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Collapse in the n‐Dimensional Nonlinear Schrödinger Equation—A Parallel with Sundman's
Author(s) -
Berkshire F. H.,
Gibbon J. D.
Publication year - 1983
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1983693229
Subject(s) - singularity , virial theorem , angular momentum , nonlinear system , nonlinear schrödinger equation , momentum (technical analysis) , classical mechanics , physics , schrödinger equation , amplitude , mathematical physics , mathematical analysis , mathematics , quantum mechanics , finance , galaxy , economics
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.