Abstract

A model of steady-state X-shaped wave generation by a superluminal (supersonic) pointlike source infinitely moving along a straight line is extended to a more realistic causal scenario of a source pulse launched at time zero and propagating rectilinearly at a constant superluminal speed. In the case of an infinitely short (delta) pulse, the new model yields an analytical solution, corresponding to the propagation-invariant X-shaped wave clipped by a droplet-shaped support, which perpetually expands along the propagation and transversal directions, thus tending the droplet-shaped wave to the X-shaped one.

© 2012 Optical Society of America

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    [CrossRef]
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  10. I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagn. Res. 19, 1–48 (1998).
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  31. E. Recami, A. Castellino, G. D. Maccarrone, and M. Rodonò, “Considerations about the apparent ≪superluminal expansions≫ observed in astrophysics,” Nuovo Cimento B 93, 119–144 (1986).
    [CrossRef]
  32. E. Recami and G. D. Maccarrone, “Solving the ≪imaginary quantities≫ problem in superluminal Lorentz transformations,” Lett. Nuovo Cimento Ser. 2, 28, 151–157 (1980).
    [CrossRef]
  33. P. Caldirola, G. D. Maccarrone, and E. Recami, “Second contribution on solving the ≪imaginary quantities≫ problem in superluminal Lorentz transformations,” Lett. Nuovo Cimento Ser. 2, 29, 241–250 (1980).
    [CrossRef]
  34. V. V. Borisov and A. B. Utkin, “The transient electromagnetic field produced by a moving pulse of line current,” J. Phys. D: Appl. Phys. 28, 614–622 (1995).
    [CrossRef]
  35. M. Zamboni-Rached and E. Recami, “Subluminal wave bullets: exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
    [CrossRef]
  36. V. V. Borisov and A. B. Utkin, “Some solutions of the wave and Maxwell’s equations,” J. Math. Phys. 35, 3624 (1994).
    [CrossRef]
  37. E. Recami, “Superluminal motions? a bird’s-eye view of the experimental situation,” Found. Phys. 31, 1119–1135(2001).
    [CrossRef]
  38. A. B. Utkin, “Riemann-Volterra time-domain technique for waveguides: a case study for elliptic geometry,” Wave Motion 49, 347–363 (2012).
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2012 (1)

A. B. Utkin, “Riemann-Volterra time-domain technique for waveguides: a case study for elliptic geometry,” Wave Motion 49, 347–363 (2012).
[CrossRef]

2011 (2)

A. B. Utkin, “Mathieu progressive waves,” Commun. Theor. Phys. 56, 733–739 (2011).
[CrossRef]

J. Vieira, S. Martins, V. Pathak, R. Fonseca, W. Mori, and L. Silva, “Magnetic control of particle injection in plasma based accelerators,” Phys. Rev. Lett. 106, 225001 (2011).
[CrossRef]

2010 (3)

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

E. Arévalo, “Boosted X waves in nonlinear optical systems,” Phys. Rev. Lett. 104, 023902 (2010).
[CrossRef]

M. Zamboni-Rached, E. Recami, and I. M. Besieris, “Cherenkov radiation versus X-shaped localized waves,” J. Opt. Soc. Am. A 27, 928–934 (2010).
[CrossRef]

2009 (1)

M. Zamboni-Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A 79, 013816 (2009).
[CrossRef]

2008 (1)

M. Zamboni-Rached and E. Recami, “Subluminal wave bullets: exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
[CrossRef]

2007 (1)

S. C. Walker and W. A. Kuperman, “Cherenkov-Vavilov formulation of X waves,” Phys. Rev. Lett. 99, 244802 (2007).
[CrossRef]

2004 (4)

M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E 69, 066606 (2004).
[CrossRef]

E. Recami, M. Zamboni-Rached, and C. A. Dartora, “Localized X-shaped field generated by a superluminal electric charge,” Phys. Rev. E 69, 027602 (2004).
[CrossRef]

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69, 036612 (2004).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, and A. M. Attiya, “Bateman conformal transformations within the framework of the bidirectional spectral representation,” Prog. Electromagn. Res. 48, 201–231 (2004).
[CrossRef]

2003 (2)

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, and H. E. Hernandez-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef]

2001 (1)

E. Recami, “Superluminal motions? a bird’s-eye view of the experimental situation,” Found. Phys. 31, 1119–1135(2001).
[CrossRef]

2000 (1)

