Abstract

Beginning with the Maxwell’s equations for an isotropic nonlinear medium, we have obtained vector slowly varying amplitude equations. The equations are a nonparaxial vector generalization of the well-known scalar 3D+1 nonlinear Schrodinger equation. We have determined the dispersion region and medium parameters necessary for experimental observation of wave propagation described by these equations. We show that these equations admit exact vortex solutions with spin l=1. For the case of two vortices, we also obtain exact analytical expressions describing their interaction. Stability and interaction properties of these vortices are also investigated numerically by a split-step Fourier method. Finally, we discuss applications of these vortices in the area of nuclear fusion and the stabilization of laser pulses.

© 2002 Optical Society of America

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References

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  1. G. A. Swartzlander, Jr. and C. T. Law, “Opical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
    [CrossRef] [PubMed]
  2. J. A. Christou, “Optical vortices in beam propagation dynamics,” Ph.D. dissertation (Australian National University, Canberra, Australia, 1999).
  3. N. N. Rozanov, V. A. Smirnov, and N. V. Vyssotina, “Numerical simulation of interaction of bright spatial solitons in medium with saturable nonlinearity,” Chaos 4, 1767–1782 (1994).
  4. A. L. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Self-focusing of intense light beams,” JETP Lett. 6, 146–148 (1967).
  5. M. D. Feit and J. A. Fleck, Jr., “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
    [CrossRef]
  6. Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. 15, 1282–1285 (1990).
    [CrossRef] [PubMed]
  7. V. Tikhonenko, J. Christou, and B. Luther-Davies, “Spiraling bright spatial solitons formed by breakup of an optical vortex in a saturable self-focusing medium,” J. Opt. Soc. Am. B 12, 2046–2052 (1995).
    [CrossRef]
  8. D. E. Edmundson and R. H. Enns, “Fully three-dimensional collisions of bistable light bullets,” Opt. Lett. 18, 1609–1611 (1993).
    [CrossRef] [PubMed]
  9. D. E. Edmundson and R. H. Enns, “Partical-like nature of colliding three dimensional optical solitons,” Phys. Rev. A 51, 2491–2498 (1995).
    [CrossRef] [PubMed]
  10. R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional optical soliton dragging logic,” Phys. Rev. A 52, 3254–3278 (1995).
    [CrossRef] [PubMed]
  11. C. T. Law and G. A. Swartzlander, Jr., “Polarized optical vortex solitons; instabilities and dynamics in Kerr nonlinear media,” Chaos 4, 1759–1766 (1994).
  12. A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Reading, Mass., 1992).
  13. P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization in third order in the electric field strength,” Phys. Rev. A 137, 801–817 (1965).
    [CrossRef]
  14. R. W. Boyd, Nonlinear Optics (Academic, New York, 1992).
  15. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Nauka, Moscow, 1978).
  16. L. de Broglie, “Sur la fréquence propre de l’électron,” C. R. Acad. Sci. 180, 498–500 (1925).
  17. A. O. Barut, “The Schrodinger and Dirac equations: linear, nonlinear and integrodifferential,” Proceedings of the International Meeting on Geometric and Algebraic Aspects of Nonlinear Field Theory S. de Filippo, (Elsevier, Amsterdam, 1989), pp. 37–51.
  18. L. M. Kovachev, “Influence of cross-phase modulation and four photon parametric mixing on the relative motion of op-tical pulses,” Opt. Quantum Electron. 23, 1091–1102 (1991).
    [CrossRef]
  19. I. Velchev, A. Dreischuh, D. Nechev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130, 385–392 (1996).
    [CrossRef]
  20. A. V. Gaponov and M. A. Miller, “Potential for charges particles in high frequency electromagnetic field,” JETP 34, 242–243 (1958).

1996 (1)

I. Velchev, A. Dreischuh, D. Nechev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130, 385–392 (1996).
[CrossRef]

1995 (3)

V. Tikhonenko, J. Christou, and B. Luther-Davies, “Spiraling bright spatial solitons formed by breakup of an optical vortex in a saturable self-focusing medium,” J. Opt. Soc. Am. B 12, 2046–2052 (1995).
[CrossRef]

D. E. Edmundson and R. H. Enns, “Partical-like nature of colliding three dimensional optical solitons,” Phys. Rev. A 51, 2491–2498 (1995).
[CrossRef] [PubMed]

R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional optical soliton dragging logic,” Phys. Rev. A 52, 3254–3278 (1995).
[CrossRef] [PubMed]

1994 (2)

C. T. Law and G. A. Swartzlander, Jr., “Polarized optical vortex solitons; instabilities and dynamics in Kerr nonlinear media,” Chaos 4, 1759–1766 (1994).

