Abstract

We address the stability of solitary waves to the complex cubic–quintic Ginzburg–Landau equation near the nonlinear Schrödinger limit. It is shown that the adiabatic method does not capture all possible instability mechanisms. The solitary wave can destabilize owing to discrete eigenvalues that move out of the continuous spectrum upon adding nonintegrable perturbations to the nonlinear Schrödinger equation. If an eigenvalue does move out of the continuous spectrum, then we say that an edge bifurcation has occurred. We present a novel analytical technique that allows us to determine whether eigenvalues arise in such a fashion, and if they do, to locate them. Using this approach, we show that Hopf bifurcations can arise in the cubic–quintic Ginzburg–Landau equation.

© 1998 Optical Society of America

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References

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  1. S. Gatz and J. Herrmann, “Soliton propagation in materials with saturable nonlinearity,” J. Opt. Soc. Am. B 8, 2296–2302 (1991).
    [CrossRef]
  2. S. Gatz and J. Herrmann, “Soliton collision and soliton fusion in dispersive materials with a linear and quadratic intensity depending refraction index change,” IEEE J. Quantum Electron. 28, 1732–1738 (1992).
    [CrossRef]
  3. C. De Angelis, “Self-trapped propagation in the nonlinear cubic–quintic Schrödinger equation: a variational approach,” IEEE J. Quantum Electron. 30, 818–821 (1994).
    [CrossRef]
  4. B. Lawrence, M. Cha, J. Kang, W. Torreullas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
    [CrossRef]
  5. B. Lawrence, M. Cha, W. Torruellas, G. Stegeman, S. Eternad, G. Baker, and F. Kajzar, “Measurement of the complex nonlinear refractive index of single crystal p-toluene sulfonate at 1064 nm,” Appl. Phys. Lett. 64, 2773–2775 (1994).
    [CrossRef]
  6. S. Gatz and J. Herrmann, “Soliton propagation and soliton collision in double-doped fibers with a non-Kerr-like nonlinear refractive-index change,” Opt. Lett. 17, 484–486 (1992).
    [CrossRef] [PubMed]
  7. J. Herrmann, “Bistable bright solitons in dispersive media with a linear and quadratic intensity-dependent refraction index change,” Opt. Commun. 87, 161–165 (1992).
    [CrossRef]
  8. A. Sombra, “Bistable pulse collisions of the cubic–quintic nonlinear Schrödinger equation,” Opt. Commun. 94, 92–98 (1992).
    [CrossRef]
  9. P. Marcq, H. Chatë, and R. Conte, “Exact solutions of the one-dimensional quintic complex Ginzburg–Landau equation,” Physica D 73, 305–317 (1994).
    [CrossRef]
  10. W. Van Saarloos and P. Hohenberg, “Fronts, pulses, sources, and sinks in the generalized complex Ginzburg–Landau equation,” Physica D 56, 303–367 (1992).
    [CrossRef]
  11. J. Soto-Crespo, N. Akhmediev, and V. Afanasjev, “Stability of the pulselike solutions of the quintic complex Ginzburg–Landau equation,” J. Opt. Soc. Am. B 13, 1439–1449 (1996).
    [CrossRef]
  12. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman & Hall, London, 1997).
  13. T. Kapitula, “Stability criterion for bright solitary waves of the perturbed cubic–quintic Schrödinger equation,” Physica D 116, 95–120 (1998).
    [CrossRef]
  14. Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fibre lasers,” Electron. Lett. 28, 1981–1983 (1992).
    [CrossRef]
  15. D. Pelinovsky, Y. Kivshar, and V. Afanasjev, “Internal modes of envelope solitons,” Physica D 116, 121–142 (1998).
    [CrossRef]
  16. T. Kapitula and B. Sandstede, “Stability of bright solitary wave solutions to perturbed nonlinear Schrödinger equations,” Physica D (to be published).
  17. Y. Kivshar, D. Pelinovsky, T. Cretegny, and M. Peyrand, “Internal modes of solitary waves,” Phys. Rev. Lett. 80, 5032–5035 (1998).
    [CrossRef]
  18. D. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
    [CrossRef] [PubMed]
  19. D. Kaup and T. Lakoba, “Variational method: How it can create false instabilities,” J. Math. Phys. 37, 3442–3462 (1996).
    [CrossRef]
  20. V. Afanasjev, P. Chu, and Y. Kivshar, “Breathing spatial solitons in non-Kerr media,” Opt. Lett. 22, 1388–1390 (1997).
    [CrossRef]
  21. A. Champneys, Yu. Kuznetsov, and B. Sandstede, “A numerical toolbox for homoclinic bifurcation analysis,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, 867–887 (1996).
    [CrossRef]
  22. E. Doedel, A. Champneys, T. Fairgrieve, Yu. Kuznetsov, B. Sandstede, and X. Wang, “AUTO97: Continuation and bifurcation software for ordinary differential equations (with HomCont),” Technical Report (Concordia University, Montreal, Quebec, Canada, 1997).

