Abstract

The first analytic verification of the stability of topological solitons corresponding to wave-front solutions of the optical parametric oscillator is given. A translational invariance allows perturbations to the system to shift the front position without affecting the underlying exponential stability of the fronts themselves. For the two-dimensional problem with a positive O(1) signal detuning, the front curvature is shown to be governed by a heat equation, so that the only stable topological solitons supported must be stripes.

© 1999 Optical Society of America

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References

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  1. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).
  2. See, for instance, the special issues on Optical Parametric Oscillation and Amplification, J. Opt. Soc. Am. B 10, 1655–1791 (1993) and Optical Parametric Devices, J. Opt. Soc. Am. B 12, 2083–2320 (1995).
  3. M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
    [CrossRef]
  4. G. L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994).
    [CrossRef] [PubMed]
  5. R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
    [CrossRef]
  6. S. Longhi, “Localized structures in optical parametric oscillation,” Phys. Scr. 56, 611–618 (1997).
    [CrossRef]
  7. K. Staliunas and V. J. Sánchez-Morcillo, “Localized structures in degenerate optical parametric oscillators,” Opt. Commun. 139, 306–312 (1997).
    [CrossRef]
  8. S. Trillo, M. Haelterman, and A. Sheppard, “Stable topological spatial solitons in optical parametric oscillators,” Opt. Lett. 22, 970–972 (1997).
    [CrossRef] [PubMed]
  9. K. Staliunas and V. J. Sánchez-Morcillo, “Spatial-localized structures in degenerate optical parametric oscillators,” Phys. Rev. A 57, 1454–1457 (1998).
    [CrossRef]
  10. S. Longhi, “Spatial solitary waves in nondegenerate optical parametric oscillators near an inverted bifurcation,” Opt. Commun. 149, 335–340 (1998).
    [CrossRef]
  11. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Walk-off and pattern selection in optical parametric oscillators,” Opt. Lett. 23, 1167–1169 (1998).
    [CrossRef]
  12. S. Trillo and M. Haelterman, “Excitation and bistability of self-trapped signal beams in optical parametric oscillators,” Opt. Lett. 23, 1514–1516 (1998).
    [CrossRef]
  13. M. Tidli, P. Mandel, and R. Lefever, “Kinetics of localized pattern formation in optical systems,” Phys. Rev. Lett. 81, 979–982 (1998).
    [CrossRef]
  14. C. Etrich, U. Peschel, and F. Lederer, “Solitary waves in quadratically nonlinear resonators,” Phys. Rev. Lett. 79, 2454–2457 (1997).
    [CrossRef]
  15. N. N. Rozanov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1996), Vol. 35, pp. 1–60.
  16. K. Staliunas and V. J. Sánchez-Morcillo, “Dynamics of phase domains in the Swift–Hohenberg equation,” Phys. Lett. A 241, 28–31 (1998).
    [CrossRef]
  17. P. Coullet, L. Gil, and D. Repaux, “Defects and subcritical bifurcations,” Phys. Rev. Lett. 62, 2957–2960 (1989).
    [CrossRef] [PubMed]
  18. V. B. Tarenko, K. Staliunas, and C. O. Weiss, “Pattern formation and localized structures in degenerate optical parametric mixing,” Phys. Rev. Lett. 81, 2236–2239 (1998).
    [CrossRef]
  19. C. Richy, K. I. Petsas, E. Giacobino, and C. Fabre, “Observation of bistability and delayed bifurcation in a triply resonant optical parametric oscillators,” J. Opt. Soc. Am. B 12, 456–461 (1995).
    [CrossRef]
  20. G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Stabilization of domain walls in optical parametric oscillators,” special issue on Pattern Formation in Nonlinear Optical System, Quant. Semiclass. Opt. 1, 133–138 (1999).
    [CrossRef]
  21. B. Friedman, Principles and Techniques of Applied Mathematics (Dover, New York, 1990), Chap. 1.
  22. G. T. Dee and W. Van Sarloos, “Bistable system with propagating fronts leading to pattern formation,” Phys. Rev. Lett. 60, 2641–2644 (1988).
    [CrossRef] [PubMed]
  23. M. Weinstein, “Modulational stability of ground states of nonlinear Schrödinger equation,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 16, 472–491 (1985).
    [CrossRef]
  24. J. N. Kutz and W. L. Kath, “Stability of pulses in nonlinear optical fibers using phase-sensitive amplifiers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 611–626 (1996).
    [CrossRef]
  25. D. Gruner, R. Kapral, and A. Lawniczak, “Nucleation, domain growth and fluctuations in a bistable chemical system,” J. Chem. Phys. 99, 3938–3945 (1993).
    [CrossRef]

