Dynamics of solitons in the model of nonlinear Schrödinger equation with an external harmonic potential: I. Bright solitons

The dynamics of nonlinear solitary waves is studied by using the model of nonlinear Schrödinger equation (NSE) with an external harmonic potential. The model allows one to analyse on the general basis a variety of nonlinear phenomena appearing both in a Bose–Einstein condensate in a magnetic trap, whose profile is described by a quadratic function of coordinates, and in 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 for 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 which the parabolic approximation is valid near potential energy minima. The conditions are found for formation of stable stationary states of antiphase solitons 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 which the amplitude of soliton oscillations in a trap exponentially increases with time is shown. It is shown that exact solutions of the problem found using the Miura transformation open up the possibility to control the dynamics of solitons. New effects are predicted, which are called the reversible and irreversible denaturation of solitons in a nonstationary harmonic potential.