Abstract

The interaction dynamics of X-waves in an AlGaAs waveguide array is theoretically considered. The nonlinear discrete diffraction dynamics of a waveguide array mediates the generation of spatio-temporal X-waves from pulsed initial conditions. The interactions between co-propagating and counter-propagating X-waves are studied. For the co-propagating case, the initial phase relation between the X-waves determine the attractive or repulsive behavior of the X-wave interaction. For the counter-propagating case, the collisions between X-waves generate a nonlinear phase-shift. These dynamics show that X-waves interact in a manner similar to solitons.

© 2010 Optical Society of America

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References

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  1. J. Lu and J. F. Greenleaf, "Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations," IEEE Trans. Ultrason. Ferrelec. Freq. contr. 39, 19-31 (1992).
    [CrossRef]
  2. E. Recami, M. Zamboni-Rached, and H. E. Hernandez-Figueroa, Localized waves (Wiley, 2007).
  3. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, "Nonlinear Electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
    [CrossRef] [PubMed]
  4. 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] [PubMed]
  5. M. Kolesik, E. M. Wright, and J. V. Moloney, "Dynamic nonlinear X-waves for femtosecond pulse propagation in water," Phys. Rev. Lett. 92, 253901 (2004).
    [CrossRef] [PubMed]
  6. D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, "Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses," Phys. Rev. Lett. 96, 193901 (2006).
    [CrossRef] [PubMed]
  7. C. Conti, and S. Trillo, "Nonspreading wave packets in three dimensions formed by an ultracold Bose gas in an optical lattice," Phys. Rev. Lett. 92, 120404 (2004).
    [CrossRef] [PubMed]
  8. S. Longhi, and D. Janner, "X-shaped waves in photonic crystals," Phys. Rev. B 70, 235123 (2004).
    [CrossRef]
  9. Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
    [CrossRef] [PubMed]
  10. D. Hudson, K. Shish, T. R. Schibli, J. N. Kutz, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, "Nonlinear femtosecond pulse reshaping in waveguide arrays," Opt. Lett. 33, 1440-1442 (2008).
    [CrossRef] [PubMed]
  11. J. N. Kutz, C. Conti, and S. Trillo, "Mode-locked X-wave lasers," Opt. Express 15, 16022-16028 (2007).
    [CrossRef] [PubMed]
  12. K. Staliunas, and M. Tlidi, "Hyperbolic Transverse Patterns in Nonlinear Optical Resonators," Phys. Rev. Lett. 94, 133902 (2005).
    [CrossRef] [PubMed]
  13. L. Mollenauer, and J. Gordon, Solitons in Optical Fibers: Fundamentals and Applications, (Springer, 2006).
  14. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete Spatial Optical Solitons in Waveguide Arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
    [CrossRef]

2008 (1)

2007 (2)

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

J. N. Kutz, C. Conti, and S. Trillo, "Mode-locked X-wave lasers," Opt. Express 15, 16022-16028 (2007).
[CrossRef] [PubMed]

2006 (1)

D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, "Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses," Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

2005 (1)

K. Staliunas, and M. Tlidi, "Hyperbolic Transverse Patterns in Nonlinear Optical Resonators," Phys. Rev. Lett. 94, 133902 (2005).
[CrossRef] [PubMed]

2004 (3)

M. Kolesik, E. M. Wright, and J. V. Moloney, "Dynamic nonlinear X-waves for femtosecond pulse propagation in water," Phys. Rev. Lett. 92, 253901 (2004).
[CrossRef] [PubMed]

C. Conti, and S. Trillo, "Nonspreading wave packets in three dimensions formed by an ultracold Bose gas in an optical lattice," Phys. Rev. Lett. 92, 120404 (2004).
[CrossRef] [PubMed]

S. Longhi, and D. Janner, "X-shaped waves in photonic crystals," Phys. Rev. B 70, 235123 (2004).
[CrossRef]

2003 (2)

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, "Nonlinear Electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

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] [PubMed]

1998 (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete Spatial Optical Solitons in Waveguide Arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete Spatial Optical Solitons in Waveguide Arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Boyd, A. R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete Spatial Optical Solitons in Waveguide Arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Bragheri, F.

D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, "Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses," Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

Christodoulides, D. N.

