical methods that will be used in this thesis are numerical continuation and augmented method. Numerical continuation is used for analyze the change in the solution of a differential equation to a parameter and is one of the main tools for plotting bifurcation diagrams. Augmented method is used to find the bifurcation point of a differential equation by adding the condition, which is existence of zero eigenvalues. Because the solution from the nonlinear Schr¨odinger equation cannot be found analytically, it is necessary to use additional numerical methods for finding solutions to these equations. The method used to find a solution is an optimization method. In the case of a one-dimensional lattice, there are five uniform solutions; one trivial solution, two upper solutions, and two lower solutions. The trivial solution will be linearly stable when the frequency parameter is negative and unstable when the parameter is positive. The uniform upper solution will always be linearly stable and the uniform lower solution will always be unstable. The linear stability of the uniform solution can be checked using the dispersion relation equation. The dispersion relation equation is obtained by finding a solution around a uniform solution with a form resembling the solution of the diffusion equation. With certain nonlinear part parameters, there is an interval frequency so that there are two stable solutions. In this bistable region, there are localized solutions with the criteria that there are sites that are in the upper solution and there are others sites that are in the trivial solution. The link between stable solutions is called front and become the main factor in the stability of localized solutions. Analysis of change of localized solutions in respect to frequency parameters can lead to snaking behavior. Snaking can be seen as a change in the stability of such a localized solution until the bifurcation diagram is snaking. Then the snaking width will be decreases with increasing coupling strength. This shows that equations will have no snaking in the continuous case. The width of this snacking will be known as pinning region. In the case of a one-dimensional diamond lattice, the site is divided into three parts to make define easier. This system has a uniform solution with linear stability which is similar to the one-dimensional lattice case. The first difference in the lattice case a one-dimensional diamond is the existence of three dispersion relation equations, with one equations form a flat-band. In the bistable region, there is a localized solution, with two types of solutions site-centred and bond-centred solutions. Snaking on the case of a one-dimensional diamond lattice will appear if the numerical continuation starts from the case site-centred. This is because the localized solution obtained must have symmetry properties. Then snaking will have additional snaking cases which referred to as inner snaking. This is due to the non-uniform front due to structural differences. Then, the pinning region for inner eating will be disappears faster than the pinning region of the main snacking for increased coupling strength. This indicates that the one-diamond structure dimensions will approach the one-dimensional structure first before becoming continuous case. Furthermore, the pinning region in the case of a one-dimensional diamond lattice larger than the pinning region in the case of a one-dimensional lattice. This is because a diamond lattice structure that is wider than a one-dimensional lattice so it is necessary greater coupling strength to eliminate snaking cases. Pinning region obtained by using the augmented method will be larger compared to doing snaking because snaking cannot be exact determines the saddle-node bifurcation point and always changes direction at smaller values.