Abstract

Analytical results, based on the virial theorem and the Furutsu–Novikov theorem, of the spatiotemporal evolution of a pulse in a nonlinear waveguide with a randomly fluctuating refractive index are presented. For initial conditions in which total collapse occurs in a homogeneous waveguide, random fluctuations postpone the collapse. Sufficiently large-amplitude and short-wavelength fluctuations can cause an initially localized pulse to spread instead of contracting. We show that the disorder can be applied to induce a high degree of controllability of the spatiotemporal extent of the pulses in the nonlinear waveguide.

© 1998 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. R. E. de Araujo and A. S. L. Gomes, Phys. Rev. A 57, 2037 (1998).
    [CrossRef]
  10. L. Bergé, E. Kuznetsov, and J. Juul. Rasmussen, Phys. Rev. E 53, R1340 (1996).
    [CrossRef]
  11. K. Ø. Rasmussen, Yu. B. Gaididei, O. Bang, and P. L. Christiansen, Phys. Lett. 204, 121 (1995).
    [CrossRef]
  12. P. L. Christiansen, Yu. Gaididei, M. Johansson, K. Ø. Rasmussen, and I. Yakimenko, Phys. Rev. E 54, 924 (1996).
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  19. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961).

1998 (1)

R. E. de Araujo and A. S. L. Gomes, Phys. Rev. A 57, 2037 (1998).
[CrossRef]

1996 (2)

L. Bergé, E. Kuznetsov, and J. Juul. Rasmussen, Phys. Rev. E 53, R1340 (1996).
[CrossRef]

P. L. Christiansen, Yu. Gaididei, M. Johansson, K. Ø. Rasmussen, and I. Yakimenko, Phys. Rev. E 54, 924 (1996).
[CrossRef]

1995 (1)

K. Ø. Rasmussen, Yu. B. Gaididei, O. Bang, and P. L. Christiansen, Phys. Lett. 204, 121 (1995).
[CrossRef]

1994 (2)

Yu. S. Kivshar and S. K. Turitsyn, Phys. Rev. E 49, R2536 (1994).
[CrossRef]

E. M. Wright, Chaos Solitons Fractals 4, 1805 (1994).
[CrossRef]

1993 (2)

1992 (1)

J. E. Rothenberg, Opt. Lett. 17, 584 (1992).

1991 (1)

1990 (1)

1986 (1)

J. Juul Rasmussen and K. Rypdal, Phys. Scr. 33, 481 (1986); K. Rypdal and J. Juul Rasmussen, Phys. Scr. 33, 498 (1986); Phys. Scr.4627 (1994); Applicable Anal. 57, 3 (1995).
[CrossRef]

1974 (1)

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, Radiophys. Quantum Electron. 14, 1062 (1974).
[CrossRef]

1964 (1)

E. A. Novikov, Zh. Eksp. Teor. Fiz. 457, 1919 (1964) [Sov. Phys. JETP 20, 1290 (1965)].

1963 (1)

K. Furutsu, J. Res. Nat. Bur. Stand. Sect. D 667, 303 (1963).

Acceves, A. B.

Andersen, D.

Bang, O.

K. Ø. Rasmussen, Yu. B. Gaididei, O. Bang, and P. L. Christiansen, Phys. Lett. 204, 121 (1995).
[CrossRef]

Bergé, L.

L. Bergé, E. Kuznetsov, and J. Juul. Rasmussen, Phys. Rev. E 53, R1340 (1996).
[CrossRef]

Christiansen, P. L.

P. L. Christiansen, Yu. Gaididei, M. Johansson, K. Ø. Rasmussen, and I. Yakimenko, Phys. Rev. E 54, 924 (1996).
[CrossRef]

K. Ø. Rasmussen, Yu. B. Gaididei, O. Bang, and P. L. Christiansen, Phys. Lett. 204, 121 (1995).
[CrossRef]

De Angelis, C.

de Araujo, R. E.

R. E. de Araujo and A. S. L. Gomes, Phys. Rev. A 57, 2037 (1998).
[CrossRef]

Desaix, M.

Flytzanis, C.

C. Flytzanis, F. Hach, M. C. Klein, D. Ricard, and P. Roussignol, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1991), Vol. XXIX, pp. 321–411.
[CrossRef]

Furutsu, K.

