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Variational principles, Lie point symmetries, and similarity solutions of the vector Maxwell equations in nonlinear optics. (English) Zbl 1076.78011
The authors formulate the vector Maxwell equations coupled to a single Lorentz oscillator with instantaneous Kerr nonlinearity in terms of Lagrangian and Hamiltonian variational principles. Four conservation laws for these equations are obtained. The traveling wave solutions and the kink solutions are investigated. It was obtained that the smooth periodic traveling wave solutions develop very steep electric field gradients as the dimensionless wave velocity tends to 1 from above, and the method of extending them into the slower velocity regime is demonstrated. It was also obtained that the kink and anti-kink solutions exist only for a restricted range of traveling wave velocities. These solutions may be used in switching devices or in optical computing.
78A60 Lasers, masers, optical bistability, nonlinear optics
35A30 Geometric theory, characteristics, transformations in context of PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q60 PDEs in connection with optics and electromagnetic theory
58E30 Variational principles in infinite-dimensional spaces
78A02 Foundations in optics and electromagnetic theory
Full Text: DOI
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