1998 (3)

T. Kapitula, “Stability criterion for bright solitary waves of the perturbed cubic–quintic Schrödinger equation,” Physica D 116, 95–120 (1998).
[CrossRef]

D. Pelinovsky, Y. Kivshar, and V. Afanasjev, “Internal modes of envelope solitons,” Physica D 116, 121–142 (1998).
[CrossRef]

Y. Kivshar, D. Pelinovsky, T. Cretegny, and M. Peyrand, “Internal modes of solitary waves,” Phys. Rev. Lett. 80, 5032–5035 (1998).
[CrossRef]

1997 (1)

1996 (3)

J. Soto-Crespo, N. Akhmediev, and V. Afanasjev, “Stability of the pulselike solutions of the quintic complex Ginzburg–Landau equation,” J. Opt. Soc. Am. B 13, 1439–1449 (1996).
[CrossRef]

D. Kaup and T. Lakoba, “Variational method: How it can create false instabilities,” J. Math. Phys. 37, 3442–3462 (1996).
[CrossRef]

A. Champneys, Yu. Kuznetsov, and B. Sandstede, “A numerical toolbox for homoclinic bifurcation analysis,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, 867–887 (1996).
[CrossRef]

1994 (4)

P. Marcq, H. Chatë, and R. Conte, “Exact solutions of the one-dimensional quintic complex Ginzburg–Landau equation,” Physica D 73, 305–317 (1994).
[CrossRef]

C. De Angelis, “Self-trapped propagation in the nonlinear cubic–quintic Schrödinger equation: a variational approach,” IEEE J. Quantum Electron. 30, 818–821 (1994).
[CrossRef]

B. Lawrence, M. Cha, J. Kang, W. Torreullas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

B. Lawrence, M. Cha, W. Torruellas, G. Stegeman, S. Eternad, G. Baker, and F. Kajzar, “Measurement of the complex nonlinear refractive index of single crystal p-toluene sulfonate at 1064 nm,” Appl. Phys. Lett. 64, 2773–2775 (1994).
[CrossRef]

1992 (6)

J. Herrmann, “Bistable bright solitons in dispersive media with a linear and quadratic intensity-dependent refraction index change,” Opt. Commun. 87, 161–165 (1992).
[CrossRef]

A. Sombra, “Bistable pulse collisions of the cubic–quintic nonlinear Schrödinger equation,” Opt. Commun. 94, 92–98 (1992).
[CrossRef]

W. Van Saarloos and P. Hohenberg, “Fronts, pulses, sources, and sinks in the generalized complex Ginzburg–Landau equation,” Physica D 56, 303–367 (1992).
[CrossRef]

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fibre lasers,” Electron. Lett. 28, 1981–1983 (1992).
[CrossRef]

S. Gatz and J. Herrmann, “Soliton collision and soliton fusion in dispersive materials with a linear and quadratic intensity depending refraction index change,” IEEE J. Quantum Electron. 28, 1732–1738 (1992).
[CrossRef]

S. Gatz and J. Herrmann, “Soliton propagation and soliton collision in double-doped fibers with a non-Kerr-like nonlinear refractive-index change,” Opt. Lett. 17, 484–486 (1992).
[CrossRef] [PubMed]

1991 (1)

1990 (1)

D. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[CrossRef] [PubMed]

Afanasjev, V.

Akhmediev, N.

Baker, G.

