We extend the Zakharov's theory for collapses of Langmuir waves into the system under harmonic potential. As a possible application, we consider the collapse of the Bose-Einstein condensate with attractive interparticle interactions under magnetic trap.
We study dynamics of a wavefunction for the nonlinear Schrödinger equation with attractive self-interaction under a harmonic potential. By use of the variance identity and a rigorous inequality, we obtain a sufficient condition for the collapse of the wavefunction. By applying this method, we investigate the dynamic stability of a trapped Bose–Einstein condensate and its dependence on the initial shape of the condensate. The result agrees qualitatively well with the results of the time-dependent variational method. We find that the anisotropy introduced in the initial state has a significant effect on the stabilization.
We consider dynamics of Bose-Einstein condensates with long-range attractive interaction proportional to 1/rb and arbitrary angular dependence. It is shown exactly that collapse of a Bose-Einstein condensate without contact interactions is possible only for $b\ensuremath{\geqslant}2$. The case b=2 is critical and requires the number of particles to exceed a critical value to allow collapse. The critical collapse in that case is a strong one, trapping into the collapsing region a finite number of particles. The case b>2 is supercritical with an expected weak collapse which traps a rapidly decreasing number of particles during approach to collapse. For b<2 a singularity at r=0 is not strong enough to allow collapse but an attractive 1/rb interaction admits stable self-trapping (a stable three-dimensional soliton solution) even in the absence of an external trapping potential.
The dynamics of nonlinear solitary waves in studied by using the model of nonlinear Schrodinger equations(NSE) with an external harmonic potential. The model allows one to analyze on the general basis a variety of nonlinear phenomena appearing both in Bose-Einstein condensate in a magnetic trap, whose profile is describe by a quadratic function of coordinates, and an nonlinear optics, physics of lasers, and biophysics. It is shown that exact solutions for a quantum-mechanical particle in a harmonic potential and solutions obtained within the framework of the adiabatic perturbation theory of bright solitons in a parabolic trap are completely identical. This fact not only proves once more that solitons behave like particles but also that they can preserve such properties in different traps for the which the parabolic approximation is valid near potential energy minima. The conditions are found for formation of stable stationary states of antiphase soliton in a harmonic potential. The interaction dynamics of solitons in nonstationary potentials is studied and the possibility of the appearance of a soliton parametric resonance at with the amplitude of soliton oscillation in a trap exponentially increases with time is shown.