Solitons In Nonlinear Optics Research Articles

Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length

We consider, by means of the variational approximation (VA) and direct numerical simulations of the Gross-Pitaevskii (GP) equation, the dynamics of 2D and 3D condensates with a scattering length containing constant and harmonically varying parts, which can be achieved with an ac magnetic field tuned to the Feshbach resonance. For a rapid time modulation, we develop an approach based on the direct averaging of the GP equation,without using the VA. In the 2D case, both VA and direct simulations, as well as the averaging method, reveal the existence of stable self-confined condensates without an external trap, in agreement with qualitatively similar results recently reported for spatial solitons in nonlinear optics. In the 3D case, the VA again predicts the existence of a stable self-confined condensate without a trap. In this case, direct simulations demonstrate that the stability is limited in time, eventually switching into collapse, even though the constant part of the scattering length is positive (but not too large). Thus a spatially uniform ac magnetic field, resonantly tuned to control the scattering length, may play the role of an effective trap confining the condensate, and sometimes causing its collapse.

Potential of interaction between two- and three-dimensional solitons

A general method to find an effective potential of interaction between far separated two-dimensional (2D) and 3D solitons is elaborated, including the case of 2D vortex solitons. The method is based on explicit calculation of the overlapping term in the full Hamiltonian of the system (without assuming that the ``tail'' of each soliton is not affected by its interaction with the other soliton). The result is obtained in an explicit form that does not contain an artificially introduced radius of the overlapping region. The potential applies to spatial and spatiotemporal solitons in nonlinear optics, where it helps to solve various dynamical problems: collisions, formation of bound states (BS's), etc. In particular, an orbiting BS of two solitons is always unstable. In the presence of weak dissipation and gain, the effective potential can also be derived, giving rise to bound states similar to those recently studied in 1D models.

Do solitons exchange conserved quantities during collisions?

The redistribution of conserved quantities among colliding solitons of the nonlinear Schr\"odinger equation is considered. An analogy with the theory of spatial solitons in nonlinear optics provides one way to calculate this redistribution. In this context, exchanges of conserved quantities among $N$ colliding solitons can be completely described from a knowledge of the case for $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$. It is shown that solitons generally exchange ${L}^{2}$ norm as they collide, with the fraction shared being small when the solitons differ significantly in velocity or amplitude. Exchanges of other conserved densities are also considered.

Solitons in nonlinear optics. I. A more accurate description of the 2π pulse in self-induced transparency

A more accurate description of the 2 pi pulse of self-induced transparency is obtained. This is an exact solution of an approximate form of the Maxwell-Bloch equations in which only backscattering is neglected; the equations are valid at densities less than about 1018 atoms cm-3. At such densities the 2 pi sech pulse of McCall and Hahn (1967, 1969) remains a good approximate well into the picosecond range. The more accurate solution is chirped by a factor proportional to the square of the ratio of the spectral width of the pulse to its carrier wave frequency. The stability of this pulse solution and other N soliton solutions of the approximate Maxwell-Bloch equations is demonstrated.