N. N. Rozanov, V. A. Smirnov, and N. V. Vyssotina, “Numerical simulation of interaction of bright spatial solitons in medium with saturable nonlinearity,” Chaos 4, 1767–1782 (1994).

1993 (1)

1992 (1)

G. A. Swartzlander, Jr. and C. T. Law, “Opical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
[CrossRef] [PubMed]

1991 (1)

L. M. Kovachev, “Influence of cross-phase modulation and four photon parametric mixing on the relative motion of op-tical pulses,” Opt. Quantum Electron. 23, 1091–1102 (1991).
[CrossRef]

1990 (1)

1988 (1)

1967 (1)

A. L. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Self-focusing of intense light beams,” JETP Lett. 6, 146–148 (1967).

1965 (1)

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization in third order in the electric field strength,” Phys. Rev. A 137, 801–817 (1965).
[CrossRef]

1958 (1)

A. V. Gaponov and M. A. Miller, “Potential for charges particles in high frequency electromagnetic field,” JETP 34, 242–243 (1958).

1925 (1)

L. de Broglie, “Sur la fréquence propre de l’électron,” C. R. Acad. Sci. 180, 498–500 (1925).

Blair, S.

R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional optical soliton dragging logic,” Phys. Rev. A 52, 3254–3278 (1995).
[CrossRef] [PubMed]

Christou, J.

de Broglie, L.

L. de Broglie, “Sur la fréquence propre de l’électron,” C. R. Acad. Sci. 180, 498–500 (1925).

Dinev, S.

I. Velchev, A. Dreischuh, D. Nechev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130, 385–392 (1996).
[CrossRef]

Dreischuh, A.

I. Velchev, A. Dreischuh, D. Nechev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130, 385–392 (1996).
[CrossRef]

Dyshko, A. L.

A. L. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Self-focusing of intense light beams,” JETP Lett. 6, 146–148 (1967).

Edmundson, D. E.

D. E. Edmundson and R. H. Enns, “Partical-like nature of colliding three dimensional optical solitons,” Phys. Rev. A 51, 2491–2498 (1995).
[CrossRef] [PubMed]

D. E. Edmundson and R. H. Enns, “Fully three-dimensional collisions of bistable light bullets,” Opt. Lett. 18, 1609–1611 (1993).
[CrossRef] [PubMed]

Enns, R. H.

D. E. Edmundson and R. H. Enns, “Partical-like nature of colliding three dimensional optical solitons,” Phys. Rev. A 51, 2491–2498 (1995).
[CrossRef] [PubMed]

D. E. Edmundson and R. H. Enns, “Fully three-dimensional collisions of bistable light bullets,” Opt. Lett. 18, 1609–1611 (1993).
[CrossRef] [PubMed]

Feit, M. D.

Fleck Jr., J. A.

Gaponov, A. V.

A. V. Gaponov and M. A. Miller, “Potential for charges particles in high frequency electromagnetic field,” JETP 34, 242–243 (1958).

Kovachev, L. M.

L. M. Kovachev, “Influence of cross-phase modulation and four photon parametric mixing on the relative motion of op-tical pulses,” Opt. Quantum Electron. 23, 1091–1102 (1991).
[CrossRef]

Law, C. T.

C. T. Law and G. A. Swartzlander, Jr., “Polarized optical vortex solitons; instabilities and dynamics in Kerr nonlinear media,” Chaos 4, 1759–1766 (1994).

G. A. Swartzlander, Jr. and C. T. Law, “Opical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
[CrossRef] [PubMed]

Lugovoi, V. N.

A. L. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Self-focusing of intense light beams,” JETP Lett. 6, 146–148 (1967).

Luther-Davies, B.

Maker, P. D.

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization in third order in the electric field strength,” Phys. Rev. A 137, 801–817 (1965).
[CrossRef]

McLeod, R.

R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional optical soliton dragging logic,” Phys. Rev. A 52, 3254–3278 (1995).
[CrossRef] [PubMed]

Miller, M. A.

A. V. Gaponov and M. A. Miller, “Potential for charges particles in high frequency electromagnetic field,” JETP 34, 242–243 (1958).

Nechev, D.

I. Velchev, A. Dreischuh, D. Nechev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130, 385–392 (1996).
[CrossRef]

Prokhorov, A. M.

