2,807 research outputs found
One-dimensional versions of three-dimensional system: Ground states for the NLS on the spatial grid
We investigate the existence of ground states for the focusing Nonlinear
Schr\"odinger Equation on the infinite three-dimensional cubic grid. We extend
the result found for the analogous two-dimensional grid by proving an
appropriate Sobolev inequality giving rise to a family of critical
Gagliardo-Nirenberg inequalities that hold for every nonlinearity power from
and , namely, from the -critical power for the same problem in
to the critical power for the same problem in .
Given the Gagliardo-Nirenberg inequality, the problem of the existence of
ground state can be treated as already done for the two-dimensional grid.Comment: 13 pages, 3 figure
Non-kirchhoff vertices and nonlinear schrodinger ground states on graphs
We review some recent results and announce some new ones on the problem of the existence of ground states for the Nonlinear Schrodinger Equation on graphs endowed with vertices where the matching condition, instead of being free (or Kirchhoff's), is non-trivially interacting. This category includes Dirac's delta conditions, delta prime, Fulop-Tsutsui, and others
Symmetries of Nonlinear PDEs on Metric Graphs and Branched Networks
This Special Issue focuses on recent progress in a new area of mathematical physics and applied analysis, namely, on nonlinear partial differential equations on metric graphs and branched networks. Graphs represent a system of edges connected at one or more branching points (vertices). The connection rule determines the graph topology. When the edges can be assigned a length and the wave functions on the edges are defined in metric spaces, the graph is called a metric graph. Evolution equations on metric graphs have attracted much attention as effective tools for the modeling of particle and wave dynamics in branched structures and networks. Since branched structures and networks appear in different areas of contemporary physics with many applications in electronics, biology, material science, and nanotechnology, the development of effective modeling tools is important for the many practical problems arising in these areas. The list of important problems includes searches for standing waves, exploring of their properties (e.g., stability and asymptotic behavior), and scattering dynamics. This Special Issue is a representative sample of the works devoted to the solutions of these and other problems
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