6 research outputs found
Percolation in Interdependent and Interconnected Networks: Abrupt Change from Second to First Order Transition
Robustness of two coupled networks system has been studied only for
dependency coupling (S. Buldyrev et. al., Nature, 2010) and only for
connectivity coupling (E. A. Leicht and R. M. D'Souza, arxiv:09070894). Here we
study, using a percolation approach, a more realistic coupled networks system
where both interdependent and interconnected links exist. We find a rich and
unusual phase transition phenomena including hybrid transition of mixed first
and second order i.e., discontinuities like a first order transition of the
giant component followed by a continuous decrease to zero like a second order
transition. Moreover, we find unusual discontinuous changes from second order
to first order transition as a function of the dependency coupling between the
two networks.Comment: 4pages,6figure
Instability of bound states of a nonlinear Schr\"odinger equation with a Dirac potential
We study analytically and numerically the stability of the standing waves for
a nonlinear Schr\"odinger equation with a point defect and a power type
nonlinearity. A main difficulty is to compute the number of negative
eigenvalues of the linearized operator around the standing waves, and it is
overcome by a perturbation method and continuation arguments. Among others, in
the case of a repulsive defect, we show that the standing wave solution is
stable in \hurad and unstable in \hu under subcritical nonlinearity.
Further we investigate the nature of instability: under critical or
supercritical nonlinear interaction, we prove the instability by blowup in the
repulsive case by showing a virial theorem and using a minimization method
involving two constraints. In the subcritical radial case, unstable bound
states cannot collapse, but rather narrow down until they reach the stable
regime (a {\em finite-width instability}). In the non-radial repulsive case,
all bound states are unstable, and the instability is manifested by a lateral
drift away from the defect, sometimes in combination with a finite-width
instability or a blowup instability