研究了具有2n+1次非线性的薛定谔方程暗孤子特性。首先,给出了静态暗孤子解的统一解析表达式,发现静态暗孤子的宽度随非线性幂次的增大而减小,其深度保持不变。其次,研究了运动暗孤子的演化行为,给出了运动暗孤子波函数随空间和时间变化的普适表达式,发现对于给定的暗孤子运动速度,孤子的密度和相移都随非线性幂次的增加而减小。研究结果表明,对于给定的非线性多方指数,运动暗孤子的能量随运动速度的增加而减小。最后,通过数值模拟验证了所得解析结果。

We study the properties of dark solitons of the nonlinear Schr?dinger equation with (2n+1)-th order nonlinearity. We give the uniform analytical expression for a static dark soliton and find that the width of the static dark soliton decreases with the increase of the nonlinear power index, and its depth remains unchanged. The evolution behavior of the moving gray soliton is studied, and the general expression of the wave function of the moving gray soliton as a function of space and time is given. It is found that if we give the speed of a moving gray soliton, the density and phase shift decrease as the nonlinear power index increases. The energy of the moving gray soliton decreases with the increase of its speed for a given nonlinear power index. Finally, the numerical simulation is given to verify the analytical results.