Defocusing

From DispersiveWiki


Informally, an equation is defocusing (or repulsive) if the nonlinearity tends to act to dissipate the solution when it is concentrated, thus (in principle) reinforcing the effects of dispersion. The opposite of defocusing is focusing, though there do exist nonlinear equations which are neither focusing nor defocusing as the nonlinearity exhibits no bias towards helping or hindering the dispersion.

Defocusing equations tend to have an definite Hamiltonian and are less likely to blow up than focusing ones. They rarely support soliton solutions; instead, solitons tend to scatter to linear solutions. Defocusing equations also often enjoy monotonicity formulae which are unavailable in the focusing case.