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Modulational instability and breathing motion in the two-dimensional nonlinear Schrödinger equation with a
Hidetsugu Sakaguchi1, Yusuke Kageyama1
1Department of Applied Science for Electronics and Materials, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka 816-8580, Japan.
Abstract:
Modulational instability and breathing motion are studied in the two-dimensional nonlinear Schrödinger (NLS) equation trapped by the one-dimensional harmonic potential. The trapping potential is uniform in the y direction and the wave function is confined in the x direction. A breathing motion appears when the initial condition is close to a stationary solution which is uniform in the y direction. The amplitude of the breathing motion is larger in the two-dimensional system than that in the corresponding one-dimensional system. Coupled equations of the one-dimensional NLS equation and two variational parameters are derived by the variational approximation to understand the amplification of the breathing motion qualitatively. On the other hand, there is a breathing solution in the x direction which is uniform in the y direction to the two-dimensional NLS equation. It is shown that the modulational instability along the y direction is suppressed when the breathing motion is sufficiently strong, even if the norm is above the critical value of the collapse.
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