We consider nonlinear optical systems describing the propagation of beams in Kerr-type nonlinear media, experiencing diffraction in transverse and longitudinal directions. The first model investigated, under nonparaxial approximation, is the complex nonlinear Helmholtz equation, recast into a coupled, real system of partial differential equations. We construct and apply its conserved vectors to determine exact solutions. This approach of a double reduction combines a point symmetry with a particular conservation law to enact a reduction and derive solutions. A second model is also studied, that is related to the Maxwell’s equations under paraxial approximation — a version of the nonlinear Schrödinger equation. We show that when the paraxial effect vanishes, a number of additional exact solutions and conservation laws are admitted.

中文翻译：

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具有类克尔非线性的平面波导中超宽光束传播的解决方案

我们考虑描述光束在克尔型非线性介质中传播的非线性光学系统，在横向和纵向上经历衍射。在非近轴近似下研究的第一个模型是复杂的非线性亥姆霍兹方程，它被重铸成一个耦合的、真实的偏微分方程系统。我们构建并应用其保守向量来确定精确的解决方案。这种双重归约方法将点对称性与特定的守恒定律相结合，以制定归约并得出解决方案。还研究了第二个模型，它与近轴近似下的麦克斯韦方程组有关——非线性薛定谔方程的一个版本。我们表明，当近轴效应消失时，会接受一些额外的精确解和守恒定律。