Through a semiquantal procedure, we study the perturbed modulation of amplitude and phase of Gross-Piteavskii equation describing trapped Bose–Einstein condensates in an optical lattice potential. By introducing quantum correctional parameters, the problem is quantized and leads to the derivation of a novel dynamical instability criterion. Additional degrees of freedom carrying quantum properties play a central role on the refine of the instability bandwidth, and, combined to the strength of optical lattice potential, entail unstable modes into full stability. A set of computational tools exhibited various features that bear instability characteristics, to confirm analytically predicted results. The quantum fluctuations thus have a stabilizing effect on the dynamics of harmonically trapped Bose–Einstein condensates in an optical lattice potential.
The modulational perturbations of the amplitude and phase for the time evolution of plane waves are studied in the semiquantum approach. Beyond the stability criterion from the nonlinear Schrödinger equation, there arise signatures of chaos in the dynamics of the associated cubic-quartic effective potential. The extended classical system described by fluctuation variables, nonlinearly coupled to the average ones, undergoes time-dependent variational approximation and foresees rooms of surprising regularity in the instability zone and chaos within the stable region of the wave number. The question of modulational instability (MI) for physical systems exhibits semiquantum chaotic features numerically observed with standard indicators. A dynamical quantum MI criterion is derived to resize the regions for stability with the correspondent momentum.
We propose a method for finding approximate analytic solutions to autonomous single degree-offreedom nonlinear oscillator equations. It consists of the harmonic balance with linearization in which Jacobian elliptic functions are used instead of circular trigonometric functions. We show that a simple change of independent variable followed by a careful choice of the form of anharmonic solution enable to obtain highly accurate approximate solutions. In particular our examples show that the proposed method is as easy to use as existing harmonic balance based methods and yet provides substantially greater accuracy.
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