J. Salo, J. Fagerholm, A. Friberg, and M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62, 4261–4275 (2000).
[CrossRef]

1998 (2)

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagn. Res. 19, 1–48 (1998).
[CrossRef]

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

1997 (1)

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[CrossRef]

1995 (1)

V. V. Borisov and A. B. Utkin, “The transient electromagnetic field produced by a moving pulse of line current,” J. Phys. D: Appl. Phys. 28, 614–622 (1995).
[CrossRef]

1994 (1)

V. V. Borisov and A. B. Utkin, “Some solutions of the wave and Maxwell’s equations,” J. Math. Phys. 35, 3624 (1994).
[CrossRef]

1993 (1)

1986 (1)

E. Recami, A. Castellino, G. D. Maccarrone, and M. Rodonò, “Considerations about the apparent ≪superluminal expansions≫ observed in astrophysics,” Nuovo Cimento B 93, 119–144 (1986).
[CrossRef]

1982 (1)

A. O. Barut, G. D. Maccarrone, and E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509–533 (1982).
[CrossRef]

1980 (2)

E. Recami and G. D. Maccarrone, “Solving the ≪imaginary quantities≫ problem in superluminal Lorentz transformations,” Lett. Nuovo Cimento Ser. 2, 28, 151–157 (1980).
[CrossRef]

P. Caldirola, G. D. Maccarrone, and E. Recami, “Second contribution on solving the ≪imaginary quantities≫ problem in superluminal Lorentz transformations,” Lett. Nuovo Cimento Ser. 2, 29, 241–250 (1980).
[CrossRef]

1978 (1)

O. Gron, “Visual appearance of superluminal bodies,” Lett. Nuovo Cimento 23, 97–100 (1978).
[CrossRef]

1975 (1)

H. Lemke, “Light from sources moving faster than light,” Lett. Nuovo Cimento 12, 342–346 (1975).
[CrossRef]

Abdel-Rahman, M.

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagn. Res. 19, 1–48 (1998).
[CrossRef]

Abdollahpour, D.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

Arévalo, E.

E. Arévalo, “Boosted X waves in nonlinear optical systems,” Phys. Rev. Lett. 104, 023902 (2010).
[CrossRef]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, 2001).

Attiya, A. M.

I. M. Besieris, A. M. Shaarawi, and A. M. Attiya, “Bateman conformal transformations within the framework of the bidirectional spectral representation,” Prog. Electromagn. Res. 48, 201–231 (2004).
[CrossRef]

Barut, A. O.

A. O. Barut, G. D. Maccarrone, and E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509–533 (1982).
[CrossRef]

Besieris, I. M.

M. Zamboni-Rached, E. Recami, and I. M. Besieris, “Cherenkov radiation versus X-shaped localized waves,” J. Opt. Soc. Am. A 27, 928–934 (2010).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, and A. M. Attiya, “Bateman conformal transformations within the framework of the bidirectional spectral representation,” Prog. Electromagn. Res. 48, 201–231 (2004).
[CrossRef]

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagn. Res. 19, 1–48 (1998).
[CrossRef]

R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
[CrossRef]

Borisov, V. V.

V. V. Borisov and A. B. Utkin, “The transient electromagnetic field produced by a moving pulse of line current,” J. Phys. D: Appl. Phys. 28, 614–622 (1995).
[CrossRef]

V. V. Borisov and A. B. Utkin, “Some solutions of the wave and Maxwell’s equations,” J. Math. Phys. 35, 3624 (1994).
[CrossRef]

Caldirola, P.

P. Caldirola, G. D. Maccarrone, and E. Recami, “Second contribution on solving the ≪imaginary quantities≫ problem in superluminal Lorentz transformations,” Lett. Nuovo Cimento Ser. 2, 29, 241–250 (1980).
[CrossRef]

Castellino, A.

E. Recami, A. Castellino, G. D. Maccarrone, and M. Rodonò, “Considerations about the apparent ≪superluminal expansions≫ observed in astrophysics,” Nuovo Cimento B 93, 119–144 (1986).
[CrossRef]

Chatzipetros, A.

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagn. Res. 19, 1–48 (1998).
[CrossRef]

Conti, C.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef]

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 2 (Wiley, 1989).

Dartora, C. A.

E. Recami, M. Zamboni-Rached, and C. A. Dartora, “Localized X-shaped field generated by a superluminal electric charge,” Phys. Rev. E 69, 027602 (2004).
[CrossRef]

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, and H. E. Hernandez-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

Di Trapani, P.