D. Hudson, K. Shish, T. R. Schibli, J. N. Kutz, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, "Nonlinear femtosecond pulse reshaping in waveguide arrays," Opt. Lett. 33, 1440-1442 (2008).
[CrossRef] [PubMed]

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

Conti, C.

J. N. Kutz, C. Conti, and S. Trillo, "Mode-locked X-wave lasers," Opt. Express 15, 16022-16028 (2007).
[CrossRef] [PubMed]

C. Conti, and S. Trillo, "Nonspreading wave packets in three dimensions formed by an ultracold Bose gas in an optical lattice," Phys. Rev. Lett. 92, 120404 (2004).
[CrossRef] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, "Nonlinear Electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

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] [PubMed]

Couairon, A.

D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, "Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses," Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

Cundiff, S. T.

Di Trapani, P.

D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, "Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses," Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

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] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, "Nonlinear Electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

Droulias, S.

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

Dubietis, A.

D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, "Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses," Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete Spatial Optical Solitons in Waveguide Arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Faccio, D.

D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, "Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses," Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

Frumker, E.

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

Hizanidis, K.

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

Hudson, D.

Janner, D.

S. Longhi, and D. Janner, "X-shaped waves in photonic crystals," Phys. Rev. B 70, 235123 (2004).
[CrossRef]

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] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, "Nonlinear Electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

Kolesik, M.

M. Kolesik, E. M. Wright, and J. V. Moloney, "Dynamic nonlinear X-waves for femtosecond pulse propagation in water," Phys. Rev. Lett. 92, 253901 (2004).
[CrossRef] [PubMed]

Kutz, J. N.

Lahini, Y.

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

Longhi, S.

S. Longhi, and D. Janner, "X-shaped waves in photonic crystals," Phys. Rev. B 70, 235123 (2004).
[CrossRef]

Moloney, J. V.

M. Kolesik, E. M. Wright, and J. V. Moloney, "Dynamic nonlinear X-waves for femtosecond pulse propagation in water," Phys. Rev. Lett. 92, 253901 (2004).
[CrossRef] [PubMed]

Morandotti, R.

D. Hudson, K. Shish, T. R. Schibli, J. N. Kutz, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, "Nonlinear femtosecond pulse reshaping in waveguide arrays," Opt. Lett. 33, 1440-1442 (2008).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete Spatial Optical Solitons in Waveguide Arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Piskarskas, A.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, "Nonlinear Electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

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] [PubMed]

Porras, M.

D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, "Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses," Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

Schibli, T. R.

Shish, K.

Silberberg, Y.

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete Spatial Optical Solitons in Waveguide Arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Staliunas, K.

K. Staliunas, and M. Tlidi, "Hyperbolic Transverse Patterns in Nonlinear Optical Resonators," Phys. Rev. Lett. 94, 133902 (2005).
[CrossRef] [PubMed]

Tlidi, M.

K. Staliunas, and M. Tlidi, "Hyperbolic Transverse Patterns in Nonlinear Optical Resonators," Phys. Rev. Lett. 94, 133902 (2005).
[CrossRef] [PubMed]

Trillo, S.

J. N. Kutz, C. Conti, and S. Trillo, "Mode-locked X-wave lasers," Opt. Express 15, 16022-16028 (2007).
[CrossRef] [PubMed]

C. Conti, and S. Trillo, "Nonspreading wave packets in three dimensions formed by an ultracold Bose gas in an optical lattice," Phys. Rev. Lett. 92, 120404 (2004).
[CrossRef] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, "Nonlinear Electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

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] [PubMed]

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] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, "Nonlinear Electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

Valiulis, G.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, "Nonlinear Electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

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] [PubMed]

Wright, E. M.

M. Kolesik, E. M. Wright, and J. V. Moloney, "Dynamic nonlinear X-waves for femtosecond pulse propagation in water," Phys. Rev. Lett. 92, 253901 (2004).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (1)

S. Longhi, and D. Janner, "X-shaped waves in photonic crystals," Phys. Rev. B 70, 235123 (2004).
[CrossRef]

Phys. Rev. Lett. (8)

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, "Nonlinear Electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

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] [PubMed]

M. Kolesik, E. M. Wright, and J. V. Moloney, "Dynamic nonlinear X-waves for femtosecond pulse propagation in water," Phys. Rev. Lett. 92, 253901 (2004).
[CrossRef] [PubMed]

D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, "Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses," Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

