Abstract

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. An exact solution with its own angular momentum in the form of a shock wave is obtained.

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References

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  1. R. W. Boyd, Nonlinear Optics (Academic, 2003).
  2. G. A. Mourou, T. Tajima, and S. V. Bulanov, Rev. Mod. Phys. 78, 309 (2006).
    [CrossRef]
  3. S. V. Bulanov, T. Zh. Esirkepov, D. Habs, F. Pegoraro, and T. Tajima, Eur. Phys. J. D 55, 483 (2009).
    [CrossRef]
  4. N. N. Rosanov, J. Exp. Theor. Phys. 76, 991 (1993).
  5. N. N. Rosanov, J. Exp. Theor. Phys. 86, 284 (1998).
    [CrossRef]
  6. J. J. Klein and B. P. Nigam, Phys. Rev. 135, B1279 (1964).
    [CrossRef]
  7. Y. J. Ding and A. E. Kaplan, Phys. Rev. Lett. 63, 2725 (1989).
    [CrossRef]
  8. M. Markund and J. Lundin, Eur. Phys. J. D 55, 319 (2009).
    [CrossRef]
  9. D. Tommasini, A. Ferrando, H. Michinel, and M. Seco, Phys. Rev. A 77, 042101 (2008).
    [CrossRef]
  10. A. M. Fedotov and N. B. Narozhny, Phys. Lett. A 362, 1 (2007).
    [CrossRef]
  11. H. Euler and K. Köckel, Naturewiss 23, 246 (1935).
    [CrossRef]
  12. W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936).
    [CrossRef]
  13. L. M. Kovachev, J. Mod. Opt. 56, 1797 (2009).
    [CrossRef]
  14. E L. Lokas, Acta Phys. Pol. B 26, 19 (1995).

2009

S. V. Bulanov, T. Zh. Esirkepov, D. Habs, F. Pegoraro, and T. Tajima, Eur. Phys. J. D 55, 483 (2009).
[CrossRef]

M. Markund and J. Lundin, Eur. Phys. J. D 55, 319 (2009).
[CrossRef]

L. M. Kovachev, J. Mod. Opt. 56, 1797 (2009).
[CrossRef]

2008

D. Tommasini, A. Ferrando, H. Michinel, and M. Seco, Phys. Rev. A 77, 042101 (2008).
[CrossRef]

2007

A. M. Fedotov and N. B. Narozhny, Phys. Lett. A 362, 1 (2007).
[CrossRef]

2006

G. A. Mourou, T. Tajima, and S. V. Bulanov, Rev. Mod. Phys. 78, 309 (2006).
[CrossRef]

1998

N. N. Rosanov, J. Exp. Theor. Phys. 86, 284 (1998).
[CrossRef]

1995

E L. Lokas, Acta Phys. Pol. B 26, 19 (1995).

1993

N. N. Rosanov, J. Exp. Theor. Phys. 76, 991 (1993).

1989

Y. J. Ding and A. E. Kaplan, Phys. Rev. Lett. 63, 2725 (1989).
[CrossRef]

1964

J. J. Klein and B. P. Nigam, Phys. Rev. 135, B1279 (1964).
[CrossRef]

1936

W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936).
[CrossRef]

1935

H. Euler and K. Köckel, Naturewiss 23, 246 (1935).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2003).

Bulanov, S. V.

S. V. Bulanov, T. Zh. Esirkepov, D. Habs, F. Pegoraro, and T. Tajima, Eur. Phys. J. D 55, 483 (2009).
[CrossRef]

G. A. Mourou, T. Tajima, and S. V. Bulanov, Rev. Mod. Phys. 78, 309 (2006).
[CrossRef]

Ding, Y. J.

Y. J. Ding and A. E. Kaplan, Phys. Rev. Lett. 63, 2725 (1989).
[CrossRef]

Esirkepov, T. Zh.

S. V. Bulanov, T. Zh. Esirkepov, D. Habs, F. Pegoraro, and T. Tajima, Eur. Phys. J. D 55, 483 (2009).
[CrossRef]

Euler, H.

W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936).
[CrossRef]

H. Euler and K. Köckel, Naturewiss 23, 246 (1935).
[CrossRef]

Fedotov, A. M.

A. M. Fedotov and N. B. Narozhny, Phys. Lett. A 362, 1 (2007).
[CrossRef]

Ferrando, A.

D. Tommasini, A. Ferrando, H. Michinel, and M. Seco, Phys. Rev. A 77, 042101 (2008).
[CrossRef]

Habs, D.