1999 (1)

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Stabilization of domain walls in optical parametric oscillators,” special issue on Pattern Formation in Nonlinear Optical System, Quant. Semiclass. Opt. 1, 133–138 (1999).
[CrossRef]

1998 (7)

K. Staliunas and V. J. Sánchez-Morcillo, “Dynamics of phase domains in the Swift–Hohenberg equation,” Phys. Lett. A 241, 28–31 (1998).
[CrossRef]

V. B. Tarenko, K. Staliunas, and C. O. Weiss, “Pattern formation and localized structures in degenerate optical parametric mixing,” Phys. Rev. Lett. 81, 2236–2239 (1998).
[CrossRef]

K. Staliunas and V. J. Sánchez-Morcillo, “Spatial-localized structures in degenerate optical parametric oscillators,” Phys. Rev. A 57, 1454–1457 (1998).
[CrossRef]

S. Longhi, “Spatial solitary waves in nondegenerate optical parametric oscillators near an inverted bifurcation,” Opt. Commun. 149, 335–340 (1998).
[CrossRef]

M. Tidli, P. Mandel, and R. Lefever, “Kinetics of localized pattern formation in optical systems,” Phys. Rev. Lett. 81, 979–982 (1998).
[CrossRef]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Walk-off and pattern selection in optical parametric oscillators,” Opt. Lett. 23, 1167–1169 (1998).
[CrossRef]

S. Trillo and M. Haelterman, “Excitation and bistability of self-trapped signal beams in optical parametric oscillators,” Opt. Lett. 23, 1514–1516 (1998).
[CrossRef]

1997 (5)

S. Trillo, M. Haelterman, and A. Sheppard, “Stable topological spatial solitons in optical parametric oscillators,” Opt. Lett. 22, 970–972 (1997).
[CrossRef] [PubMed]

C. Etrich, U. Peschel, and F. Lederer, “Solitary waves in quadratically nonlinear resonators,” Phys. Rev. Lett. 79, 2454–2457 (1997).
[CrossRef]

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

S. Longhi, “Localized structures in optical parametric oscillation,” Phys. Scr. 56, 611–618 (1997).
[CrossRef]

K. Staliunas and V. J. Sánchez-Morcillo, “Localized structures in degenerate optical parametric oscillators,” Opt. Commun. 139, 306–312 (1997).
[CrossRef]

1996 (1)

J. N. Kutz and W. L. Kath, “Stability of pulses in nonlinear optical fibers using phase-sensitive amplifiers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 611–626 (1996).
[CrossRef]

1995 (1)

1994 (1)

G. L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994).
[CrossRef] [PubMed]

1993 (2)

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
[CrossRef]

D. Gruner, R. Kapral, and A. Lawniczak, “Nucleation, domain growth and fluctuations in a bistable chemical system,” J. Chem. Phys. 99, 3938–3945 (1993).
[CrossRef]

1989 (1)

P. Coullet, L. Gil, and D. Repaux, “Defects and subcritical bifurcations,” Phys. Rev. Lett. 62, 2957–2960 (1989).
[CrossRef] [PubMed]

1988 (1)

G. T. Dee and W. Van Sarloos, “Bistable system with propagating fronts leading to pattern formation,” Phys. Rev. Lett. 60, 2641–2644 (1988).
[CrossRef] [PubMed]

1985 (1)

M. Weinstein, “Modulational stability of ground states of nonlinear Schrödinger equation,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 16, 472–491 (1985).
[CrossRef]

Baboiu, D. M.

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Brambilla, M.

G. L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994).
[CrossRef] [PubMed]

Colet, P.

Coullet, P.

P. Coullet, L. Gil, and D. Repaux, “Defects and subcritical bifurcations,” Phys. Rev. Lett. 62, 2957–2960 (1989).
[CrossRef] [PubMed]

Cross, M. C.

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
[CrossRef]

Dee, G. T.

G. T. Dee and W. Van Sarloos, “Bistable system with propagating fronts leading to pattern formation,” Phys. Rev. Lett. 60, 2641–2644 (1988).
[CrossRef] [PubMed]

Etrich, C.

