Nonlinear waves in the Pitaevskii-Gross equation

Wei Hua, Xue‐Shen Liu, Peizhu Ding

Iva Březinová, Lee A. Collins, Katharina Ludwig +2 more

The Gross-Pitaevskii equation (GPE) plays an important role in the description of Bose-Einstein condensates (BECs) at the mean-field level. The GPE belongs to the class of nonlinear Schr\"odinger equations which are known to feature dynamical instability and collapse for attractive nonlinear interactions. We show that the GPE with repulsive nonlinear interactions typical for BECs features chaotic wave dynamics. We find positive Lyapunov exponents for BECs expanding in periodic and aperiodic smooth external potentials, as well as disorder potentials. Our analysis demonstrates that wave chaos characterized by the exponential divergence of nearby initial wave functions is to be distinguished from the notion of nonintegrability of nonlinear wave equations. We discuss the implications of these observations for the limits of applicability of the GPE, the problem of Anderson localization, and the properties of the underlying many-body dynamics.

H. A. Ishkhanyan, В. П. Крайнов

Above-barrier reflection of cold alkali atoms by resonant laser light was considered analytically within the Gross-Pitaevskii approximation. Correction for the reflection coefficient because of a weak nonlinearity of the stationary Schrödinger equation has been derived using multiscale analysis as a form of perturbation theory. The nonlinearity adds spatial harmonics to linear incident and reflecting waves. It was shown that the role of nonlinearity increases when the kinetic energy of an atom is nearly to the height of the potential barrier. Results are compared to the known numerical derivations for wave functions of the Gross-Pitaevskii equation with the step potential.

Manjun Ma, Zhe Huang

Wei Hua, Xue‐Shen Liu, Peizhu Ding

We solve the time-independent Gross-Pitaevskii equation that describes the dilute Bose-condensed atoms in harmonic trap at T =0 by symplectic shooting method (SSM). Both the repulsive nonlinearity and the attractive nonlinearity cases are studied, and the bounded state eigenvalues as well as the corresponding wave functions are presented.

Claude M. Dion, Eric Cancès B