Simple, exact analytical solutions of Maxwell's equations are given for the TE-type self-guided modes of a medium that has a power-law dependence on intensity I(q) for the continuum values of q. An analytical criterion shows that such spatial solitons are stable for q < 2 only. Our derivation is novel in that solitons are borrowed from the known modes of the sech(2) profile (linear) waveguide, rather than by solving the nonlinear wave equation. The results reveal the change in soliton propagation as the nonlinear medium itself changes.
The ability to solve novel complex problems predicts success in a wide range of areas. Recent research suggests that the ability to cognitively segment complex problems into smaller parts constrains nonverbal reasoning in adults. This study aimed to test whether cognitively segmenting problems improves nonverbal reasoning performance for children as it does for adults. 80 children aged 7-10 years completed two versions of a modified traditional matrix reasoning task in which demands on working memory, integration, and processing speed were minimised, such that the only significant requirement was to break each problem into its constituent parts. In one version of the task, participants were presented with a traditional 2x2 matrix and asked to draw the missing matrix item into a response box below. In a second version, the problem was broken down into its component features across three separate cells, reducing the need for participants to segment the problem. As with adults, performance was better in the condition in which the problems were separated into component parts. Children with lower fluid intelligence did not benefit more in the separated condition than children with higher fluid intelligence, and there was no evidence that segmenting problems was more beneficial for younger than older children. This study demonstrates that cognitive segmentation is a critical component of complex problem-solving for children, as it is for adults. By forcing children to focus their attention on separate parts of a complex visual problem, their performance can be dramatically improved.
We demonstrate, in both two and three dimensions, how a self-guided beam in a non-Kerr medium is split into two beams on weak illumination. We also provide an elegant physical explanation that predicts the universal character of the observed phenomenon. Possible applications of our findings to guiding light with light are also discussed.
The physics of nonlinear couplers is dictated by its normal modes, modes that are found from linear (axially uniform) couplers. Elementary power-flow arguments establish whether the mode is stable or unstable. These facts provide the bifurcation diagram that fully characterizes nonlinear coupling.
Poynting's vector theorem describes power flow on twin-core couplers with arbitrary axial perturbations, both linear and nonlinear. This bypasses the coupled amplitude formulation and unifies nonlinear two-mode interactions in general.
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