B. Lawrence, M. Cha, J. Kang, W. Torreullas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

B. Lawrence, M. Cha, W. Torruellas, G. Stegeman, S. Eternad, G. Baker, and F. Kajzar, “Measurement of the complex nonlinear refractive index of single crystal p-toluene sulfonate at 1064 nm,” Appl. Phys. Lett. 64, 2773–2775 (1994).
[CrossRef]

Cha, M.

B. Lawrence, M. Cha, W. Torruellas, G. Stegeman, S. Eternad, G. Baker, and F. Kajzar, “Measurement of the complex nonlinear refractive index of single crystal p-toluene sulfonate at 1064 nm,” Appl. Phys. Lett. 64, 2773–2775 (1994).
[CrossRef]

B. Lawrence, M. Cha, J. Kang, W. Torreullas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Champneys, A.

A. Champneys, Yu. Kuznetsov, and B. Sandstede, “A numerical toolbox for homoclinic bifurcation analysis,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, 867–887 (1996).
[CrossRef]

Chatë, H.

P. Marcq, H. Chatë, and R. Conte, “Exact solutions of the one-dimensional quintic complex Ginzburg–Landau equation,” Physica D 73, 305–317 (1994).
[CrossRef]

Chu, P.

Conte, R.

P. Marcq, H. Chatë, and R. Conte, “Exact solutions of the one-dimensional quintic complex Ginzburg–Landau equation,” Physica D 73, 305–317 (1994).
[CrossRef]

Cretegny, T.

Y. Kivshar, D. Pelinovsky, T. Cretegny, and M. Peyrand, “Internal modes of solitary waves,” Phys. Rev. Lett. 80, 5032–5035 (1998).
[CrossRef]

De Angelis, C.

C. De Angelis, “Self-trapped propagation in the nonlinear cubic–quintic Schrödinger equation: a variational approach,” IEEE J. Quantum Electron. 30, 818–821 (1994).
[CrossRef]

Etemad, S.

B. Lawrence, M. Cha, J. Kang, W. Torreullas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Eternad, S.

B. Lawrence, M. Cha, W. Torruellas, G. Stegeman, S. Eternad, G. Baker, and F. Kajzar, “Measurement of the complex nonlinear refractive index of single crystal p-toluene sulfonate at 1064 nm,” Appl. Phys. Lett. 64, 2773–2775 (1994).
[CrossRef]

Gatz, S.

Herrmann, J.

J. Herrmann, “Bistable bright solitons in dispersive media with a linear and quadratic intensity-dependent refraction index change,” Opt. Commun. 87, 161–165 (1992).
[CrossRef]

S. Gatz and J. Herrmann, “Soliton collision and soliton fusion in dispersive materials with a linear and quadratic intensity depending refraction index change,” IEEE J. Quantum Electron. 28, 1732–1738 (1992).
[CrossRef]

S. Gatz and J. Herrmann, “Soliton propagation and soliton collision in double-doped fibers with a non-Kerr-like nonlinear refractive-index change,” Opt. Lett. 17, 484–486 (1992).
[CrossRef] [PubMed]

S. Gatz and J. Herrmann, “Soliton propagation in materials with saturable nonlinearity,” J. Opt. Soc. Am. B 8, 2296–2302 (1991).
[CrossRef]

Hohenberg, P.

W. Van Saarloos and P. Hohenberg, “Fronts, pulses, sources, and sinks in the generalized complex Ginzburg–Landau equation,” Physica D 56, 303–367 (1992).
[CrossRef]

Kajzar, F.

B. Lawrence, M. Cha, W. Torruellas, G. Stegeman, S. Eternad, G. Baker, and F. Kajzar, “Measurement of the complex nonlinear refractive index of single crystal p-toluene sulfonate at 1064 nm,” Appl. Phys. Lett. 64, 2773–2775 (1994).
[CrossRef]

Kang, J.

B. Lawrence, M. Cha, J. Kang, W. Torreullas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Kapitula, T.

T. Kapitula, “Stability criterion for bright solitary waves of the perturbed cubic–quintic Schrödinger equation,” Physica D 116, 95–120 (1998).
[CrossRef]

Kaup, D.

