三阶非线性效应对边界限制的自聚焦振荡型响应函数系统中二次孤子的影响

Influence of cubic nonlinearity effect on quadratic solitons in boundary-constrained self-focusing oscillatory response function system

面向边界限制的自聚焦振荡型响应函数系统建立了二阶和三阶非线性共同作用时的二次孤子理论模型.在此基础上,通过数值模拟对有三阶非线性效应和无三阶非线性效应情况下的孤子解进行对比,研究了三阶非线性效应对该系统中二次孤子的影响.结果表明,在孤子波形方面,三阶非线性影响较小,仅轻微改变其横向分布.但是在孤子的存在区间方面,三阶非线性导致孤子只存在于强非局域和一般非局域情况,在弱非局域情况下找不到相应的孤子.此外,三阶非线性的存在还减少了孤子拓展半周期的个数.而在孤子稳定性方面,三阶非线性则缩减了孤子的稳定区间.

In this paper,we theoretically study the influence of cubic nonlinearity effect on quadratic solitons in the boundary-constrained self-focusing oscillatory response function system.Based on the Newton iteration approach,we numerically solve the nonlinear coupled-wave equations with both quadratic and cubic nonlinearity.Moreover,a family of quadratic solitons is obtained.By comparing the quadratic solitons with both quadratic and cubic nonlinearity with those with only quadratic nonlinearity,we find that the cubic nonlinearity changes the transverse distribution of the soliton profiles only slightly.However,because of the existence of the cubic nonlinearity,quadratic solitons can be found only in the strongly nonlocal case and general nonlocal case,and they cannot be found in the weakly nonlocal case,in which the quadratic solitons with only quadratic nonlinearity can be found.In addition,the existence of cubic nonlinearity reduces the number of extended half-periods of the quadratic solitons.Through the linear stability analysis of the obtained soliton solutions,it is found that the stability intervals of solitons are also shrunk due to the existence of the cubic nonlinearity.The results of the linear stability analysis are verified by the numerical simulations of soliton propagations through using the split-step Fourier method.