Abstract

We study the propagation of stationary waves in disordered nonlinear media described by the nonlinear Schrödinger equation and show that, for given boundary conditions and a given coherent wave incident on the sample, the number of solutions of the equation increases exponentially with sample size. We also discuss the ballistic case, the sensitivity of the solutions to the change of external parameters, the similarity of this problem to the problem of spin glasses, and time-dependent solutions.

© 2004 Optical Society of America

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References

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  1. L. Landau, E. Lifshitz, and L. P. Pitaevski, Electrodynamics of Continuous Media, 2nd ed. (Elsevier, New York, 1985).
  2. V. E. Zakharov, V. S. Lvov, and G. Falkovich, Kolmogorov Spectra of Turbulence (Springer-Verlag, Berlin, 1992).
    [CrossRef]
  3. B. B. Kadomzev, Collective Phenomena in Plasma (Nauka, Moscow, 1976).
  4. A. Zyuzin and B. Spivak, “Langevin description of mesoscopic fluctuations in disordered media,” Sov. Phys. JETP 66, 560–565 (1987).
  5. S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
    [CrossRef] [PubMed]
  6. B. Spivak and A. Zyuzin, “Mesoscopic fluctuations of current density in disordered conductors,” in Mesoscopic Phenomena in Solids, B. Altshuler, P. Lee, and R. Webb, eds., Vol. 30 of Modern Problems in Condensed Matter Sciences (North-Holland, Amsterdam, 1991), pp. 37–80.
    [CrossRef]
  7. B. Spivak and A. Zyuzin, “Mesoscopic sensitivity of speckles in disordered nonlinear media to changes of disordered potential,” Phys. Rev. Lett. 84, 1970–1073 (2000).
    [CrossRef] [PubMed]
  8. P. A. Lee and A. D. Stone, “Universal conductance fluctuations in metals,” Phys. Rev. Lett. 55, 1622–1625 (1985).
    [CrossRef] [PubMed]
  9. A. Abrikosov, L. Gorkov, and I. Dzyialoshinski, Methods of Quantum Field Theory in Statistical Physics (Dover, New York, 1977).
  10. B. Altshuler and B. Spivak, “Change of random potential realization and conductivity of small size sample,” JETP Lett. 42, 447–449 (1986).
  11. B. D. Simons and B. L. Altshuler, “Universalities in the spectra of disordered and chaotic systems,” Phys. Rev. B 48, 5422–5425 (1993).
    [CrossRef]
  12. S. Feng, P. Lee, and A. D. Stone, “Sensitivity of the conductance of a disordered metal to the motion of a single atom,” Phys. Rev. Lett. 56, 1960–1063 (1986).
    [CrossRef] [PubMed]
  13. A. M. Finkelshtein, “Electron liquid in disordered conductors,” Sov. Sci. Rev., Sect. A 14, 1–42 (1990).
  14. S. E. Skipetrov, “Temporal fluctuations of waves in weakly nonlinear disordered media,” Phys. Rev. E 63, 056614 (2001).
    [CrossRef]
  15. S. E. Skipetrov and R. Maynard, Diffusive Waves in Nonlinear Disordered Media, Vol. 107 of NATO Science Series on Mathematics, Physics and Chemistry, B. A. van Tiggelin and S. E. Skipetrov, eds. (Kluwer Academic, Dordrecht, The Netherlands, 2003).
  16. S. E. Skipetrov, “Langevin description of speckle dynamics in nonlinear disordered media,” Phys. Rev. E 67, 016601 (2003).
    [CrossRef]
  17. S. E. Skipetrov, “Instability of speckle patterns in random media with noninstantaneous Kerr nonlinearity,” Opt. Lett. 28, 646–648 (2003).
    [CrossRef] [PubMed]

2003 (2)

S. E. Skipetrov, “Langevin description of speckle dynamics in nonlinear disordered media,” Phys. Rev. E 67, 016601 (2003).
[CrossRef]

S. E. Skipetrov, “Instability of speckle patterns in random media with noninstantaneous Kerr nonlinearity,” Opt. Lett. 28, 646–648 (2003).
[CrossRef] [PubMed]