D. Kaup and T. Lakoba, “Variational method: How it can create false instabilities,” J. Math. Phys. 37, 3442–3462 (1996).
[CrossRef]

D. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[CrossRef] [PubMed]

Kivshar, Y.

D. Pelinovsky, Y. Kivshar, and V. Afanasjev, “Internal modes of envelope solitons,” Physica D 116, 121–142 (1998).
[CrossRef]

Y. Kivshar, D. Pelinovsky, T. Cretegny, and M. Peyrand, “Internal modes of solitary waves,” Phys. Rev. Lett. 80, 5032–5035 (1998).
[CrossRef]

V. Afanasjev, P. Chu, and Y. Kivshar, “Breathing spatial solitons in non-Kerr media,” Opt. Lett. 22, 1388–1390 (1997).
[CrossRef]

Kodama, Y.

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fibre lasers,” Electron. Lett. 28, 1981–1983 (1992).
[CrossRef]

Kuznetsov, Yu.

A. Champneys, Yu. Kuznetsov, and B. Sandstede, “A numerical toolbox for homoclinic bifurcation analysis,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, 867–887 (1996).
[CrossRef]

Lakoba, T.

D. Kaup and T. Lakoba, “Variational method: How it can create false instabilities,” J. Math. Phys. 37, 3442–3462 (1996).
[CrossRef]

Lawrence, B.

B. Lawrence, M. Cha, J. Kang, W. Torreullas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

B. Lawrence, M. Cha, W. Torruellas, G. Stegeman, S. Eternad, G. Baker, and F. Kajzar, “Measurement of the complex nonlinear refractive index of single crystal p-toluene sulfonate at 1064 nm,” Appl. Phys. Lett. 64, 2773–2775 (1994).
[CrossRef]

Marcq, P.

P. Marcq, H. Chatë, and R. Conte, “Exact solutions of the one-dimensional quintic complex Ginzburg–Landau equation,” Physica D 73, 305–317 (1994).
[CrossRef]

Meth, J.

B. Lawrence, M. Cha, J. Kang, W. Torreullas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Pelinovsky, D.

D. Pelinovsky, Y. Kivshar, and V. Afanasjev, “Internal modes of envelope solitons,” Physica D 116, 121–142 (1998).
[CrossRef]

Y. Kivshar, D. Pelinovsky, T. Cretegny, and M. Peyrand, “Internal modes of solitary waves,” Phys. Rev. Lett. 80, 5032–5035 (1998).
[CrossRef]

Peyrand, M.

Y. Kivshar, D. Pelinovsky, T. Cretegny, and M. Peyrand, “Internal modes of solitary waves,” Phys. Rev. Lett. 80, 5032–5035 (1998).
[CrossRef]

Romagnoli, M.

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fibre lasers,” Electron. Lett. 28, 1981–1983 (1992).
[CrossRef]

Sandstede, B.

A. Champneys, Yu. Kuznetsov, and B. Sandstede, “A numerical toolbox for homoclinic bifurcation analysis,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, 867–887 (1996).
[CrossRef]

Sombra, A.

A. Sombra, “Bistable pulse collisions of the cubic–quintic nonlinear Schrödinger equation,” Opt. Commun. 94, 92–98 (1992).
[CrossRef]

Soto-Crespo, J.

Stegeman, G.

B. Lawrence, M. Cha, J. Kang, W. Torreullas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

B. Lawrence, M. Cha, W. Torruellas, G. Stegeman, S. Eternad, G. Baker, and F. Kajzar, “Measurement of the complex nonlinear refractive index of single crystal p-toluene sulfonate at 1064 nm,” Appl. Phys. Lett. 64, 2773–2775 (1994).
[CrossRef]

Torreullas, W.

B. Lawrence, M. Cha, J. Kang, W. Torreullas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Torruellas, W.

B. Lawrence, M. Cha, W. Torruellas, G. Stegeman, S. Eternad, G. Baker, and F. Kajzar, “Measurement of the complex nonlinear refractive index of single crystal p-toluene sulfonate at 1064 nm,” Appl. Phys. Lett. 64, 2773–2775 (1994).
[CrossRef]

Van Saarloos, W.

W. Van Saarloos and P. Hohenberg, “Fronts, pulses, sources, and sinks in the generalized complex Ginzburg–Landau equation,” Physica D 56, 303–367 (1992).
[CrossRef]

Wabnitz, S.