A. L. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Self-focusing of intense light beams,” JETP Lett. 6, 146–148 (1967).

Rozanov, N. N.

N. N. Rozanov, V. A. Smirnov, and N. V. Vyssotina, “Numerical simulation of interaction of bright spatial solitons in medium with saturable nonlinearity,” Chaos 4, 1767–1782 (1994).

Silberberg, Y.

Smirnov, V. A.

N. N. Rozanov, V. A. Smirnov, and N. V. Vyssotina, “Numerical simulation of interaction of bright spatial solitons in medium with saturable nonlinearity,” Chaos 4, 1767–1782 (1994).

Swartzlander Jr., G. A.

C. T. Law and G. A. Swartzlander, Jr., “Polarized optical vortex solitons; instabilities and dynamics in Kerr nonlinear media,” Chaos 4, 1759–1766 (1994).

G. A. Swartzlander, Jr. and C. T. Law, “Opical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
[CrossRef] [PubMed]

Terhune, R. W.

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization in third order in the electric field strength,” Phys. Rev. A 137, 801–817 (1965).
[CrossRef]

Tikhonenko, V.

Velchev, I.

I. Velchev, A. Dreischuh, D. Nechev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130, 385–392 (1996).
[CrossRef]

Vyssotina, N. V.

N. N. Rozanov, V. A. Smirnov, and N. V. Vyssotina, “Numerical simulation of interaction of bright spatial solitons in medium with saturable nonlinearity,” Chaos 4, 1767–1782 (1994).

Wagner, K.

R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional optical soliton dragging logic,” Phys. Rev. A 52, 3254–3278 (1995).
[CrossRef] [PubMed]

C. R. Acad. Sci. (1)

L. de Broglie, “Sur la fréquence propre de l’électron,” C. R. Acad. Sci. 180, 498–500 (1925).

Chaos (2)

C. T. Law and G. A. Swartzlander, Jr., “Polarized optical vortex solitons; instabilities and dynamics in Kerr nonlinear media,” Chaos 4, 1759–1766 (1994).

N. N. Rozanov, V. A. Smirnov, and N. V. Vyssotina, “Numerical simulation of interaction of bright spatial solitons in medium with saturable nonlinearity,” Chaos 4, 1767–1782 (1994).

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

JETP (1)

A. V. Gaponov and M. A. Miller, “Potential for charges particles in high frequency electromagnetic field,” JETP 34, 242–243 (1958).

JETP Lett. (1)

A. L. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Self-focusing of intense light beams,” JETP Lett. 6, 146–148 (1967).

Opt. Commun. (1)

I. Velchev, A. Dreischuh, D. Nechev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130, 385–392 (1996).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

L. M. Kovachev, “Influence of cross-phase modulation and four photon parametric mixing on the relative motion of op-tical pulses,” Opt. Quantum Electron. 23, 1091–1102 (1991).
[CrossRef]

Phys. Rev. A (3)

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization in third order in the electric field strength,” Phys. Rev. A 137, 801–817 (1965).
[CrossRef]

D. E. Edmundson and R. H. Enns, “Partical-like nature of colliding three dimensional optical solitons,” Phys. Rev. A 51, 2491–2498 (1995).
[CrossRef] [PubMed]

R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional optical soliton dragging logic,” Phys. Rev. A 52, 3254–3278 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

G. A. Swartzlander, Jr. and C. T. Law, “Opical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
[CrossRef] [PubMed]

Other (5)

J. A. Christou, “Optical vortices in beam propagation dynamics,” Ph.D. dissertation (Australian National University, Canberra, Australia, 1999).

R. W. Boyd, Nonlinear Optics (Academic, New York, 1992).

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Nauka, Moscow, 1978).

A. O. Barut, “The Schrodinger and Dirac equations: linear, nonlinear and integrodifferential,” Proceedings of the International Meeting on Geometric and Algebraic Aspects of Nonlinear Field Theory S. de Filippo, (Elsevier, Amsterdam, 1989), pp. 37–51.

A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Reading, Mass., 1992).

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Figures (2)

Fig. 1
Fig. 1

(a) Initial condition of the first component |A1y(x, y, z=0, t=0)| of the coupled-vortex solutions (32). (b) Same component after propagating a normalized time t=1/2 |A1y(x, y, z=0, t=1/2)| according to the coupled Eqs. (12) and (13). The second component A2y has group velocity α=1, the same evolution characteristic, and also propagates without separation or change of amplitude.

