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Collapse of solutions of the n‐dimensional nonlinear Schrödinger equation are studied using the integrals of the motion and an equation corresponding to the Lagrange‐Jacobi virial equation of classical mechanics. There are strong parallels with collapse in the classical N‐body problem and in particular with the results of K. F. Sundman. Collapse occurs when the amplitude of the solution becomes singular as the initial data collapse to the center of mass in finite time. In some cases the singularity is inevitable (for negative energy), but in others only a necessary condition for collapse can be derived, involving the angular momentum.

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SUMMARYCertain theoretical features of high altitude lee waves for some simple two-dimensional models are described. It is shown that some of these allow an intense local radiation of lee waves in a nearly vertical beam into the high stratosphere (similar to the leakage upwards of short radio waves in a familiar electromagnetic theory), Some associated quasi-resonance phenomena are noted.

Lee waves in the lower stratosphere are estimated by a consideration of natural wave 'modes' which have complex wavenumber and hence decay with distance in the downstream direction. For simple two layer troposphere-stratosphere models the ' leaky modes ' can be evaluated explicitly and compared with computed waves. When there is a layer of neutral stability in the upper troposphere the natural complex wavenumbers are responsible for some quasi-resonance phenomena.