SNAKING PHENOMENON ON STUB LATTICE
The nonlinear Schr¨odinger equation is an equation that is often applied invarious fields of science, both in continuous and discrete cases. These two cases can produce a snaking phenomenon, with the main factor being in the discrete case is the Laplacian part. This research will examine the phen...
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| Format: | Final Project |
| Language: | Indonesia |
| Online Access: | https://digilib.itb.ac.id/gdl/view/85862 |
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| Institution: | Institut Teknologi Bandung |
| Language: | Indonesia |
| Summary: | The nonlinear Schr¨odinger equation is an equation that is often applied invarious
fields of science, both in continuous and discrete cases. These two cases can
produce a snaking phenomenon, with the main factor being in the discrete case
is the Laplacian part. This research will examine the phenomenon of snaking on
discrete nonlinear Schr¨odinger equation with one-dimensional stub lattice. The
linear stability of the uniform solution can be tested using the relation equation
dispersion. This equation is obtained by finding a solution around the uniform
solution which has a form similar to the solution of the diffusion equation. Analysis
of changes in localized solutions to frequency parameters can show snaking
behavior. This snaking is seen as a change in the stability of the localized solution
produces a snaking bifurcation diagram. The snaking width will decrease along
with increasing coupling strength, shows that equation will not have snaking in
the continuous case. This snaking width is called as a pinning region. In the case
of a one-dimensional stub lattice, the site is divided into three section to simplify
definitions. This system has a uniform solution with linear stability similar to
the one-dimensional lattice case. Main differences in a one-dimensional stub
lattice there are three dispersion relation equations. In the bistable region, there
are localized solutions of two types: c-centred and ac-centred. Snaking in the
case of one-dimensional stub lattices occurs in both types this solution, because
the localized solution obtained has symmetry properties. Snaking will also show
an additional case called switchback, namely interaction convoluted caused by
non-uniformity of the front due to structural differences. Pinning regions for
switchbacks will disappear faster in comparison with the pinning region of the
main snaking as the coupling strength increases. Besides, the pinning region
in a one-dimensional stub lattice is larger than in a one-dimensional stub lattice
dimensions, as the slightly wider stub lattice structure requires couplings higher
strength to eliminate snaking cases. |
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