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fibre lasers,” Electron. Lett. 28, 1981–1983 (1992).
[CrossRef]

Appl. Phys. Lett. (1)

B. Lawrence, M. Cha, W. Torruellas, G. Stegeman, S. Eternad, G. Baker, and F. Kajzar, “Measurement of the complex nonlinear refractive index of single crystal p-toluene sulfonate at 1064 nm,” Appl. Phys. Lett. 64, 2773–2775 (1994).
[CrossRef]

Electron. Lett. (2)

B. Lawrence, M. Cha, J. Kang, W. Torreullas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fibre lasers,” Electron. Lett. 28, 1981–1983 (1992).
[CrossRef]

IEEE J. Quantum Electron. (2)

S. Gatz and J. Herrmann, “Soliton collision and soliton fusion in dispersive materials with a linear and quadratic intensity depending refraction index change,” IEEE J. Quantum Electron. 28, 1732–1738 (1992).
[CrossRef]

C. De Angelis, “Self-trapped propagation in the nonlinear cubic–quintic Schrödinger equation: a variational approach,” IEEE J. Quantum Electron. 30, 818–821 (1994).
[CrossRef]

Int. J. Bifurcation Chaos Appl. Sci. Eng. (1)

A. Champneys, Yu. Kuznetsov, and B. Sandstede, “A numerical toolbox for homoclinic bifurcation analysis,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, 867–887 (1996).
[CrossRef]

J. Math. Phys. (1)

D. Kaup and T. Lakoba, “Variational method: How it can create false instabilities,” J. Math. Phys. 37, 3442–3462 (1996).
[CrossRef]

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

Opt. Commun. (2)

J. Herrmann, “Bistable bright solitons in dispersive media with a linear and quadratic intensity-dependent refraction index change,” Opt. Commun. 87, 161–165 (1992).
[CrossRef]

A. Sombra, “Bistable pulse collisions of the cubic–quintic nonlinear Schrödinger equation,” Opt. Commun. 94, 92–98 (1992).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

D. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

Y. Kivshar, D. Pelinovsky, T. Cretegny, and M. Peyrand, “Internal modes of solitary waves,” Phys. Rev. Lett. 80, 5032–5035 (1998).
[CrossRef]

Physica D (4)

D. Pelinovsky, Y. Kivshar, and V. Afanasjev, “Internal modes of envelope solitons,” Physica D 116, 121–142 (1998).
[CrossRef]

P. Marcq, H. Chatë, and R. Conte, “Exact solutions of the one-dimensional quintic complex Ginzburg–Landau equation,” Physica D 73, 305–317 (1994).
[CrossRef]

W. Van Saarloos and P. Hohenberg, “Fronts, pulses, sources, and sinks in the generalized complex Ginzburg–Landau equation,” Physica D 56, 303–367 (1992).
[CrossRef]

T. Kapitula, “Stability criterion for bright solitary waves of the perturbed cubic–quintic Schrödinger equation,” Physica D 116, 95–120 (1998).
[CrossRef]

Other (3)

N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman & Hall, London, 1997).

T. Kapitula and B. Sandstede, “Stability of bright solitary wave solutions to perturbed nonlinear Schrödinger equations,” Physica D (to be published).

E. Doedel, A. Champneys, T. Fairgrieve, Yu. Kuznetsov, B. Sandstede, and X. Wang, “AUTO97: Continuation and bifurcation software for ordinary differential equations (with HomCont),” Technical Report (Concordia University, Montreal, Quebec, Canada, 1997).

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Figures (4)

Fig. 1
Fig. 1

(a) Spectrum for the NLS. (b) If d2=O(|α|), then it is possible that the spectrum of the CQGL looks as shown. Upon changing d2, a Hopf instability can occur before the instability induced by radiation modes.

Fig. 2
Fig. 2

(a) (d1, d3) plane. Whenever (d1, d3) is in region II, then Hopf bifurcations are not possible, and the wave is stable. (b) If (d1, d3) is in region I, then the situation in the (d2, α) plane is as shown. Whenever (d2, α) is in the shaded region in (b), then the solitary wave is stable. The wave destabilizes in a Hopf bifurcation if (d2, α) crosses through the solid line into the nonshaded region.