M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E 69, 066606 (2004).
[CrossRef]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef]

Fagerholm, J.

J. Salo, J. Fagerholm, A. Friberg, and M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62, 4261–4275 (2000).
[CrossRef]

Fonseca, R.

J. Vieira, S. Martins, V. Pathak, R. Fonseca, W. Mori, and L. Silva, “Magnetic control of particle injection in plasma based accelerators,” Phys. Rev. Lett. 106, 225001 (2011).
[CrossRef]

Friberg, A.

J. Salo, J. Fagerholm, A. Friberg, and M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62, 4261–4275 (2000).
[CrossRef]

Gron, O.

O. Gron, “Visual appearance of superluminal bodies,” Lett. Nuovo Cimento 23, 97–100 (1978).
[CrossRef]

Hernandez-Figueroa, H. E.

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, and H. E. Hernandez-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 2 (Wiley, 1989).

Jedrkiewicz, O.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef]

Kuperman, W. A.

S. C. Walker and W. A. Kuperman, “Cherenkov-Vavilov formulation of X waves,” Phys. Rev. Lett. 99, 244802 (2007).
[CrossRef]

Lemke, H.

H. Lemke, “Light from sources moving faster than light,” Lett. Nuovo Cimento 12, 342–346 (1975).
[CrossRef]

Maccarrone, G. D.

E. Recami, A. Castellino, G. D. Maccarrone, and M. Rodonò, “Considerations about the apparent ≪superluminal expansions≫ observed in astrophysics,” Nuovo Cimento B 93, 119–144 (1986).
[CrossRef]

A. O. Barut, G. D. Maccarrone, and E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509–533 (1982).
[CrossRef]

P. Caldirola, G. D. Maccarrone, and E. Recami, “Second contribution on solving the ≪imaginary quantities≫ problem in superluminal Lorentz transformations,” Lett. Nuovo Cimento Ser. 2, 29, 241–250 (1980).
[CrossRef]

E. Recami and G. D. Maccarrone, “Solving the ≪imaginary quantities≫ problem in superluminal Lorentz transformations,” Lett. Nuovo Cimento Ser. 2, 28, 151–157 (1980).
[CrossRef]

Martins, S.

J. Vieira, S. Martins, V. Pathak, R. Fonseca, W. Mori, and L. Silva, “Magnetic control of particle injection in plasma based accelerators,” Phys. Rev. Lett. 106, 225001 (2011).
[CrossRef]

Mori, W.

J. Vieira, S. Martins, V. Pathak, R. Fonseca, W. Mori, and L. Silva, “Magnetic control of particle injection in plasma based accelerators,” Phys. Rev. Lett. 106, 225001 (2011).
[CrossRef]

Nobrega, K. Z.

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, and H. E. Hernandez-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

Papazoglou, D. G.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

Pathak, V.

J. Vieira, S. Martins, V. Pathak, R. Fonseca, W. Mori, and L. Silva, “Magnetic control of particle injection in plasma based accelerators,” Phys. Rev. Lett. 106, 225001 (2011).
[CrossRef]

Piskarskas, A.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef]

Porras, M. A.

M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E 69, 066606 (2004).
[CrossRef]

Recami, E.

M. Zamboni-Rached, E. Recami, and I. M. Besieris, “Cherenkov radiation versus X-shaped localized waves,” J. Opt. Soc. Am. A 27, 928–934 (2010).
[CrossRef]

M. Zamboni-Rached and E. Recami, “Subluminal wave bullets: exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
[CrossRef]

E. Recami, M. Zamboni-Rached, and C. A. Dartora, “Localized X-shaped field generated by a superluminal electric charge,” Phys. Rev. E 69, 027602 (2004).
[CrossRef]

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, and H. E. Hernandez-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

E. Recami, “Superluminal motions? a bird’s-eye view of the experimental situation,” Found. Phys. 31, 1119–1135(2001).
[CrossRef]

E. Recami, A. Castellino, G. D. Maccarrone, and M. Rodonò, “Considerations about the apparent ≪superluminal expansions≫ observed in astrophysics,” Nuovo Cimento B 93, 119–144 (1986).
[CrossRef]

A. O. Barut, G. D. Maccarrone, and E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509–533 (1982).
[CrossRef]

P. Caldirola, G. D. Maccarrone, and E. Recami, “Second contribution on solving the ≪imaginary quantities≫ problem in superluminal Lorentz transformations,” Lett. Nuovo Cimento Ser. 2, 29, 241–250 (1980).
[CrossRef]

E. Recami and G. D. Maccarrone, “Solving the ≪imaginary quantities≫ problem in superluminal Lorentz transformations,” Lett. Nuovo Cimento Ser. 2, 28, 151–157 (1980).
[CrossRef]

E. Recami, Tachyons, Monopoles and Related Topics (Elsevier, 1978).

E. Recami, Classical Tachyons and Possible Applications, Rivista Nuovo Cimento Series 3, Vol. 9, Number 6 (Editrice Compositori, 1986).

