Motions in a Bose condensate. V. Stability of solitary wave solutions of non-linear Schrodinger equations in two and three dimensions

For pt.IV see ibid., vol.15, no.8, p.2599-619 (1982). The non-linear Schrodinger equation has previously been solved in two dimensions (2D) and three dimensions (3D) to give sequences of solitary waves, i.e. finite amplitude disturbances that propagate without change of form. The 3D dispersion relationship (the plot of energy, E, against quasimomentum, p) has two branches for p>pmin and none for p<pmin. The lower energy branch consists largely of disturbances possessing circulation and resembling large vortex rings in the limit p to infinity . The upper energy branch consists of rarefaction pulses that are governed as p to infinity by the Kadomtsev-Petviashvili (KP) equation. The 2D dispersion relationship (where E and p refer to unit length) is a single branch of solitary waves extending from p=0 to p= infinity , the latter resembling vortex pairs and the former being rarefaction pulses governed by the KP equation. In this paper new integral properties of the solitary waves are derived, and are tested against previous numerical work. Their forms in the KP limit are compared with the results of Iordanskii and Smirnov (1978). The stability of solitary waves is examined both analytically and numerically. The analysis compares E for a solitary wave with E for a neighbouring state of the same p, obtained from a solitary wave by coordinate stretching. On this basis it is suggested that the lower branch of the 3D waves and the entire 2D sequence is stable to these disturbances, whereas the upper branch 3D solutions appear to be unstable. The implied creation of circulation if an upper branch solution descends in energy to a lower branch wave is examined, and is shown not to violate Kelvin's theorem. The tensor virial theorem is derived. Numerical work is presented that supports the analytical arguments.