基于二维广义非线性薛定谔方程的逆级联现象  

Inverse cascade based on a two-dimensional generalized nonlinear Schrödinger equation

作　　者：崔少燕 郁明阳[2,3] CUI Shao-yan;YU Ming-yang(School of Mathematics and Statistics Science,Ludong University,Yantai 264025;Institute for Fusion Theory and Simulation,Zhejiang University,Hangzhou 310007;College of Engineering Physics,Shenzhen Technology University,Shenzhen 518118)

机构地区：[1]鲁东大学数学与统计科学学院,烟台264025 [2]浙江大学聚变理论与模拟中心,杭州310007 [3]深圳技术大学工程物理学院,深圳518118

出　　处：《核聚变与等离子体物理》2023年第3期352-358,共7页Nuclear Fusion and Plasma Physics

基　　金：国家自然科学基金(11105065,11371183);上海光机所强场激光物理国家重点实验室开放基金(2011GB105000)。

摘　　要：在外部势场和初始空间局部有限振幅脉冲给定的情况下,基于二维广义非线性薛定谔方程i∂_(t)E+p▽^(2)E+[V(x,y)+q|E|^(2)]E=0,研究了波的不稳定性随时间的演化。波的不稳定性演化主要依赖于方程里的群色散系数p=p_(r)+ip_(i)和非线性系数q=q_(r)+iq_(i),数值结果发现系统会出现调制不稳定性、波坍缩、逆级联以及整个空间的湍流态。当p=3.5+0.5i,q=q_(r)+iq_(i)时,数值结果发现系统在出现部分逆级联之后波场的能量主要聚集在波矢量k空间中半径|k|≥100的短波区域,同时形成了以区域中心为圆心,半径|k|≈100的圆形相对稳定区域。当粘性阻尼系数p_(i)在[0.1,1.0]时,数值结果发现产生逆级联的部分区域随着p_(i)(<1.1)的不断增大而逐渐缩小,相对稳定的圆域半径从零开始逐渐增大。因此,粘性系数p_(i)相当于系统中的一个调节开关,调整它在一定范围(p_(i)≤1)的取值可以控制逆级联的进程与最终结果。For a fixed external potential field,the evolution of a highly localized finite-amplitude initial pulse is investigated by numerically solving the corresponding two-dimensional generalized nonlinear Schrödinger equation i∂_(t)E+p▽^(2)E+[V(x,y)+q|E|^(2)]E=0.The evolution depends crucially on the complex group dispersion coefficient p and the complex nonlinearity coefficient q of the equation.It is found that,in general,the wave field first suffers modulational instability,followed by abrupt collapse into a turbulent state containing the shortest wavelength modes allowed in the system.The latter is in turn followed by inverse cascade of the shortest wavelength modes back to the longer wavelength ones,until a statistically stationary turbulent state is reached.For p=3.5+0.5i and q=8.0+0.9i,it is found that the energy is mainly concentrated in the wave vector region|k|≥100 of the energy spectrum,i.e.,the inverse cascade is limited to the shorter wavelength modes with|k|≥100.Furthermore,for the imaginary part p_(i)(which represents viscous damping)of the group dispersion coefficient in the range 0.1<p_(i)<1.0,it is found that the region of inverse cascade gradually shrinks with increase of p_(i)(<1.1).That is,in the range considered,the viscous damping coefficient p_(i)acts like a control switch,and it can regulate the degree of inverse cascade and the final results.

关 键 词：非线性薛定谔方程 逆级联 湍流 粘性阻尼 

分 类 号：O5342[理学—等离子体物理]