Fig. 3
Fig. 3

Curves in the (d2, α) plane are shown where a Hopf bifurcation occurs: (a) =0.05 and (b) =0.1. The solid, dashed, and dotted curves are for d3=2.2, d3=4.0, and d3 =7.0, respectively.

Fig. 4
Fig. 4

(a) Hopf curve for d2=0 in the (, α) plane. (b) Δ =λ*-ω as a function of . The solid, dashed, and dotted curves are for d3=2.2, d3=4.0, and d3=7.0, respectively.

Equations (51)

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

iϕx+ϕtt+4|ϕ|2ϕ=0,
Φ(x, t)=ω2 sech(ωt) exp(iωx).
iϕx+ϕtt+4|ϕ|2ϕ=i(d1ϕtt+d2ϕ+d3|ϕ|2ϕ
+d4|ϕ|4ϕ)-3α|ϕ|4ϕ
d4<0,d3>d1>0,(d3-d1)2>245 d2d4,
ω±=54|d4| d3-d1±(d3-d1)2-245 d2d4.
Φ(x, t)=ω2 sech(ωt)exp(iωx)1-14 αω1-12 sech2(ωt)+iω2 sech(ωt)×0tθ(τ)dτ+O(α2+2),
θ(t)=-12ω tanh(ωt)d2+ωd1-110 ω2d4 sech2(ωt),
ρrot=-C1d1+O(2),
ρamp=C2215 ω±2d4-d2+O(2),
ρ=iω+d2+(i-d1)k2,
iϕx+ϕtt+4|ϕ|2ϕ+αf(|ϕ|2)ϕ
=i[d1ϕtt+R(ϕ, ϕ*)],
ρ*=iω+d2+O(α2+2).
iϕx+ϕtt+4ϕ2ψ+3αϕ3ψ2
=i(d1ϕtt+d2ϕ+d3ϕ2ψ+d4ϕ3ψ2),
-iψx+ψtt+4ϕψ2+3αϕ2ψ3
=i(d1ψtt+d2ψ+d3ϕψ2+d4ϕ2ψ3),
iσ3Px+LP=0,
σ3=100-1
L0=(tt-ω)1001+4|Φ|22112,
Lα=-2ω|Φ|22112+|Φ|4138813,
L=-iσ3(d1tt+d2)1001+d3|Φ|22112+d4|Φ|43223-8|Φ|20tθ(τ)dτ0110,
(L+iρσ3)P=0,
ρ=iλ,
(L-λσ3)P=0.
E(λ)=42ωω-λ,
arg(ω-λ)-π2, 3π2.
αE(ω)=-22ω-P+LαP+dt,
E(ω)=-22ω-P+(L+id2σ3)P+dt.
P+(t)=01-sech2(ωt)11
λ*=ω-id2-[˜E(ω)˜]232ω+O(|˜|3).
αE(ω)=-22ω-168ω2 |Φ|8-132ω |Φ|6+37|Φ|4-4ω|Φ|2dt=-223 ω2,
E(ω)=i22ω-(8d1-2d3)|Φ|2+-24ω d1+8ω d3-3d4|Φ|4+12ω d4|Φ|6dt=i 223 ω2d3+95 ωd4.
λ*=ω1-ω236 α2+O(|α|3)
Im λ*=-d2+O[(|α|+)].
d2=O(|α|+).
ω=-52d4 (d3-d1),
E(ω)=i 223 ω92 d1-52 d3.
A=2.5(1.8d1-d3),
˜E(ω)˜=223 ω(-ωα+iA).
3π4<arg(-ωα+iA)<7π4,
ωα>A.
α>|d4|d3-d1 (1.8d1-d3).
λ*=ω1+136 (A22-ω2α2)+O(3+|α|3)+i118 ω2Aα-d2+O(α2+2).
α>D1(1.8d1-d3)
Im λ*=[D2(1.8d1-d3)α-d2+O(α2+2)].
D1=|d4|d3-d1,D2=125144 d3-d1d42.
α>-|1.8d1-d3|D1.
d2=-|1.8d1-d3|D2α.
α>×|1.8d1-d3|D1>0.

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