A-Special-Non-linear-Equation-of-Mathematical-Physics-–the-Equation-of-the-Curvilinear-Electromagnetic-Wave.pdfVIP

1 A Special Non-linear Equation of Mathematical Physics – the Equation of the Curvilinear Electromagnetic Wave Alexander G. Kyriakos Saint-Petersburg State Institute of Technology, St. Petersburg, Russia Present address: Athens, Greece, e-mail: agkyriak@ Abstract A special non-linear equation of curvilinear electromagnetic wave is presented. The particularity of this equation lies in the fact that in matrix form it is mathematically equivalent to the Dirac electron equation. It is shown that the solution of this equation is the motion of the plane- polarized electromagnetic wave on a circular trajectory. It is also shown, that such twirled wave can be considered as a massive charge particle with a spin of one half, similar to electron. The non-linear equation and its Lagrangian of these EM particles are founded. 1.0. Basic supposition Let us suppose that under some conditions a linear electromagnetic wave (photon) is able to twirl and move along the closed curvilinear trajectory. As is known according to Maxwell-Lorentz electromagnetic theory [1] those conditions appear when electromagnetic wave propagates in the very strong external electromagnetic field. Below we translate this supposition on the mathematics language and show that in matrix form the non-linear equations of such curvilinear waves mathematically fully coincides with quantum equations. 2.0. Linear electromagnetic wave equation in the matrix form Let us consider the plane electromagnetic wave moving, for example, on y - axis (fig.1): Fig. 1 where the electric and magnetic fields can be written in the complex form as: