两个非线性几何光学模型涡旋解的存在性
发布时间：2018-08-07 14:52
【摘要】：光学涡旋在现代光学物理中有着重要的作用.本文研究了来自于非线性几何光学中的两个模型,第一个模型是由两束激光耦合得到的Schrodinger方程组,我们首先利用直接变分法和山路引理建立了该模型的一系列稳态解的存在性定理;其次,利用约束变分法建立了该模型的径向对称正解的存在性定理,并给出了波传播常数β的下界估计;最后,利用山路引理建立了鞍点解的存在性定理.第二个模型是两束光耦合传播的非线性Kerr模型,我们证明了该模型无非零稳态解,并利用约束变分法建立了该模型涡旋解的存在性定理及参数λ的范围.
[Abstract]:Optical vortex plays an important role in modern optical physics. In this paper, two models from nonlinear geometric optics are studied. The first model is the Schrodinger equations obtained by coupling two laser beams. In this paper, we first establish a series of existence theorems of the steady-state solutions of the model by using direct variational method and mountain pass Lemma, and then, by using the constrained variational method, we establish the existence theorems of the radial symmetric positive solutions of the model. The lower bound estimate of wave propagation constant 尾 is given. Finally, the existence theorem of saddle point solution is established by using mountain pass Lemma. The second model is a nonlinear Kerr model for the coupled propagation of two beams of light. We prove that the model has zero steady-state solution and establish the existence theorem of vortex solution and the range of parameter 位 by using the constrained variational method.
【学位授予单位】：河南大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175;O435
本文编号：2170380
[Abstract]:Optical vortex plays an important role in modern optical physics. In this paper, two models from nonlinear geometric optics are studied. The first model is the Schrodinger equations obtained by coupling two laser beams. In this paper, we first establish a series of existence theorems of the steady-state solutions of the model by using direct variational method and mountain pass Lemma, and then, by using the constrained variational method, we establish the existence theorems of the radial symmetric positive solutions of the model. The lower bound estimate of wave propagation constant 尾 is given. Finally, the existence theorem of saddle point solution is established by using mountain pass Lemma. The second model is a nonlinear Kerr model for the coupled propagation of two beams of light. We prove that the model has zero steady-state solution and establish the existence theorem of vortex solution and the range of parameter 位 by using the constrained variational method.
【学位授予单位】：河南大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175;O435
【参考文献】
相关硕士学位论文 前1条
1 陈筱;非线性几何光学中涡旋解的存在性[D];河南大学;2014年
，本文编号：2170380
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