Modulational instability and breathing motion in the two-dimensional nonlinear Schr¨odinger equation with a one-dimensional harmonic potential

摘要

Modulational instability and breathing motion are studied in the two-dimensional nonlinear Schr¨odinger (NLS)nequation trapped by the one-dimensional harmonic potential. The trapping potential is uniform in the y directionnand the wave function is confined in the x direction. A breathing motion appears when the initial condition isnclose to a stationary solution which is uniform in the y direction. The amplitude of the breathing motion is largernin the two-dimensional system than that in the corresponding one-dimensional system. Coupled equations of thenone-dimensional NLS equation and two variational parameters are derived by the variational approximation tonunderstand the amplification of the breathing motion qualitatively. On the other hand, there is a breathing solutionnin the x direction which is uniform in the y direction to the two-dimensional NLS equation. It is shown thatnthe modulational instability along the y direction is suppressed when the breathing motion is sufficiently strong,neven if the norm is above the critical value of the collapse.