arXiv:1208.3616v3 [physics.optics] 5 Sep 2014 Electromagnetic shock wave in nonlinear vacuum: Exact solution Lubomir M. Kovachev, 1,* Daniela A. Georgieva, 2 and Kamen L. Kovachev 1 1 Institute of Electronics, Bulgarian Academy of Sciences, Tzarigradcko shossee 72,1784 Sofia, Bulgaria, 2 Faculty of Applied Mathematics and Computer Science, Technical University of Sofia, 8, Kliment Ohridski Blvd., 1000 Sofia, Bulgaria * Corresponding author: lubomirkovach@yahoo.com Compiled March 1, 2018 An analytical approach to the theory of electromagnetic waves in nonlinear vacuum is developed. The evolution of the pulse is governed by a system of nonlinear wave vector equations. Exact solution with its own angular momentum in form of a shock wave is obtained. c  2018 Optical Society of America OCIS codes: 190.5940, 260.5950. Contemporary hight-power laser facilities can gener- ate optical pulses with intensities of the order of 10 22 W/cm 2 . At the same time the critical power for obser- vation self-action effects due to virtual electron-positron pairs is of order [1–3] P cr = λ 2 /8n 0 n 2 =2.5 − 4.4 × 10 24 W, at a wavelength 1 µm. Thus, for a laser pulse with waist r ⊥ =1 mm the corresponding intensity be- comes I vac cr = P cr /r 2 ⊥ =2.5 − 4.4 × 10 26 W/cm 2 , which is above the range of the new high-power lasers. The non- linear addition to the refractive index in vacuum depends also on the magnetic field. That is why new different nonlinear effects can be expected. There are not only self-action effects, but also vacuum birefringence [4, 6], different kinds of four wave interaction [5,7,8] and higher order harmonic generation [9]. In this paper we shall in- vestigate the self-action effect only for intensities of the order of I vac cr . Euler, Heisenberg and Kockel [10,11] predicted intrin- sic nonlinearity of the electromagnetic vacuum due to the electron-positron nonlinear polarization. The clas- sical field-dependent nonlinear vacuum dielectric tensor can be written in the form ǫ ik = δ ik + 7e 4 ¯ h 45πm 4 c 7  2  |  E| 2 −|  B| 2  δ ik +7B i B k  , (1) where a complex form of presenting of the electrical E i and magnetic B i components is used. Note that the term containing B i B k vanishes, when a localized electromag- netic wave with only one magnetic component B l is in- vestigated. The dielectric response relevant to such op- tical pulse is thus ǫ ik = δ ik + 14e 4 ¯ h 45πm 4 c 7  |  E| 2 −|  B| 2  δ ik . (2) In the case when the spectral width of a pulse △k z ex- ceeds the values of the main wave-vector, i.e. △k z ≃ k 0 , the system of amplitude equations can be reduced to wave type [12] and in nonlinear vacuum becomes Δ  E − 1 c 2 ∂ 2  E ∂t 2 + γ (|  E| 2 −|  B| 2 )|  E =0 Δ  B − 1 c 2 ∂ 2  B ∂t 2 + γ (|  E| 2 −|  B| 2 )|  B =0, (3) where γ = 7k 2 0 e 4 ¯ h 90πm 4 c 7 and  E,  B are the amplitude func- tions. Initially, we can write the components of the elec- trical and magnetic fields as a vector sum of circular and linear components E z ; E c = iE x − E y ; B l = −B z . Thus (3) is transformed in the following scalar system of equations ΔE z − 1 c 2 ∂ 2 E z ∂t 2 + γ (|E z | 2 + |E c | 2 −|B l | 2 )E z =0 ΔE c − 1 c 2 ∂ 2 E c ∂t 2 + γ (|E z | 2 + |E c | 2 −|B l | 2 )E c =0 (4) ΔB l − 1 c 2 ∂ 2 B l ∂t 2 + γ (|E z | 2 + |E c | 2 −|B l | 2 )B l =0. Let us now parameterize the 3D +1 space-time through pseudospherical coordinates (r, τ, θ, ϕ): z = r cosh(τ ) cos(θ), y = r cosh(τ ) sin(θ) sin(ϕ), x = r cosh(τ ) sin(θ) cos(ϕ) and ict = r sinh(τ ), where r =  x 2 + y 2 + z 2 − c 2 t 2 . After calculations the correspond- ing d’Alambert operator in pseudospherical coordinates becomes [13] △− 1 c 2 ∂ 2 ∂t 2 = 3 r ∂ ∂r + ∂ 2 ∂r 2 − 1 r 2 ∂ 2 ∂τ 2 − 2 tanh τ r 2 ∂ ∂τ + 1 r 2 cosh 2 τ △ θ,ϕ , (5) where with △ θ,ϕ is denoted the angular part of the usual Laplace operator △ θ,ϕ = 1 sin θ ∂ ∂θ  sin θ ∂ ∂θ  + 1 sin 2 θ ∂ 2 ∂ϕ 2 . (6) The system of equations (4) in pseudo-spherical coordi- nates becomes 1