Fig. 2
Fig. 2

(a) Initial condition of the Gaussian amplitude (41) |A(x, y, z=0, t=0)|. (b) Same amplitude after propagating a normalized time t=1/10 |A(x, y, z=0, t=1/10)| according to the 3D+1 nonlinear Schrodinger equation (40). As seen from Fig. 2 for a normalized time one-fifth of Fig. 1, there is significant self-focusing of the Gaussian pulse.

Equations (101)

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×E=-1cBt,
×B=1cDt,
·D=0,
·B=0,
B=H,D=Pl+4πPnl,
Pl=0[1+4πχ(1)(t-τ)]E(τ, x, y, z)dτ=0ε0(t-τ)E(τ, x, y, z)dτ,
Pnl(3)=A(E·E*)E+12B(E·E)E*,
Pnl(3)=n˜2(E·E*)E,
n˜2=2πn02A+12B
ΔE-(·E)-1c22Dt2=0,
ΔE-1c22Dt2=0.
E(x, y, z, t)=i=1nAi(x, y, z, t)exp i(ωi t-kiz),
i=1, 2 ,..n,
2iα1A1t+ΔA1+β12A1z2+γ1(|A1|2+2|A2|2)A1
=0,
2iα2A2t-λ2A2z+ΔA2+β22A2z2+γ2(|A2|2
+2|A1|2)A2=0,
βi=β=1,γi=γ=1,αi=102,λ2=λ,i=1,2,
2iα1A1t+ΔA1+(|A1|2+2|A2|2)A1=0,
2iα2A2t-λA2z+ΔA2+(|A2|2+2|A1|2)A2=0.
A1(x, y, z, t)=j=x,y,zjAj(x, y, z)exp-iα12t,
A2(x, y, z, t)=j=x,y,zjBj(x, y, z)exp iα2λz-t2.
α12Al+ΔAl+j=x,y,z(|Aj|2+2|Bj|2)Al=0,
l=x, y, z,
η2Bl+ΔBl+j=x,y,z(|Bj|2+2|Aj|2)Bl=0,
η2=α22+α22λ2=α22(1+λ2).
α12Al+ΔrAl+1r2Δθ,φAl+j=x, y,z(|Aj|2+2|Bj|2)Al
=0,l=x, y, z,
η2Bl+ΔrBl+1r2Δθ,φBl+j=x, y,z(|Bj|2+2|Aj|2)Bl
=0,
Δr=1r2rr2r,
Δθ,φ=1sin θθsin θθ+1sin2 θ2φ2
Aj(r, θ, φ, t)=R1(r)jYji(θ, φ),
Bj(r, θ, φ, t)=R2(r)jYji(θ, φ),
j=x, y, z,i=1, 0, -1,
|Yx1(θ, φ)|2+|Yy-1(θ, φ)|2+|Yz0(θ, φ)|2=const.
r2ΔrR1R1+r2(α12+|R1|2+2|R2|2)
=-Δθ,φYjiYji=l(l+1),
r2ΔrR2R2+r2(α22+|R2|2+2|R1|2)=-Δθ,ϕYjiYji=l(l+1),i=1, -1, 0; j=x, y, z,
Yji=Ylm(θ, φ)=ΘlmΦm=4π3  2l+(l-m)!4π(l+m)! Plm(cos θ)exp(imφ),
i=1, -1, 0,j=x, y, z,
l=0, 1, 2 ,, m=0,±1,±2 ,, |m|<l.
Yx1=sin θ cos φ,
Yy-1=sin θ sin φ,l=1,
Yz0=cos θ,
R1=1i 23exp(iα1r)r,
R2=1i 23exp(iηr)r.
A1=Re1i 23exp(ia1r)r sin θ cos φ exp(-ia1t)+Re1i 23exp(ia1r)r sin θ sin φ exp(-ia1t)y+Re1i 23exp(ia1r)r cos θ exp(-ia1t)z=23sin(α1r)r cos(α1t)sin θ cos φx+23sin(α1r)r cos(α1t)sin θ sin φy+23sin(α1r)r cos(α1t)cos θz,
