Conservation laws of femtosecond pulse propagation described by generalized nonlinear Schrödinger equation with cubic nonlinearity

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.