S. V. Bulanov, T. Zh. Esirkepov, D. Habs, F. Pegoraro, and T. Tajima, Eur. Phys. J. D 55, 483 (2009).
[CrossRef]

Heisenberg, W.

W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936).
[CrossRef]

Kaplan, A. E.

Y. J. Ding and A. E. Kaplan, Phys. Rev. Lett. 63, 2725 (1989).
[CrossRef]

Klein, J. J.

J. J. Klein and B. P. Nigam, Phys. Rev. 135, B1279 (1964).
[CrossRef]

Köckel, K.

H. Euler and K. Köckel, Naturewiss 23, 246 (1935).
[CrossRef]

Kovachev, L. M.

L. M. Kovachev, J. Mod. Opt. 56, 1797 (2009).
[CrossRef]

Lokas, E L.

E L. Lokas, Acta Phys. Pol. B 26, 19 (1995).

Lundin, J.

M. Markund and J. Lundin, Eur. Phys. J. D 55, 319 (2009).
[CrossRef]

Markund, M.

M. Markund and J. Lundin, Eur. Phys. J. D 55, 319 (2009).
[CrossRef]

Michinel, H.

D. Tommasini, A. Ferrando, H. Michinel, and M. Seco, Phys. Rev. A 77, 042101 (2008).
[CrossRef]

Mourou, G. A.

G. A. Mourou, T. Tajima, and S. V. Bulanov, Rev. Mod. Phys. 78, 309 (2006).
[CrossRef]

Narozhny, N. B.

A. M. Fedotov and N. B. Narozhny, Phys. Lett. A 362, 1 (2007).
[CrossRef]

Nigam, B. P.

J. J. Klein and B. P. Nigam, Phys. Rev. 135, B1279 (1964).
[CrossRef]

Pegoraro, F.

S. V. Bulanov, T. Zh. Esirkepov, D. Habs, F. Pegoraro, and T. Tajima, Eur. Phys. J. D 55, 483 (2009).
[CrossRef]

Rosanov, N. N.

N. N. Rosanov, J. Exp. Theor. Phys. 86, 284 (1998).
[CrossRef]

N. N. Rosanov, J. Exp. Theor. Phys. 76, 991 (1993).

Seco, M.

D. Tommasini, A. Ferrando, H. Michinel, and M. Seco, Phys. Rev. A 77, 042101 (2008).
[CrossRef]

Tajima, T.

S. V. Bulanov, T. Zh. Esirkepov, D. Habs, F. Pegoraro, and T. Tajima, Eur. Phys. J. D 55, 483 (2009).
[CrossRef]

G. A. Mourou, T. Tajima, and S. V. Bulanov, Rev. Mod. Phys. 78, 309 (2006).
[CrossRef]

Tommasini, D.

D. Tommasini, A. Ferrando, H. Michinel, and M. Seco, Phys. Rev. A 77, 042101 (2008).
[CrossRef]

Acta Phys. Pol. B

E L. Lokas, Acta Phys. Pol. B 26, 19 (1995).

Eur. Phys. J. D

S. V. Bulanov, T. Zh. Esirkepov, D. Habs, F. Pegoraro, and T. Tajima, Eur. Phys. J. D 55, 483 (2009).
[CrossRef]

M. Markund and J. Lundin, Eur. Phys. J. D 55, 319 (2009).
[CrossRef]

J. Exp. Theor. Phys.

N. N. Rosanov, J. Exp. Theor. Phys. 76, 991 (1993).

N. N. Rosanov, J. Exp. Theor. Phys. 86, 284 (1998).
[CrossRef]

J. Mod. Opt.

L. M. Kovachev, J. Mod. Opt. 56, 1797 (2009).
[CrossRef]

Naturewiss

H. Euler and K. Köckel, Naturewiss 23, 246 (1935).
[CrossRef]

Phys. Lett. A

A. M. Fedotov and N. B. Narozhny, Phys. Lett. A 362, 1 (2007).
[CrossRef]

Phys. Rev.

J. J. Klein and B. P. Nigam, Phys. Rev. 135, B1279 (1964).
[CrossRef]

Phys. Rev. A

D. Tommasini, A. Ferrando, H. Michinel, and M. Seco, Phys. Rev. A 77, 042101 (2008).
[CrossRef]

Phys. Rev. Lett.

Y. J. Ding and A. E. Kaplan, Phys. Rev. Lett. 63, 2725 (1989).
[CrossRef]

Rev. Mod. Phys.