C. Conti, and S. Trillo, "Nonspreading wave packets in three dimensions formed by an ultracold Bose gas in an optical lattice," Phys. Rev. Lett. 92, 120404 (2004).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete Spatial Optical Solitons in Waveguide Arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

K. Staliunas, and M. Tlidi, "Hyperbolic Transverse Patterns in Nonlinear Optical Resonators," Phys. Rev. Lett. 94, 133902 (2005).
[CrossRef] [PubMed]

Other (3)

L. Mollenauer, and J. Gordon, Solitons in Optical Fibers: Fundamentals and Applications, (Springer, 2006).

J. Lu and J. F. Greenleaf, "Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations," IEEE Trans. Ultrason. Ferrelec. Freq. contr. 39, 19-31 (1992).
[CrossRef]

E. Recami, M. Zamboni-Rached, and H. E. Hernandez-Figueroa, Localized waves (Wiley, 2007).

Supplementary Material (4)

» Media 1: MOV (2284 KB)     
» Media 2: MOV (2388 KB)     
» Media 3: MOV (2223 KB)     
» Media 4: MOV (2484 KB)     

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Figures (8)

Fig. 1
Fig. 1

Pseudo-color plot of in-phase pulses interacting. The final result is a pair of closely-spaced but distinct X-waves. The white and green dotted lines indicate the center of the X-wave at Z = 0 in the 0th and 1st waveguide respectively. The white and green dashed lines represent the center of the X-wave at the present value of Z. Notice that the final separation is slightly larger than the initial separation of the pulses. ( Media 1)

Fig. 2
Fig. 2

Pseudo-color plot of out of-phase pulses interacting. In this case, the X-waves attract and form a pair of X-waves with a negligible delay in time. The white and green dotted lines denote the center of X-wave in the 0th and 1st waveguides at Z = 0, and the dashed lines represent the center of the X-wave at the present value of Z. ( Media 2)

Fig. 3
Fig. 3

Plot of the X-wave separation as a function of phase-difference for three different initial separations. The dashed lines show the initial separation and the solid lines of the same color show the final X-wave separation. Regardless of initial separation, X-waves with small phase-differences repel and X-waves with phase-differences near π attract.

Fig. 4
Fig. 4

Plot of the final X-wave separation for a pair of in-phase X-waves, shown in black, and π out-of-phase X-waves, shown in blue with an initial separation of ΔT = 1. For sufficiently high initial powers, the in-phase solutions repel and the out-of-phase solutions attract.

Fig. 5
Fig. 5

Pseudo-color plot of two unequally sized X-waves with Δθ = 0. The X-wave in waveguide 0 has amplitude 2.0 while the pulse in waveguide 1 has amplitude 1. The larger pulse dominates the dynamics and the attracts the smaller pulse. ( Media 3)

Fig. 6
Fig. 6

Pseudo-color plot of the collision of counter-propagating X-waves. The top series of plots shows the propagation of both the forward and backward X-waves at three snapshots in time. The bottom series of plots follows the propagation of the forward X-wave as it interacts with the backward X-wave. ( Media 4)

Fig. 7
Fig. 7

Plot of the phase shift in radians and time delay of the interacting X-waves. The magnitude of the phase shift is dependent upon the size of the X-wave, but the time-delay remains zero for all initial conditions.

Fig. 8
Fig. 8

Plot the phase shift of counter-propagating X-waves as a function of initial separation. The different colors correspond to initial conditions with η +, η = 1.50, 1.25, 1.00, 0.75, and 0.50 for blue, green, red, teal, and purple respectively.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

i A n Z + D 2 2 A n T 2 + c ( A n 1 + A n + 1 ) + γ | A n | 2 A n = 0 ,
A 0 ( 0 , T ) = η 0 sech ( T ) and A 1 ( 0 , T ) = η 1 sech ( T + Δ T ) × exp ( i Δ θ ) .
i A n Z + i σ d A n d T + D 2 2 A n T 2 + γ ( | A n | 2 + 2 | B n | 2 ) A n + c ( A n 1 + A n + 1 ) = 0
i B n Z i σ d B n d T + D 2 2 A n T 2 + γ ( 2 | A n | 2 + | B n | 2 ) B n + c ( B n 1 + B n + 1 ) = 0
A 0 ( Z , 0 ) = η + sech ( Z + Δ Z ) and B 0 ( Z , 0 ) = η sech ( Z Δ Z ) .

Metrics