Abstract We derive conservation laws for so-called generalized nonlinear Schrodinger equation (GNLSE), which describes a propagation of super-short femtosecond pulse in a medium with cubic nonlinear response in the framework of slowly-evolving-wave approximation (SEWA). We take into account the beam diffraction, the pulse spreading due to second order dispersion, the pulse self-steepening, as well as mixed derivatives of the pulse envelope. Such nonlinear interaction of the laser pulse with a medium is widely investigated by many authors because various substances manifest a cubic nonlinear response of medium in various laser systems. However, until present time the conservation laws (integrals of motion) of the GNLSE are absent. For their deriving we propose a novel transform of the GNLSE. It results in an equation containing neither the derivative of a term describing the nonlinear response of medium nor mixed derivatives of a complex amplitude. In new variables, the femtosecond pulse propagation is described by three equations containing only the linear differential operators. Using this transform, the conservation laws for a problem under consideration are found out. We claim that for avoiding a non-physical modulation instability of a laser pulse propagation it is necessary to satisfy to a spectral invariant at the frequency, which is singular one in the Fourier space. This frequency is inherent to the GNLSE. The conservation laws allow developing the conservative finite-difference schemes that preserve difference analogs of these laws.

中文翻译：

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具有三次非线性的广义非线性薛定谔方程描述的飞秒脉冲传播守恒定律

摘要 我们推导出了所谓广义非线性薛定谔方程 (GNLSE) 的守恒定律，该方程描述了超短飞秒脉冲在慢波近似 (SEWA) 框架中具有三次非线性响应的介质中的传播。我们考虑了光束衍射、由于二阶色散引起的脉冲扩展、脉冲自陡化以及脉冲包络的混合导数。许多作者广泛研究了激光脉冲与介质的这种非线性相互作用，因为各种物质在各种激光系统中表现出介质的三次非线性响应。然而，直到现在，GNLSE 的守恒定律（运动积分）都不存在。对于他们的推导，我们提出了 GNLSE 的新变换。它导致方程既不包含描述介质非线性响应的项的导数，也不包含复振幅的混合导数。在新变量中，飞秒脉冲传播由三个仅包含线性微分算子的方程描述。使用这种变换，可以找出所考虑问题的守恒定律。我们声称，为了避免激光脉冲传播的非物理调制不稳定性，必须满足频率处的光谱不变量，这在傅立叶空间中是奇异的。该频率是 GNLSE 固有的。守恒定律允许开发保守的有限差分方案，以保留这些定律的差分类似物。