C. Etrich, U. Peschel, and F. Lederer, “Solitary waves in quadratically nonlinear resonators,” Phys. Rev. Lett. 79, 2454–2457 (1997).
[CrossRef]

Fabre, C.

Firth, W. J.

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Stabilization of domain walls in optical parametric oscillators,” special issue on Pattern Formation in Nonlinear Optical System, Quant. Semiclass. Opt. 1, 133–138 (1999).
[CrossRef]

Fuerst, R. A.

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Giacobino, E.

Gil, L.

P. Coullet, L. Gil, and D. Repaux, “Defects and subcritical bifurcations,” Phys. Rev. Lett. 62, 2957–2960 (1989).
[CrossRef] [PubMed]

Gruner, D.

D. Gruner, R. Kapral, and A. Lawniczak, “Nucleation, domain growth and fluctuations in a bistable chemical system,” J. Chem. Phys. 99, 3938–3945 (1993).
[CrossRef]

Haelterman, M.

Hohenberg, P. C.

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
[CrossRef]

Kapral, R.

D. Gruner, R. Kapral, and A. Lawniczak, “Nucleation, domain growth and fluctuations in a bistable chemical system,” J. Chem. Phys. 99, 3938–3945 (1993).
[CrossRef]

Kath, W. L.

J. N. Kutz and W. L. Kath, “Stability of pulses in nonlinear optical fibers using phase-sensitive amplifiers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 611–626 (1996).
[CrossRef]

Kutz, J. N.

J. N. Kutz and W. L. Kath, “Stability of pulses in nonlinear optical fibers using phase-sensitive amplifiers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 611–626 (1996).
[CrossRef]

Lawniczak, A.

D. Gruner, R. Kapral, and A. Lawniczak, “Nucleation, domain growth and fluctuations in a bistable chemical system,” J. Chem. Phys. 99, 3938–3945 (1993).
[CrossRef]

Lawrence, B.

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Lederer, F.

C. Etrich, U. Peschel, and F. Lederer, “Solitary waves in quadratically nonlinear resonators,” Phys. Rev. Lett. 79, 2454–2457 (1997).
[CrossRef]

Lefever, R.

M. Tidli, P. Mandel, and R. Lefever, “Kinetics of localized pattern formation in optical systems,” Phys. Rev. Lett. 81, 979–982 (1998).
[CrossRef]

Longhi, S.

S. Longhi, “Spatial solitary waves in nondegenerate optical parametric oscillators near an inverted bifurcation,” Opt. Commun. 149, 335–340 (1998).
[CrossRef]

S. Longhi, “Localized structures in optical parametric oscillation,” Phys. Scr. 56, 611–618 (1997).
[CrossRef]

Lugiato, L. A.

G. L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994).
[CrossRef] [PubMed]

Mandel, P.

M. Tidli, P. Mandel, and R. Lefever, “Kinetics of localized pattern formation in optical systems,” Phys. Rev. Lett. 81, 979–982 (1998).
[CrossRef]

Oppo, G. L.

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Stabilization of domain walls in optical parametric oscillators,” special issue on Pattern Formation in Nonlinear Optical System, Quant. Semiclass. Opt. 1, 133–138 (1999).
[CrossRef]

G. L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994).
[CrossRef] [PubMed]

Peschel, U.

C. Etrich, U. Peschel, and F. Lederer, “Solitary waves in quadratically nonlinear resonators,” Phys. Rev. Lett. 79, 2454–2457 (1997).
[CrossRef]

Petsas, K. I.

Repaux, D.

P. Coullet, L. Gil, and D. Repaux, “Defects and subcritical bifurcations,” Phys. Rev. Lett. 62, 2957–2960 (1989).
[CrossRef] [PubMed]

Richy, C.

San Miguel, M.

Sánchez-Morcillo, V. J.

K. Staliunas and V. J. Sánchez-Morcillo, “Spatial-localized structures in degenerate optical parametric oscillators,” Phys. Rev. A 57, 1454–1457 (1998).
[CrossRef]

K. Staliunas and V. J. Sánchez-Morcillo, “Dynamics of phase domains in the Swift–Hohenberg equation,” Phys. Lett. A 241, 28–31 (1998).
[CrossRef]

K. Staliunas and V. J. Sánchez-Morcillo, “Localized structures in degenerate optical parametric oscillators,” Opt. Commun. 139, 306–312 (1997).
[CrossRef]

Santagiustina, M.

Scroggie, A. J.