A2=Re1i 23exp(iηr)r sin θ cos φ×expia2λz-t2x+Re1i 23exp(iηr)r sin θ sin φ×expia2λz-t2y+Re1i 23exp(iηr)r cos θ expia2λz-t2z=23sin(ηr)r cosα2t2-λzsin θ cos φx+23sin(ηr)r cosα2t2-λzsin θ sin φy+23sin(ηr)r cosα2t2-λzcos θz.
Δr1t=0,
Δr2t=λ+P2z,
2Δr1t2=1N1 r,θ,φ|A1(r, θ, φ)|2r[|A2(r+Δr, θ, φ)|2]r2 sin θdrdθdφ,
2Δr2t2=1N2 r,θ,φ|A2(r, θ, φ)|2r[|A1(r+Δr, θ, φ)|2]r2 sin θdrdθdφ,
Ni=r,θ,φWlocr2 sin θdrdθdφ=2π20+ Re(Ai)·Re(A1*)dr
P2z=12iNi-A2zzA2z*-A2z*zA2zdz
Δri=1Ni r,θ,φAi(r, θ, φ, t)rA1*(r, θ, φ, t)r2×sin θ drdθdφ
2Δr1t2=2Δr2t2=-2Δr2.
V(Δr)=-Δr2Δr2 dr=-2Δr.
2iαAt+ΔA+|A|2A=0.
A(x, y, z, t=0)=A0 exp[-(x2+y2+z2)/2].
γir02ki2n˜2|Ai|2α2n˜2|Ai|21.
α=102103.
n˜2|Ai|210-410-6,
βi=kivi2ki=-1.
ε0(ϖ)constϖ,
ε(ωi)=1-ωp2ωi2,
ki(ωi)=ωi2-ωp2c.
kivi2ki=-ωp2ωi2-1,ωi2ωp2.
ki2(ϖ)=ϖ2ε0(ϖ)c2.
ε0(ω)=1+4π×constωresω-ωres+ωresω+ωres.
ε0(ω)1+4π×const×ωresω-ωres1ϖ
ki2(ϖ)=ϖ2ε0(ϖ)c2ϖc2,
ki(ϖ)ϖc.
·D=0.
ε0(ω)=0[1+4πχ(1)(τ)]exp(-iωτ)dτ,
·D=·(ε0E+n˜2|E|2E)=0.
·E=n˜2ε0(|E|2)·E1+n˜2ε0|E|2=ln1+n˜2ε0|E|2·E.
Ei(x, y, z, t)0,i=x, y, z,x, y, z±,
Ei(x, y, z, t)E0=const,x, y, zc=const,
E=E0E, x=r0x, y=r0y, z=r0z,
·E=[ln(1+α|E|2)]·E,
0<|Ei|1,0<|Ei|xi<1,i=x, y, z,
0[ln(1+α|E|2)]<10-6.
·E0
E(x, y, z, t)=i=1nAi(x, y, z, t)exp i(ωjt-kjz),
Ai(x, y, z, t)=-Ai(x, y, z, ω-ωi)×exp[ω-ωi)t]dω,
Ai(x, y, z, t)t=i-(ω-ωi)Ai(x, y, z, t)×exp[i(ω-ωi)t]dω,
2Ai(x, y, z, t)t2=--(ω-ωi)2(x, y, z,t)×exp[i(ω-ωi)t]dω.
0ε0(t-τ)exp(iωτ)dτ=ε0(ω)exp(iωt)
Pil=0ε0(t-τ)exp[i(ωiτ-kiz)]-+ Ai(x, y, z, ω-ωi)exp[i(ω-ωi)τ]dωdτ=exp(-ikiz)-+ Ai(x, y, z, ω-ωi)0ε0(t-τ)×exp(iωτ)dτdω.
Pil(x, y, z, t)=exp[i(ωit-kiz)]-+ε0(ω)exp[i(ω-ωi)t]Ai(x, y, z, ω-ωi)dω.
1c22Pii(x, y, z, t)t2=-exp[i(ωit-kiz)]×-+ω2ε0(ω)c2 exp[i(ω-ωi)t]×Ai(x, y, z, ω-ωi)dω.
k2(ω)=ωε0(ω)c2=i=lnki2(ωi)+[ki2(ωi)]ωi(ω-ωi)+122[ki2(ω)]ωi2(ω-ωi)2+.
1c22Pil(x, y, z, t)t2=-ki2Ai-2ikiviAit+kiki+1vi22Ait2+×exp[i(ωit-kiz)].
-iAit+viAiz=vi2kiΔAi-vi2ki+lkivi22Ait2+n˜2kivi2|A|2+2ij|Aj|2Ai=0.
vi2kiki=-1,
t=t, z=z-v1t.
iA1t+v12k1ΔA1-v13k122A1z2+n˜2k1v12
×|A1|2+2j1|Aj|2A1=0,
iAit-ΔviA2z+vi2kiΔAi-vi3ki22Aiz2+n˜2kivi2
×|Ai|2+2ji|Aj|2Ai=0.

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