2001 (1)

S. E. Skipetrov, “Temporal fluctuations of waves in weakly nonlinear disordered media,” Phys. Rev. E 63, 056614 (2001).
[CrossRef]

2000 (1)

B. Spivak and A. Zyuzin, “Mesoscopic sensitivity of speckles in disordered nonlinear media to changes of disordered potential,” Phys. Rev. Lett. 84, 1970–1073 (2000).
[CrossRef] [PubMed]

1993 (1)

B. D. Simons and B. L. Altshuler, “Universalities in the spectra of disordered and chaotic systems,” Phys. Rev. B 48, 5422–5425 (1993).
[CrossRef]

1990 (1)

A. M. Finkelshtein, “Electron liquid in disordered conductors,” Sov. Sci. Rev., Sect. A 14, 1–42 (1990).

1988 (1)

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

1987 (1)

A. Zyuzin and B. Spivak, “Langevin description of mesoscopic fluctuations in disordered media,” Sov. Phys. JETP 66, 560–565 (1987).

1986 (2)

B. Altshuler and B. Spivak, “Change of random potential realization and conductivity of small size sample,” JETP Lett. 42, 447–449 (1986).

S. Feng, P. Lee, and A. D. Stone, “Sensitivity of the conductance of a disordered metal to the motion of a single atom,” Phys. Rev. Lett. 56, 1960–1063 (1986).
[CrossRef] [PubMed]

1985 (1)

P. A. Lee and A. D. Stone, “Universal conductance fluctuations in metals,” Phys. Rev. Lett. 55, 1622–1625 (1985).
[CrossRef] [PubMed]

Altshuler, B.

B. Altshuler and B. Spivak, “Change of random potential realization and conductivity of small size sample,” JETP Lett. 42, 447–449 (1986).

Altshuler, B. L.

B. D. Simons and B. L. Altshuler, “Universalities in the spectra of disordered and chaotic systems,” Phys. Rev. B 48, 5422–5425 (1993).
[CrossRef]

Feng, S.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

S. Feng, P. Lee, and A. D. Stone, “Sensitivity of the conductance of a disordered metal to the motion of a single atom,” Phys. Rev. Lett. 56, 1960–1063 (1986).
[CrossRef] [PubMed]

Finkelshtein, A. M.

A. M. Finkelshtein, “Electron liquid in disordered conductors,” Sov. Sci. Rev., Sect. A 14, 1–42 (1990).

Kane, C.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Lee, P.

S. Feng, P. Lee, and A. D. Stone, “Sensitivity of the conductance of a disordered metal to the motion of a single atom,” Phys. Rev. Lett. 56, 1960–1063 (1986).
[CrossRef] [PubMed]

Lee, P. A.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

P. A. Lee and A. D. Stone, “Universal conductance fluctuations in metals,” Phys. Rev. Lett. 55, 1622–1625 (1985).
[CrossRef] [PubMed]

Simons, B. D.

B. D. Simons and B. L. Altshuler, “Universalities in the spectra of disordered and chaotic systems,” Phys. Rev. B 48, 5422–5425 (1993).
[CrossRef]

Skipetrov, S. E.

S. E. Skipetrov, “Langevin description of speckle dynamics in nonlinear disordered media,” Phys. Rev. E 67, 016601 (2003).
[CrossRef]

S. E. Skipetrov, “Instability of speckle patterns in random media with noninstantaneous Kerr nonlinearity,” Opt. Lett. 28, 646–648 (2003).
[CrossRef] [PubMed]

S. E. Skipetrov, “Temporal fluctuations of waves in weakly nonlinear disordered media,” Phys. Rev. E 63, 056614 (2001).
[CrossRef]

Spivak, B.

B. Spivak and A. Zyuzin, “Mesoscopic sensitivity of speckles in disordered nonlinear media to changes of disordered potential,” Phys. Rev. Lett. 84, 1970–1073 (2000).
[CrossRef] [PubMed]

A. Zyuzin and B. Spivak, “Langevin description of mesoscopic fluctuations in disordered media,” Sov. Phys. JETP 66, 560–565 (1987).

