In this article, we consider a one-dimensional model of electromagnetic pulse propagation in isotropic media, taking into account a nonlinearity of the third order. We introduce a method for Maxwell's equation transformation on the basis of a complete set of projecting operators. The operators correspond to wave dispersion branches including the direction of propagation. As the simplest result of applying the method, we derive a system of equations describing one-dimensional dynamics of pulses of opposite directions of propagation without dispersion account. The corresponding self-action equations are extracted. Next we introduce a dispersion and show how the operators change.Through applications in such a manner, the generalized short pulse equations (SPE) of Shafer and Wayne are obtained for both propagation directions. The effects of the unidirectional pulse interaction are discussed.
We present the application of projection operator methods to solving the problem of the propagation and interaction of short optical pulses of different polarizations and directions in a nonlinear dispersive medium. We restrict ourselves by the caseof one-dimensional theory, taking into account material dispersion and Kerr nonlinearity. The construction of operators is delivered in two variants: for the Cauchy problem and for the given boundary regime at the initial point of the propagation half-space x > 0. As a result, we derive a system of four first-order differential equations that describe the interaction of four specified modes. In the construction of projection operators, we use an expansion with respect to a small parameter that characterizes the material dispersion, amplitude, and pulse profile. The results are compared with the vector short pulse equations (VSPE) as well as ours previous, the one for opposite propagated pulses.while material equations in operator form D y ¼E y , D z 1 E z ,H y ¼ B y , andH z ¼ B z close the description.
We consider the propagation of electromagnetic pulses in isotropic media taking a third-order nonlinearity into account. We develop a method for transforming Maxwell's equations based on a complete set of projection operators corresponding to wave-dispersion branches (in a waveguide or in matter) with the propagation direction taken into account. The most important result of applying the method is a system of equations describing the one-dimensional dynamics of pulses propagating in opposite directions without accounting for dispersion. We derive the corresponding self-action equations. We thus introduce dispersion in the media and show how the operators change. We obtain generalized Schäfer-Wayne short-pulse equations accounting for both propagation directions. In the three-dimensional problem, we focus on optic fibers with dispersive matter, deriving and numerically solving equations of the waveguide-mode interaction. We discuss the effects of the interaction of unidirectional pulses. For the coupled nonlinear Schrödinger equations, we discuss a concept of numerical integrability and apply the developed calculation schemes.
In this article we consider one dimensional model of an ultra short pulse propagation in isotropic dispersionless media taking into account a nonlinearity of the third order. We introduce a method for Maxwell's equations transformation based on a complete set of projecting operators. The operators generally correspond wave dispersion branches. As a simplest result of the method application we derive a system of equations describing dynamics of ultrashort pulses of opposite directions of propagation. We show that in such way the generalized Short Pulse Equations of Shafer and Wayne is obtained if the only directed wave is initialized. The effects of the unidirectional pulses interaction are traced.
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