K. Furutsu, J. Res. Nat. Bur. Stand. Sect. D 667, 303 (1963).

Gaididei, Yu.

P. L. Christiansen, Yu. Gaididei, M. Johansson, K. Ø. Rasmussen, and I. Yakimenko, Phys. Rev. E 54, 924 (1996).
[CrossRef]

Gaididei, Yu. B.

K. Ø. Rasmussen, Yu. B. Gaididei, O. Bang, and P. L. Christiansen, Phys. Lett. 204, 121 (1995).
[CrossRef]

Gomes, A. S. L.

R. E. de Araujo and A. S. L. Gomes, Phys. Rev. A 57, 2037 (1998).
[CrossRef]

Hach, F.

C. Flytzanis, F. Hach, M. C. Klein, D. Ricard, and P. Roussignol, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1991), Vol. XXIX, pp. 321–411.
[CrossRef]

Johansson, M.

P. L. Christiansen, Yu. Gaididei, M. Johansson, K. Ø. Rasmussen, and I. Yakimenko, Phys. Rev. E 54, 924 (1996).
[CrossRef]

Kivshar, Yu. S.

Yu. S. Kivshar and S. K. Turitsyn, Phys. Rev. E 49, R2536 (1994).
[CrossRef]

Klein, M. C.

C. Flytzanis, F. Hach, M. C. Klein, D. Ricard, and P. Roussignol, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1991), Vol. XXIX, pp. 321–411.
[CrossRef]

Klyatskin, V. I.

V. I. Klyatskin, Stochastic Equations and Waves in Random Media (Nauka, Moscow, 1980) in Russian.

Konotop, V. V.

V. V. Konotop and L. Va?quez, Nonlinear Random Waves (World Scientific, Singapore, 1994).

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961).

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961).

Kuznetsov, E.

L. Bergé, E. Kuznetsov, and J. Juul. Rasmussen, Phys. Rev. E 53, R1340 (1996).
[CrossRef]

Lisak, M.

Novikov, E. A.

E. A. Novikov, Zh. Eksp. Teor. Fiz. 457, 1919 (1964) [Sov. Phys. JETP 20, 1290 (1965)].

Petrishchev, V. A.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, Radiophys. Quantum Electron. 14, 1062 (1974).
[CrossRef]

Rasmussen, J. Juul

J. Juul Rasmussen and K. Rypdal, Phys. Scr. 33, 481 (1986); K. Rypdal and J. Juul Rasmussen, Phys. Scr. 33, 498 (1986); Phys. Scr.4627 (1994); Applicable Anal. 57, 3 (1995).
[CrossRef]

Rasmussen, J. Juul.

L. Bergé, E. Kuznetsov, and J. Juul. Rasmussen, Phys. Rev. E 53, R1340 (1996).
[CrossRef]

Rasmussen, K. Ø.

P. L. Christiansen, Yu. Gaididei, M. Johansson, K. Ø. Rasmussen, and I. Yakimenko, Phys. Rev. E 54, 924 (1996).
[CrossRef]

K. Ø. Rasmussen, Yu. B. Gaididei, O. Bang, and P. L. Christiansen, Phys. Lett. 204, 121 (1995).
[CrossRef]

Ricard, D.

C. Flytzanis, F. Hach, M. C. Klein, D. Ricard, and P. Roussignol, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1991), Vol. XXIX, pp. 321–411.
[CrossRef]

Rothenberg, J. E.

J. E. Rothenberg, Opt. Lett. 17, 584 (1992).

Roussignol, P.

C. Flytzanis, F. Hach, M. C. Klein, D. Ricard, and P. Roussignol, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1991), Vol. XXIX, pp. 321–411.
[CrossRef]

Rypdal, K.

J. Juul Rasmussen and K. Rypdal, Phys. Scr. 33, 481 (1986); K. Rypdal and J. Juul Rasmussen, Phys. Scr. 33, 498 (1986); Phys. Scr.4627 (1994); Applicable Anal. 57, 3 (1995).
[CrossRef]

Silberberg, Y.

Talanov, V. I.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, Radiophys. Quantum Electron. 14, 1062 (1974).
[CrossRef]

Turitsyn, S. K.

Yu. S. Kivshar and S. K. Turitsyn, Phys. Rev. E 49, R2536 (1994).
[CrossRef]

S. K. Turitsyn, Opt. Lett. 18, 110 (1993).
[CrossRef]

Vazquez, L.