Reivelt, K.

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69, 036612 (2004).
[CrossRef]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[CrossRef]

Rodonò, M.

E. Recami, A. Castellino, G. D. Maccarrone, and M. Rodonò, “Considerations about the apparent ≪superluminal expansions≫ observed in astrophysics,” Nuovo Cimento B 93, 119–144 (1986).
[CrossRef]

Saari, P.

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69, 036612 (2004).
[CrossRef]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[CrossRef]

P. Saari, “Superluminal localized waves of electromagnetic field in vacuo,” in Proceedings of International Conference “Time’s Arrows, Quantum Measurement and Superluminal Behaviour”, D. Mugnai, A. Ranfagni, and L. S. Shulman, eds. (CNR, 2001), pp. 37–48 (arXiv:physics/0103054v1).

Saghafi, S.

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

Salo, J.

J. Salo, J. Fagerholm, A. Friberg, and M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62, 4261–4275 (2000).
[CrossRef]

Salomaa, M.

J. Salo, J. Fagerholm, A. Friberg, and M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62, 4261–4275 (2000).
[CrossRef]

Shaarawi, A. M.

I. M. Besieris, A. M. Shaarawi, and A. M. Attiya, “Bateman conformal transformations within the framework of the bidirectional spectral representation,” Prog. Electromagn. Res. 48, 201–231 (2004).
[CrossRef]

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagn. Res. 19, 1–48 (1998).
[CrossRef]

R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

Silva, L.

J. Vieira, S. Martins, V. Pathak, R. Fonseca, W. Mori, and L. Silva, “Magnetic control of particle injection in plasma based accelerators,” Phys. Rev. Lett. 106, 225001 (2011).
[CrossRef]

Smirnov, V. I.

V. I. Smirnov, Course of Higher Mathematics, Vol. 4, Part 2 (Pergamon, 1964).

Suntsov, S.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

Trillo, S.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef]

Trull, J.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef]

Tzortzakis, S.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

Utkin, A. B.

A. B. Utkin, “Riemann-Volterra time-domain technique for waveguides: a case study for elliptic geometry,” Wave Motion 49, 347–363 (2012).
[CrossRef]

A. B. Utkin, “Mathieu progressive waves,” Commun. Theor. Phys. 56, 733–739 (2011).
[CrossRef]

V. V. Borisov and A. B. Utkin, “The transient electromagnetic field produced by a moving pulse of line current,” J. Phys. D: Appl. Phys. 28, 614–622 (1995).
[CrossRef]

V. V. Borisov and A. B. Utkin, “Some solutions of the wave and Maxwell’s equations,” J. Math. Phys. 35, 3624 (1994).
[CrossRef]

A. B. Utkin, “Electromagnetic waves generated by line current pulses,” in Wave Propagation, A. Petrin, ed. (InTech, 2011), pp. 483–508.

Valiulis, G.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef]

Vieira, J.

J. Vieira, S. Martins, V. Pathak, R. Fonseca, W. Mori, and L. Silva, “Magnetic control of particle injection in plasma based accelerators,” Phys. Rev. Lett. 106, 225001 (2011).
[CrossRef]

Walker, S. C.

S. C. Walker and W. A. Kuperman, “Cherenkov-Vavilov formulation of X waves,” Phys. Rev. Lett. 99, 244802 (2007).
[CrossRef]

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, 2001).

Zamboni-Rached, M.

M. Zamboni-Rached, E. Recami, and I. M. Besieris, “Cherenkov radiation versus X-shaped localized waves,” J. Opt. Soc. Am. A 27, 928–934 (2010).
[CrossRef]

M. Zamboni-Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A 79, 013816 (2009).
[CrossRef]

M. Zamboni-Rached and E. Recami, “Subluminal wave bullets: exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
[CrossRef]

E. Recami, M. Zamboni-Rached, and C. A. Dartora, “Localized X-shaped field generated by a superluminal electric charge,” Phys. Rev. E 69, 027602 (2004).
[CrossRef]

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, and H. E. Hernandez-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

Ziolkowski, R. W.