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Stabilization of domain walls in optical parametric oscillators,” special issue on Pattern Formation in Nonlinear Optical System, Quant. Semiclass. Opt. 1, 133–138 (1999).
[CrossRef]

Sheppard, A.

Staliunas, K.

K. Staliunas and V. J. Sánchez-Morcillo, “Spatial-localized structures in degenerate optical parametric oscillators,” Phys. Rev. A 57, 1454–1457 (1998).
[CrossRef]

K. Staliunas and V. J. Sánchez-Morcillo, “Dynamics of phase domains in the Swift–Hohenberg equation,” Phys. Lett. A 241, 28–31 (1998).
[CrossRef]

V. B. Tarenko, K. Staliunas, and C. O. Weiss, “Pattern formation and localized structures in degenerate optical parametric mixing,” Phys. Rev. Lett. 81, 2236–2239 (1998).
[CrossRef]

K. Staliunas and V. J. Sánchez-Morcillo, “Localized structures in degenerate optical parametric oscillators,” Opt. Commun. 139, 306–312 (1997).
[CrossRef]

Stegeman, G. I.

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Tarenko, V. B.

V. B. Tarenko, K. Staliunas, and C. O. Weiss, “Pattern formation and localized structures in degenerate optical parametric mixing,” Phys. Rev. Lett. 81, 2236–2239 (1998).
[CrossRef]

Tidli, M.

M. Tidli, P. Mandel, and R. Lefever, “Kinetics of localized pattern formation in optical systems,” Phys. Rev. Lett. 81, 979–982 (1998).
[CrossRef]

Torruellas, W. E.

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Trillo, S.

S. Trillo and M. Haelterman, “Excitation and bistability of self-trapped signal beams in optical parametric oscillators,” Opt. Lett. 23, 1514–1516 (1998).
[CrossRef]

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

S. Trillo, M. Haelterman, and A. Sheppard, “Stable topological spatial solitons in optical parametric oscillators,” Opt. Lett. 22, 970–972 (1997).
[CrossRef] [PubMed]

Van Sarloos, W.

G. T. Dee and W. Van Sarloos, “Bistable system with propagating fronts leading to pattern formation,” Phys. Rev. Lett. 60, 2641–2644 (1988).
[CrossRef] [PubMed]

Wabnitz, S.

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Walgraef, D.

Weinstein, M.

M. Weinstein, “Modulational stability of ground states of nonlinear Schrödinger equation,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 16, 472–491 (1985).
[CrossRef]

Weiss, C. O.

V. B. Tarenko, K. Staliunas, and C. O. Weiss, “Pattern formation and localized structures in degenerate optical parametric mixing,” Phys. Rev. Lett. 81, 2236–2239 (1998).
[CrossRef]

J. Chem. Phys. (1)

D. Gruner, R. Kapral, and A. Lawniczak, “Nucleation, domain growth and fluctuations in a bistable chemical system,” J. Chem. Phys. 99, 3938–3945 (1993).
[CrossRef]

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

Opt. Commun. (2)

S. Longhi, “Spatial solitary waves in nondegenerate optical parametric oscillators near an inverted bifurcation,” Opt. Commun. 149, 335–340 (1998).
[CrossRef]

K. Staliunas and V. J. Sánchez-Morcillo, “Localized structures in degenerate optical parametric oscillators,” Opt. Commun. 139, 306–312 (1997).
[CrossRef]

Opt. Lett. (3)

Phys. Lett. A (1)

K. Staliunas and V. J. Sánchez-Morcillo, “Dynamics of phase domains in the Swift–Hohenberg equation,” Phys. Lett. A 241, 28–31 (1998).
[CrossRef]

Phys. Rev. A (2)

K. Staliunas and V. J. Sánchez-Morcillo, “Spatial-localized structures in degenerate optical parametric oscillators,” Phys. Rev. A 57, 1454–1457 (1998).
[CrossRef]

G. L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994).
[CrossRef] [PubMed]

Phys. Rev. Lett. (6)

R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

M. Tidli, P. Mandel, and R. Lefever, “Kinetics of localized pattern formation in optical systems,” Phys. Rev. Lett. 81, 979–982 (1998).
[CrossRef]

C. Etrich, U. Peschel, and F. Lederer, “Solitary waves in quadratically nonlinear resonators,” Phys. Rev. Lett. 79, 2454–2457 (1997).
[CrossRef]