V. V. Konotop and L. Va?quez, Nonlinear Random Waves (World Scientific, Singapore, 1994).

Vlasov, S. N.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, Radiophys. Quantum Electron. 14, 1062 (1974).
[CrossRef]

Wright, E. M.

E. M. Wright, Chaos Solitons Fractals 4, 1805 (1994).
[CrossRef]

Yakimenko, I.

P. L. Christiansen, Yu. Gaididei, M. Johansson, K. Ø. Rasmussen, and I. Yakimenko, Phys. Rev. E 54, 924 (1996).
[CrossRef]

Chaos Solitons Fractals (1)

E. M. Wright, Chaos Solitons Fractals 4, 1805 (1994).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Res. Nat. Bur. Stand. Sect. D (1)

K. Furutsu, J. Res. Nat. Bur. Stand. Sect. D 667, 303 (1963).

Opt. Lett. (4)

Phys. Lett. (1)

K. Ø. Rasmussen, Yu. B. Gaididei, O. Bang, and P. L. Christiansen, Phys. Lett. 204, 121 (1995).
[CrossRef]

Phys. Rev. A (1)

R. E. de Araujo and A. S. L. Gomes, Phys. Rev. A 57, 2037 (1998).
[CrossRef]

Phys. Rev. E (3)

L. Bergé, E. Kuznetsov, and J. Juul. Rasmussen, Phys. Rev. E 53, R1340 (1996).
[CrossRef]

P. L. Christiansen, Yu. Gaididei, M. Johansson, K. Ø. Rasmussen, and I. Yakimenko, Phys. Rev. E 54, 924 (1996).
[CrossRef]

Yu. S. Kivshar and S. K. Turitsyn, Phys. Rev. E 49, R2536 (1994).
[CrossRef]

Phys. Scr. (1)

J. Juul Rasmussen and K. Rypdal, Phys. Scr. 33, 481 (1986); K. Rypdal and J. Juul Rasmussen, Phys. Scr. 33, 498 (1986); Phys. Scr.4627 (1994); Applicable Anal. 57, 3 (1995).
[CrossRef]

Radiophys. Quantum Electron. (1)

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, Radiophys. Quantum Electron. 14, 1062 (1974).
[CrossRef]

Zh. Eksp. Teor. Fiz. (1)

E. A. Novikov, Zh. Eksp. Teor. Fiz. 457, 1919 (1964) [Sov. Phys. JETP 20, 1290 (1965)].

Other (4)

C. Flytzanis, F. Hach, M. C. Klein, D. Ricard, and P. Roussignol, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1991), Vol. XXIX, pp. 321–411.
[CrossRef]

V. I. Klyatskin, Stochastic Equations and Waves in Random Media (Nauka, Moscow, 1980) in Russian.

V. V. Konotop and L. Va?quez, Nonlinear Random Waves (World Scientific, Singapore, 1994).

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961).

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Equations (18)

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izψ+2ψ+fψ2,x,z,tψ=0,
fψ2,x,z,t=ψ2+ηx,z,t,
ηx,z=0,ηx1,z1ηx2,z2ηxn,zn=Knx1,x2,,xnδz1-z2×δz2-z3δzn-1-zn,
Xz=1N-dx-dtxρx,z,t,Iz=1N-dx-dtx2+t2ρx,z,t.
Nd2dz2Xz=2-dx-dtρx,z,txηx,z,
N4d2dz2Iz=2H+-dx-dtρx,z,txxηx,z,
H=-dx-dttψ2+xψ2-12ψ4
ddzH=-dx-dtηx,zzρx,z,t.
ηx,zRη=12n=11n!-dx1×-dxnKn+1x,x1,xn×δnRδηx1,zδηxn,z,
zρ=τ, zτ=Lψ,ψ*-2xρxη,
δδηx,zzρx,z,t=-2xρx,z,txδx-x,δδηx,zρx,z,t=0,δnδηx1,zδηxn,zzρx,z,t=0  for n2.
Nd2/dz2Iz¯=8Hz¯,d/dzHz¯=κN,
Iz¯=B2+4z2+43κz3,
zcB21+κ123/2B,
κcr=43/2B3
ηx,z,t=η coskx+azcoskx+α,
ηx,z=nanxgz-zn,
κ=w2/lA0.

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