Commun. Theor. Phys. (1)

A. B. Utkin, “Mathieu progressive waves,” Commun. Theor. Phys. 56, 733–739 (2011).
[CrossRef]

Found. Phys. (1)

E. Recami, “Superluminal motions? a bird’s-eye view of the experimental situation,” Found. Phys. 31, 1119–1135(2001).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, and H. E. Hernandez-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

J. Math. Phys. (1)

V. V. Borisov and A. B. Utkin, “Some solutions of the wave and Maxwell’s equations,” J. Math. Phys. 35, 3624 (1994).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Phys. D: Appl. Phys. (1)

V. V. Borisov and A. B. Utkin, “The transient electromagnetic field produced by a moving pulse of line current,” J. Phys. D: Appl. Phys. 28, 614–622 (1995).
[CrossRef]

Lett. Nuovo Cimento (4)

E. Recami and G. D. Maccarrone, “Solving the ≪imaginary quantities≫ problem in superluminal Lorentz transformations,” Lett. Nuovo Cimento Ser. 2, 28, 151–157 (1980).
[CrossRef]

P. Caldirola, G. D. Maccarrone, and E. Recami, “Second contribution on solving the ≪imaginary quantities≫ problem in superluminal Lorentz transformations,” Lett. Nuovo Cimento Ser. 2, 29, 241–250 (1980).
[CrossRef]

O. Gron, “Visual appearance of superluminal bodies,” Lett. Nuovo Cimento 23, 97–100 (1978).
[CrossRef]

H. Lemke, “Light from sources moving faster than light,” Lett. Nuovo Cimento 12, 342–346 (1975).
[CrossRef]

Nuovo Cimento A (1)

A. O. Barut, G. D. Maccarrone, and E. Recami, “On the shape of tachyons,” Nuovo Cimento A 71, 509–533 (1982).
[CrossRef]

Nuovo Cimento B (1)

E. Recami, A. Castellino, G. D. Maccarrone, and M. Rodonò, “Considerations about the apparent ≪superluminal expansions≫ observed in astrophysics,” Nuovo Cimento B 93, 119–144 (1986).
[CrossRef]

Phys. Rev. A (3)

M. Zamboni-Rached and E. Recami, “Subluminal wave bullets: exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
[CrossRef]

M. Zamboni-Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A 79, 013816 (2009).
[CrossRef]

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

Phys. Rev. E (4)

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69, 036612 (2004).
[CrossRef]

J. Salo, J. Fagerholm, A. Friberg, and M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62, 4261–4275 (2000).
[CrossRef]

M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E 69, 066606 (2004).
[CrossRef]

E. Recami, M. Zamboni-Rached, and C. A. Dartora, “Localized X-shaped field generated by a superluminal electric charge,” Phys. Rev. E 69, 027602 (2004).
[CrossRef]

Phys. Rev. Lett. (6)

S. C. Walker and W. A. Kuperman, “Cherenkov-Vavilov formulation of X waves,” Phys. Rev. Lett. 99, 244802 (2007).
[CrossRef]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

E. Arévalo, “Boosted X waves in nonlinear optical systems,” Phys. Rev. Lett. 104, 023902 (2010).
[CrossRef]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[CrossRef]

J. Vieira, S. Martins, V. Pathak, R. Fonseca, W. Mori, and L. Silva, “Magnetic control of particle injection in plasma based accelerators,” Phys. Rev. Lett. 106, 225001 (2011).
[CrossRef]

Prog. Electromagn. Res. (2)

I. M. Besieris, A. M. Shaarawi, and A. M. Attiya, “Bateman conformal transformations within the framework of the bidirectional spectral representation,” Prog. Electromagn. Res. 48, 201–231 (2004).
[CrossRef]

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Prog. Electromagn. Res. 19, 1–48 (1998).
[CrossRef]

Wave Motion (1)

A. B. Utkin, “Riemann-Volterra time-domain technique for waveguides: a case study for elliptic geometry,” Wave Motion 49, 347–363 (2012).
[CrossRef]

Other (9)

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 2 (Wiley, 1989).

V. I. Smirnov, Course of Higher Mathematics, Vol. 4, Part 2 (Pergamon, 1964).

A. B. Utkin, “Electromagnetic waves generated by line current pulses,” in Wave Propagation, A. Petrin, ed. (InTech, 2011), pp. 483–508.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, 2001).