B. Altshuler and B. Spivak, “Change of random potential realization and conductivity of small size sample,” JETP Lett. 42, 447–449 (1986).

Stone, A. D.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

S. Feng, P. Lee, and A. D. Stone, “Sensitivity of the conductance of a disordered metal to the motion of a single atom,” Phys. Rev. Lett. 56, 1960–1063 (1986).
[CrossRef] [PubMed]

P. A. Lee and A. D. Stone, “Universal conductance fluctuations in metals,” Phys. Rev. Lett. 55, 1622–1625 (1985).
[CrossRef] [PubMed]

Zyuzin, A.

B. Spivak and A. Zyuzin, “Mesoscopic sensitivity of speckles in disordered nonlinear media to changes of disordered potential,” Phys. Rev. Lett. 84, 1970–1073 (2000).
[CrossRef] [PubMed]

A. Zyuzin and B. Spivak, “Langevin description of mesoscopic fluctuations in disordered media,” Sov. Phys. JETP 66, 560–565 (1987).

JETP Lett. (1)

B. Altshuler and B. Spivak, “Change of random potential realization and conductivity of small size sample,” JETP Lett. 42, 447–449 (1986).

Opt. Lett. (1)

Phys. Rev. B (1)

B. D. Simons and B. L. Altshuler, “Universalities in the spectra of disordered and chaotic systems,” Phys. Rev. B 48, 5422–5425 (1993).
[CrossRef]

Phys. Rev. E (2)

S. E. Skipetrov, “Langevin description of speckle dynamics in nonlinear disordered media,” Phys. Rev. E 67, 016601 (2003).
[CrossRef]

S. E. Skipetrov, “Temporal fluctuations of waves in weakly nonlinear disordered media,” Phys. Rev. E 63, 056614 (2001).
[CrossRef]

Phys. Rev. Lett. (4)

B. Spivak and A. Zyuzin, “Mesoscopic sensitivity of speckles in disordered nonlinear media to changes of disordered potential,” Phys. Rev. Lett. 84, 1970–1073 (2000).
[CrossRef] [PubMed]

P. A. Lee and A. D. Stone, “Universal conductance fluctuations in metals,” Phys. Rev. Lett. 55, 1622–1625 (1985).
[CrossRef] [PubMed]

S. Feng, P. Lee, and A. D. Stone, “Sensitivity of the conductance of a disordered metal to the motion of a single atom,” Phys. Rev. Lett. 56, 1960–1063 (1986).
[CrossRef] [PubMed]

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

A. Zyuzin and B. Spivak, “Langevin description of mesoscopic fluctuations in disordered media,” Sov. Phys. JETP 66, 560–565 (1987).

Sov. Sci. Rev., Sect. A (1)

A. M. Finkelshtein, “Electron liquid in disordered conductors,” Sov. Sci. Rev., Sect. A 14, 1–42 (1990).

Other (6)

B. Spivak and A. Zyuzin, “Mesoscopic fluctuations of current density in disordered conductors,” in Mesoscopic Phenomena in Solids, B. Altshuler, P. Lee, and R. Webb, eds., Vol. 30 of Modern Problems in Condensed Matter Sciences (North-Holland, Amsterdam, 1991), pp. 37–80.
[CrossRef]

L. Landau, E. Lifshitz, and L. P. Pitaevski, Electrodynamics of Continuous Media, 2nd ed. (Elsevier, New York, 1985).

V. E. Zakharov, V. S. Lvov, and G. Falkovich, Kolmogorov Spectra of Turbulence (Springer-Verlag, Berlin, 1992).
[CrossRef]

B. B. Kadomzev, Collective Phenomena in Plasma (Nauka, Moscow, 1976).

A. Abrikosov, L. Gorkov, and I. Dzyialoshinski, Methods of Quantum Field Theory in Statistical Physics (Dover, New York, 1977).

