高阶非线性光传输方程的精确对称性和近似对称性
Exact and approximate symmetries for light propagation equations with higher order nonlinearity
摘要：
首次构造了（1+1）维程函方程的精确解析解，其中折射率是强度的饱和函数。证明了解的坍缩性，得到了自聚焦位置的解析表达式，并基于近似的Lie对称群构造了任意非线性折射率的程函方程的解.精确解和近似解之间的比较。得到了任意折射率和初始光强分布的（1+2）维非线性薛定谔方程的近似解。考虑了由克尔折射和多光子电离组成的折射率的一种特殊情况。证明了束流不仅可以在束轴处发生坍缩，而且可以在束轴周围的离轴环形区域内发生坍缩，得到了区分这两种情况的解析条件，并给出了自聚焦位置的显式表达式。
英文摘要：
For the first time exact analytical solutions to the eikonal equations in (1+1) dimensions with a refractive index being a saturated function of intensity are constructed. It is demonstrated that the solutions exhibit collapse; an explicit analytical expression for the self-focusing position, where the intensity tends to infinity, is found. Based on an approximated Lie symmetry group, solutions to the eikonal equations with arbitrary nonlinear refractive index are constructed. Comparison between exact and approximate solutions is presented. Approximate solutions to the nonlinear Schrodinger equation in (1+2) dimensions with arbitrary refractive index and initial intensity distribution are obtained. A particular case of refractive index consisting of Kerr refraction and multiphoton ionization is considered. It is demonstrated that the beam collapse can take place not only at the beam axis but also in an off-axis ring region around it. An analytical condition distinguishing these two cases is obtained and explicit formula for the self-focusing position is presented.
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