Autosolitons in trapped Bose-Einstein condensates with two- and three-body inelastic processes

In this paper, we consider the conditions for the existence of autosolitons, in trapped Bose-Einstein conden- sates with attractive atomic interactions. First, the variational approach is employed to estimate the stationary solutions for the three-dimensional Gross-Pitaevskii equation with trap potential, linear atomic feeding from the thermal cloud, and two- and three-body inelastic processes. Next, by using exact numerical calculations, we show that the variational approach gives reliable analytical results. We also discuss the possible observation of The existence of envelope autosolitons in the one- dimensional ~1D! case, in a homogeneous nonlinear medium with dissipation and amplification, was revealed by Pereira and Stenflo @1#. They found the exact solution for autosoli- tons with arbitrary growth and damping strengths in the per- turbed nonlinear Schrodinger equation ~NLSE!. Later, auto- solitons were discovered in nonlinear fiber optics, namely, in fibers with amplifiers and distributed filters~the latter corre- sponds to the frequency-dependent damping in the nonlinear Schrodinger equation !@ 2,3# and also for waves on the sur- face of deep water @4#. Correspondingly, in a 2D homoge- neous medium with amplification and nonlinear damping, the possibility of existence of a 2D analog of the Pereira- Stenflo solitons was recently shown by a variational ap- proach @5#. Autosolitons in a weakly dispersive nonlinear media, described by the Korteweg-deVries equation, have been studied in Refs. @6,7#. The autosolitons can be distinguished from ordinary soli- tons. The latter exist in a conservative media and are origi- nated from the balance between the nonlinear and dispersive effects of the wave propagation. The properties of these gen- erated solitons are defined by the initial conditions ~their number, parameters such as amplitudes, widths, etc. !@ 6#, with the solutions characterized by their corresponding prop- erties. As for to the autosolitons, they can be generated in nonconservative media when effects of amplification and dissipation are present. For the existence of autosolitons, one should add to the equilibrium condition between nonlinearity and dispersion the requirement of a balance between ampli- fication, frequency-dependent damping, and nonlinear dissi- pation. In distinction from ordinary solitons, the properties of the autosoliton, as a rule, are fixed by the coefficients of the perturbed NLSE and by any initial perturbation that is at- tracted to this point ~attractor in the space of coefficients!. Mathematically, the problem is described by the NLSE with complex parameters. If the imaginary parts are large, the equation is equivalent to the so-called complex Ginzburg- Landau equation. An interesting limit is represented by the NLSE with small complex coefficients.