Sakaguchi Hidetsugu, Kageyama Yusuke
Department of Applied Science for Electronics and Materials, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka 816-8580, Japan.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Nov;88(5):053203. doi: 10.1103/PhysRevE.88.053203. Epub 2013 Nov 18.
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.
研究了一维谐振子势阱捕获的二维非线性薛定谔（NLS）方程中的调制不稳定性和呼吸运动。捕获势在y方向上是均匀的，波函数在x方向上受到限制。当初始条件接近y方向上均匀的定态解时，会出现呼吸运动。二维系统中呼吸运动的振幅比相应一维系统中的大。通过变分近似推导了一维NLS方程和两个变分参数的耦合方程，以定性地理解呼吸运动的放大。另一方面，对于二维NLS方程，存在y方向上均匀的x方向呼吸解。结果表明，当呼吸运动足够强时，即使范数高于坍缩的临界值，沿y方向的调制不稳定性也会受到抑制。