E. Recami, Tachyons, Monopoles and Related Topics (Elsevier, 1978).

E. Recami, Classical Tachyons and Possible Applications, Rivista Nuovo Cimento Series 3, Vol. 9, Number 6 (Editrice Compositori, 1986).

H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves (Wiley, 2008).

D. Burgarella, M. Livio, and C. P. O’Dea, eds., Astrophysical Jets: Proceedings of the Astrophysical Jets Meeting, Baltimore, 1992 May 12–14 (Cambridge University, 1993).

P. Saari, “Superluminal localized waves of electromagnetic field in vacuo,” in Proceedings of International Conference “Time’s Arrows, Quantum Measurement and Superluminal Behaviour”, D. Mugnai, A. Ranfagni, and L. S. Shulman, eds. (CNR, 2001), pp. 37–48 (arXiv:physics/0103054v1).

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Figures (6)

Fig. 1.
Fig. 1.

Integration path for the general solution [Eq. (10)].

Fig. 2.
Fig. 2.

Instantiation of the general solution for the observation point ρ, z, where z>zc.

Fig. 3.
Fig. 3.

Areas of case representation of ψ(τ,ρ,z).

Fig. 4.
Fig. 4.

Areas of case representation of ψ(τ,ρ,ξ).

Fig. 5.
Fig. 5.

(a) Contour plots illustrating the shape of the wave function support (solid isolines τ˜1=τ˜) and the boundary between the cases in Eqs. (16) and (17) (dashed isolines τ˜2=τ˜), (b) a snapshot of ψ˜, clipped by ψ˜=10 in the vicinity of its singularities, taken at τ˜=1.

Fig. 6.
Fig. 6.

Dynamics of the droplet-shaped wave propagation (the points on the vertex and the cone surface, in which ψ˜ diverges, are omitted).

Equations (21)

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ψ(ρ,ζ)=const×βζ2(β21)ρ2.
[τ2ρ1ρ(ρρ)z2]ψ(τ,ρ,z)=S(τ,ρ,z),
ψ=0,S=0forτ<0,
j(τ,ρ,z)=ceβδ(ρ)2πρh(z)h(βτz)f(βτz),
ϱ(τ,ρ,z)=1c0τdτzj=eδ(ρ)2πρ[h(z)h(βτz)f(βτz)βh(τ)δ(z)0τdτf(βτz)]=1cβj+(e)δ(ρ)2πρh(τ)δ(z)0βτdξf(ξ),
A(0)=0τdτzA(3)=0τdτzψ.
(τ2z2+s2)Ψ=μ02πceβh(z)h(βτz)f(βτz),
Ψ=0forτ<0.
ψ(τ,s,z)=μ04πceβzτz+τdz0τ|zz|dτJ0(s(ττ)2(zz)2)×h(z)h(βτz)f(βτz).
0dssJ0(sρ)J0(sρ)=1ρδ(ρρ),
δ(g(τ))=iδ(ττi)|τg(τi)|
ψ(τ,ρ,z)=μ04πρceβzτz+τdz0τ|zz|dτh(z)×h(βτz)f(βτz)δ(ττ+ρ2+(zz)2)ρ2+(zz)2.
ψ={0<τ<rμ04πceβ0z2dzf(β(τr)z)rr<τ<;
ψ={0<τ<τtμ04πceβz1z2dzf(β(τr)z)rτt<τ<rμ04πceβ0z2dzf(β(τr)z)rr<τ<,
τ1,2=τz2,1/β=γ2(βξξ2(β21)ρ2)
τ1,2r(τ1,2,ρ,ξ)=0;
ψ(τ,ρ,ξ)=0,
ψ(τ,ρ,ξ)=μ04πceβ2τ1τdτf(β[τr(τ,ρ,ξ)])r(τ,ρ,ξ),
ψ(τ,ρ,ξ)=μ04πceβ2τ1τ2dτf(β[τr(τ,ρ,ξ)])r(τ,ρ,ξ);
ψ={μ04πceβτ1τdτδ(τr(τ,ρ,ξ))ττ1<τ<τ2μ04πceβτ1τ2dτδ(τr(τ,ρ,ξ))ττ2<τ<.
ψ˜(τ˜,ρ˜,ξ˜)={12βξ˜2(β21)ρ˜2τ˜1<τ˜<τ˜2βξ˜2(β21)ρ˜2τ˜2<τ˜<.

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