Collapse in the n‐Dimensional Nonlinear Schrödinger Equation—A Parallel with Sundman's

Abstract

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.

Optimization of optical beam steering in nonlinear Kerr media by spatial phase modulation

“…)], one can see that the phase modulation breaks an input beam into multiple subbeams propagating at different angles1 7 (Raman-Nath scattering):…”

“…The three conservation laws associated with the NLSE and corresponding to the conservation of the wave action M, the transverse momentum P, and the energy H (or the Hamiltonian) can be obtained either from Noether's theorem 7 1 0 or directly from Eq. (1).…”

“…7 '1 0 However, catastrophic self-focusing or beam collapse can be easily avoided in practice by the choice of a medium length shorter than the critical self-focusing distance. Thus with proper choice of the medium length the beam-steering technique would work well even in a bulk Kerr medium.…”

Modulational instability and nonlinear evolution of two-dimensional electrostatic wave packets in ultra-relativistic degenerate dense plasmas

“…Note that since the evolution equations are of the DS II-type, we can not derive the criteria that are sufficient to ensure collapse by the Virial theorem or else 11 . However, we will present some conditions according to Berkshire and Gibbon 23 . To that end, we first see that the integrals of motion are the wave action N and the Hamiltonian H where…”

“…23 , the conditions for the collapse to occur in a finite time are d 2 I/dτ 2 ≥ 0, C 1 (0) < 0, C 2 (0) > 0 and that the angular momentum integral vanishes, J (τ = 0) = 0. Furthermore, if C 1 = 0, i.e.…”