Geodesic
Multimode systems of nonlinear equations: Derivation, integrability,
Teoretičeskaâ i matematičeskaâ fizika, Tome 168 (2011) no. 1, pp. 138-150

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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.
Keywords: projection-operator method, multimode waveguide, coupled nonlinear Schrödinger equations, short-pulse equation. 
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     author = {M. Kuszner and S. B. Leble and B. Reichel},
     title = {Multimode systems of nonlinear equations: {Derivation,} integrability,},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     publisher = {mathdoc},
     volume = {168},
     number = {1},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2011_168_1_a10/}
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M. Kuszner; S. B. Leble; B. Reichel. Multimode systems of nonlinear equations: Derivation, integrability,. Teoretičeskaâ i matematičeskaâ fizika, Tome 168 (2011) no. 1, pp. 138-150. http://geodesic.mathdoc.fr/item/TMF_2011_168_1_a10/