S. E. Skipetrov and R. Maynard, Diffusive Waves in Nonlinear Disordered Media, Vol. 107 of NATO Science Series on Mathematics, Physics and Chemistry, B. A. van Tiggelin and S. E. Skipetrov, eds. (Kluwer Academic, Dordrecht, The Netherlands, 2003).

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Figures (2)

Fig. 1
Fig. 1

(a) Graphical solution of Eq. (7). Wavy curve corresponds to Fi(u¯i), two straight lines correspond to γ-1i2/3u¯i at different values of γ. Dashed line indicates critical instability point at which two solutions of Eq. (7) appear for the first time; the solutions are marked a and b. (b) Solid curves correspond to the intersection F1(u¯1, u¯2) and γ-1u¯1, dashed curves, to the intersection of F2(u¯1, u¯2) and γ-122/3u¯2.

Fig. 2
Fig. 2

(a) Diagrams of n(r). Solid lines correspond to Green functions of Eq. (1) with β=0, dashed lines, to πδ(r-r)/lm2. (b), (c) Diagrams of Eq. (24). Inner solid lines describe Green functions that correspond to {u¯i}; outer solid lines correspond to {u¯i+Δu¯i}.

Equations (38)

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-12m 2r2-+u(r)+βn(r)ϕ(r)=0.
j(r)=-D n(r)r,divj(r)=0,
|βn0|2k/lm.
βn(r)=DL i=1i1/3u¯ini(r)
D 22rni(r)=Eini(r),
-12m 2r2-+u(r)+β DL i=1i1/3u¯ini(r)ϕ(r)=0.
i2/3u¯iγ=Fi({u¯i}).
γ=3n0β2Ll3/2,
Fi({u¯i})=ki1/3l1/2n0L drn[r, {u(r)}, {u¯i}]ni(r),
δFi({u¯i})δFj({u¯i})=δij,
[Fi({u¯i+Δu¯i})-Fi({u¯i})]2(Δu¯n)2
Δu¯i1,
Fiu¯r×Fju¯s[(r/s)1/3+(s/r)1/3]
×1(|i-j|+|r-s|)2/3.
NγI 1I i-2/3exp23γ3/2.
NI=2γ×2-2/3γ.
n(r)n(r)=n02/m2k2|r-r|2,
l(NL)=k2(βn0)2m.
γ(NL)=32 (n0β)4 Lmk3/2.
(δχ(NL))2=βk22dsdsδn(r)δn(r).
Fi({u¯i})Fj({u¯i+Δu¯i})
=i1/3j1/3lk2L2n02 drdrni(r)nj(r)n(r, {u¯l})×n(r1, {u¯l+Δu¯l}).
div δj(r)=0,
δj(r)=-D rδn(r)+JL[r, {u(r)}, {u¯i}].
JiL(r, {u(r}, {u¯i})JjL[r, {u(r)}, {u¯i+Δu¯i}]
=2πl3m2|ϕ(r, {u¯l})ϕ*(r, {u¯l+Δu¯l})|2δijδ(r-r).
n(r, {u¯l})n(r, {u¯l+Δu¯l})
=2πl3m2 dr1 d(r, r1)dr1 d(r, r1)dr1×|ϕ(r, {u¯l})ϕ*(r, {u¯l+Δu¯l})|2,
(r, r)=l nl(r)nl(r)El.
D 22r+i DL i i1/3Δu¯ini(r)
×ϕ(r, {u¯l})ϕ*(r, {u¯l+Δu¯l})=0.
ϕ(r, {u¯l})ϕ*(r, {u¯l+Δu¯l})
=n01+i DL ii1/3 ni(r)EiΔu¯i-D2L dr(r, r)i i2/3[Δu¯ini(r)]2Ei2+.
Δθexp-23γ3/2
β(r)=β0 αU(r-rα),
ϕα=β0U0R3 βG(rα, rβ)|ϕβ|2ϕβ,
G(rα, rβ)exp(ik|rα-rβ|+iδ(rα, rβ)|rα-rβ|1/2|rα-rβ|l,
τi[1+γgi({u¯i})]tu¯i(t)=γi-2/3F({u¯i})-u¯i.

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