G. A. Mourou, T. Tajima, and S. V. Bulanov, Rev. Mod. Phys. 78, 309 (2006).
[CrossRef]

Z. Phys.

W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936).
[CrossRef]

Other

R. W. Boyd, Nonlinear Optics (Academic, 2003).

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Figures (2)

Fig. 1.
Fig. 1.

Time evolution of the intensity profile I of the spherically symmetric analytical solution (17) of the linear wave equation (18) ( r 0 = 1 and c = 1 ). The initially ( t = 0 ) localized amplitude function [Fig. 1(a)] decreases with the generation of outside and inside fronts [Fig. 1(b)], while the energy density distributes over the whole space for a finite time ( t = 10 ).

Fig. 2.
Fig. 2.

Time evolution of the intensity (16) of the solution (15) of the nonlinear system of equations (4) in Euler’s vacuum ( c = 1 ) for t = 0 and t = 10 correspondingly. The nonlinear wave demonstrates entirely different evolution than the linear spherical one: the shock wave preserves its amplitude maximum and self compresses in r direction.

Equations (18)

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ϵ i k = δ i k + 7 e 4 45 π m 4 c 7 [ 2 ( | E⃗ | 2 | B⃗ | 2 ) + 7 B i B k ] ,
ϵ i k = δ i k + 14 e 4 45 π m 4 c 7 ( | E⃗ | 2 | B⃗ | 2 ) .
Δ E⃗ 1 c 2 2 E⃗ t 2 + γ ( | E⃗ | 2 | B⃗ | 2 ) | E⃗ = 0 , Δ B⃗ 1 c 2 2 B⃗ t 2 + γ ( | E⃗ | 2 | B⃗ | 2 ) | B⃗ = 0 ,
Δ E z 1 c 2 2 E z t 2 + γ ( | E z | 2 + | E c | 2 | B l | 2 ) E z = 0 , Δ E c 1 c 2 2 E c t 2 + γ ( | E z | 2 + | E c | 2 | B l | 2 ) E c = 0 , Δ B l 1 c 2 2 B l t 2 + γ ( | E z | 2 + | E c | 2 | B l | 2 ) B l = 0.
Δ 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 Δ θ , φ ,
Δ θ , φ = 1 sin θ θ ( sin θ θ ) + 1 sin 2 θ 2 φ 2 .
3 r E z r + 2 E z r 2 1 r 2 2 E z τ 2 2 tanh τ r 2 E z τ + 1 r 2 cosh 2 τ Δ θ , φ E z + γ ( | E z | 2 + | E c | 2 | B l | 2 ) E z = 0 , 3 r E c r + 2 E c r 2 1 r 2 2 E c τ 2 2 tanh τ r 2 E c τ + 1 r 2 cosh 2 τ Δ θ , φ E c + γ ( | E z | 2 + | E c | 2 | B l | 2 ) E c = 0 , 3 r B l r + 2 B l r 2 1 r 2 2 B l τ 2 2 tanh τ r 2 B l τ + 1 r 2 cosh 2 τ Δ θ , φ B l + γ ( | E z | 2 + | E c | 2 | B l | 2 ) B l = 0.
E i ( r , τ , θ , φ ) = R ( r ) T i ( τ ) Y i ( θ , φ ) , B l ( r , τ , θ , φ ) = R ( r ) T l ( τ ) Y l ( θ , φ ) ,
| T z | 2 | Y z | 2 + | T c | 2 | Y c | 2 | T l | 2 | Y l | 2 = const .
3 r R r + 2 R r 2 A i r 2 R + γ | R | 2 R = 0 ,
R = sech ( ln ( r α ) ) r ,
cosh 2 τ d 2 T i d τ 2 + 2 sinh τ cosh τ d T i d τ + ( C i A i cosh 2 τ ) T i , = 0 ,
Δ θ , φ Y i Y i = 2 ,
E z ( r , τ , θ ) = sech ( ln ( r ± 2 ) ) r cosh τ cos θ , E c ( r , τ , θ , φ ) = sech ( l n ( r ± 2 ) ) r cosh τ sin θ exp ( i φ ) , B l ( r , τ ) = sech ( l n ( r ± 2 ) ) r sinh τ .
E z = 2 z r 4 + 1 , E c = 2 ( x + i y ) r 4 + 1 , B l = 2 c t r 4 + 1 ,
I ( x , y , z , t ) = 4 ( x 2 + y 2 + z 2 + c 2 t 2 ) [ ( x 2 + y 2 + z 2 c 2 t 2 ) 2 + 1 ] 2 .
E ( x , y , z , t ) = 1 / [ r 2 r 0 2 + ( 1 + i c t r 0 ) 2 ]
Δ E = 1 c 2 2 E t 2 ,

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