G. T. Dee and W. Van Sarloos, “Bistable system with propagating fronts leading to pattern formation,” Phys. Rev. Lett. 60, 2641–2644 (1988).
[CrossRef] [PubMed]

P. Coullet, L. Gil, and D. Repaux, “Defects and subcritical bifurcations,” Phys. Rev. Lett. 62, 2957–2960 (1989).
[CrossRef] [PubMed]

V. B. Tarenko, K. Staliunas, and C. O. Weiss, “Pattern formation and localized structures in degenerate optical parametric mixing,” Phys. Rev. Lett. 81, 2236–2239 (1998).
[CrossRef]

Phys. Scr. (1)

S. Longhi, “Localized structures in optical parametric oscillation,” Phys. Scr. 56, 611–618 (1997).
[CrossRef]

Quant. Semiclass. Opt. (1)

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Stabilization of domain walls in optical parametric oscillators,” special issue on Pattern Formation in Nonlinear Optical System, Quant. Semiclass. Opt. 1, 133–138 (1999).
[CrossRef]

Rev. Mod. Phys. (1)

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (1)

J. N. Kutz and W. L. Kath, “Stability of pulses in nonlinear optical fibers using phase-sensitive amplifiers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 611–626 (1996).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. (1)

M. Weinstein, “Modulational stability of ground states of nonlinear Schrödinger equation,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 16, 472–491 (1985).
[CrossRef]

Other (4)

B. Friedman, Principles and Techniques of Applied Mathematics (Dover, New York, 1990), Chap. 1.

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

See, for instance, the special issues on Optical Parametric Oscillation and Amplification, J. Opt. Soc. Am. B 10, 1655–1791 (1993) and Optical Parametric Devices, J. Opt. Soc. Am. B 12, 2083–2320 (1995).

N. N. Rozanov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1996), Vol. 35, pp. 1–60.

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Figures (6)

Fig. 1
Fig. 1

Supercritical bifurcation of the neutrally stable front solution that arises from the onset of instability from the trivial solution of the signal field (Fu=Fv=F=0).

Fig. 2
Fig. 2

Numerical solution to Eqs. (1) in one dimension depicting the formation of the steady-state front solution of the signal field. Here we have taken Δ1=1, Δ2=1, ρ=0.5, and S=Sc+0.5 and demonstrated stability of the front, starting with two arbitrary initial conditions.

Fig. 3
Fig. 3

Numerical solutions to Eqs. (1) in one dimension with the same parameters as Fig. 2, showing the exponential decay of a given initial condition to the steady-state solution (Us). The theoretically predicted decay is given by the second discrete mode of L+.

Fig. 4
Fig. 4

Numerical solution to Eqs. (1) under the influence of slight spatial walk-off in one dimension. The evolution depicts the formation of a steady-state front solution that drifts to the left or right, depending on the sign of the spatial walk-off. Here we have taken Δ1=1, Δ2=1, ρ=0.5, and S=Sc+0.5 with the mismatch [δUx in Eqs. (1)] parameter being δ=±0.1.

Fig. 5
Fig. 5

Numerical solution to Eqs. (1) depicting the formation of a uniform solution to the signal field intensity with two stripes (fronts). We begin with white noise and notice that the complicated structures slowly straighten out and form the uniform state with the straight fronts depicted at time t=100. The grid size is x, y[-20, 20].

Fig. 6
Fig. 6

Numerical solution to Eqs. (1) depicting the formation of a uniform solution to the signal field intensity with two stripes (fronts). We begin with white noise and notice that the complicated structures slowly straighten out and form the uniform state with the straight fronts depicted at time t=100. The grid size is x, y[-20, 20].

Equations (15)

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

Ut=i2Uxx+VU*-(1+iΔ1)U+Fu,
Vt=i2ρVxx-U2-(α-iΔ2)V+S+Fv,
U=0,
V=S/(α+iΔ2).
Sc=(α+iΔ2)(1+iΔ1),
U=0+u(τ, ξ),
V=S/(α+iΔ2)+2v(τ, ξ),
ϕτ-ϕζζ+ϕ3-2γϕ=F.
ϕ=ϕ0+ϕ˜=2γ tanh(γζ)+ϕ˜,
L+ψ=(λ+3)ψ,
ϕ=2γ tanh(γz),
ζ0=-δτ,
ζτ0=34-F sech2 zdz.
ϕ=2γ tanhγ1+(ζη0)2z1/2,
ζτ0=ζηη0.

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