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Bifurcating bright and dark solitary waves for the perturbed cubic-quintic nonlinear Schrödinger equation

Published online by Cambridge University Press:  14 November 2011

Todd Kapitula
Todd Kapitula
Affiliation:
Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, U.S.A., E-mail: kapitula@math.unm.edu
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Abstract

The existence of bright and dark multi-bump solitary waves for Ginzburg–Landau type perturbations of the cubic-quintic Schrodinger equation is considered. The waves in question are not perturbations of known analytic solitary waves, but instead arise as a bifurcation from a heteroclinic cycle in a three-dimensional ODE phase space. Using geometric singular perturbation techniques, regions in parameter space for which 1-bump bright and dark solitary waves will bifurcate are identified. The existence of N-bump dark solitary waves (N ≧ 1) is shown via an application of the Exchange Lemma with Exponentially Small Error. N-bump bright solitary waves are shown to exist as a consequence of the work of Kapitula and Maier-Paape.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 3 , 1998 , pp. 585 - 629
Copyright
Copyright © Royal Society of Edinburgh 1998

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References

1Alexander, J. and Jones, C. K. R. T.. Existence and stability of asymptotically oscillatory double pulses. J. Reine Angew. Math. 446 (1994), 49–79.Google Scholar
2Angelis, C. De. Self-trapped propagation in the nonlinear cubic-quintic Schrödinger equation: a variational approach. IEEE J. Quantum Elect. 30 (1994), 818–21.CrossRefGoogle Scholar
3Boling, G. and Yaping, W.. Orbital stability of solitary waves for the nonlinear derivative Schrödinger equation. J. Differential Equations 123 (1995), 35–55.Google Scholar
4Bouard, A. De. Instability of stationary bubbles. SIAM J. Math. Anal. 26 (1995), 566–82.CrossRefGoogle Scholar
5Doelman, A.. Breaking the hidden symmetry in the Ginzburg–Landau equation. Phys. D 97 (1996), 398–428.CrossRefGoogle Scholar
6Doelman, A. and Eckhaus, W.. Periodic and quasi-periodic solutions of degenerate modulation equations. Phys. D 53 (1991), 249–66.CrossRefGoogle Scholar
7Duan, J. and Holmes, P.. Fronts, domain walls and pulses in a generalized Ginzburg–Landau equation. Proc. Edinburgh Math. Soc. 38 (1995) 77–97.CrossRefGoogle Scholar
8Fenichel, N.. Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21 (1973), 193–226.CrossRefGoogle Scholar
9Gatz, S. and Herrmann, J.. Soliton propagation in materials with saturable nonlinearity. J. Opt. Soc. Amer. B 8 (1991), 2296–302.CrossRefGoogle Scholar
10Gatz, S. and Herrmann, J.. Soliton collision and soliton fusion in dispersive materials with a linear and quadratic intensity depending refraction index change. IEEE J. Quantum Elect. 28(1992), 1732–8.CrossRefGoogle Scholar
11Gatz, S. and Herrmann, J.. Soliton propagation and soliton collision in double-doped fibers with a non-Kerr-like nonlinear refractive-index change. Optics Lett. 17 (1992), 484–6.CrossRefGoogle ScholarPubMed
12Grillakis, M., Shatah, J. and Strauss, W.. Stability theory of solitary waves in the presence of symmetry, I. J. Funct. Anal. 74 (1987), 160–97.CrossRefGoogle Scholar
13Herrmann, J.. Bistable bright solitons in dispersive media with a linear and quadratic intensitydependent refraction index change. Optics Comm. 87 (1992), 161–5.CrossRefGoogle Scholar
14Jones, C. K. R. T.. Geometric singular perturbation theory. In Lecture Notes in Mathematics 1609, ed. Johnson, R. (New York: Springer, 1995).Google Scholar
15Jones, C. K. R. T., Kaper, T. and Kopell, N.. Tracking invariant manifolds up to exponentially small errors. SIAM J. Math. Anal. 27 (1996), 558–77.CrossRefGoogle Scholar
16Jones, C. K. R. T., Kapitula, T. and Powell, J.. Nearly real fronts in a Ginzburg–Landau equation. Proc. Roy. Soc. Edinburgh Sect. A 116 (1990), 193–206.CrossRefGoogle Scholar
17Jones, C. K. R. T. and Kopell, N.. Tracking invariant manifolds with differential forms in singularly perturbed systems. J. Differential Equations 108 (1994), 64–88.CrossRefGoogle Scholar
18Kapitula, T.. Singular heteroclinic orbits for degenerate modulation equations. Phys. D 82 (1995), 36–59.CrossRefGoogle Scholar
19Kapitula, T.. Stability criterion for bright solitary waves of the perturbed cubic-quintic Schrödinger equation. (To appear in Physica D.)Google Scholar
20Kapitula, T. and Maier-Paape, S.. Spatial dynamics of time periodic solutions for the Ginzburg–Landau equation. Z. Angew. Math. Phys. 47 (1996), 265–305.CrossRefGoogle Scholar
21Krupa, M., Sandstede, B. and Szmolyan, P.. Fast and slow waves in the Fitzhugh–Nagumo equation. J. Differential Equations 133 (1997), 49–97.CrossRefGoogle Scholar
22Lawrence, B., Cha, M., Kang, J., Toreullas, W., Stegeman, G., Baker, G., Meth, J. and Etemad, S.. Large purely refractive nonlinear index of singlecrystal P-toluene sulphonate (PTS) at 1600 nm. Elect. Lett. 30 (1994), 447–8.CrossRefGoogle Scholar
23Lawrence, B., Cha, M., Torruellas, W., Stegeman, G., Eternad, S., Baker, G. and Kajzar, F.. Measurement of the complex nonlinear refractive index of single crystal p-toluene sulfonate at 1064 nm. Appl. Phys. Lett. 64 (1994), 2773–5.CrossRefGoogle Scholar
24Malomed, B. and Nepomnyashchy, A.. Kinks and solitons in the generalized Ginzburg–Landau equation. Phys. Rev. A 42 (1990), 6009–14.CrossRefGoogle ScholarPubMed
25Marcq, P., Chatë, H. and Conte, R.. Exact solutions of the one-dimensional quintic complex Ginzburg–Landau equation. Phys. D 73 (1994), 305.CrossRefGoogle Scholar
26Nii, S.. An extension of the stability index for the traveling-wave solutions and its application for bifurcations. SI AM J. Appl. Math. 28 (1997), 402–33.Google Scholar
27Pelinovsky, D., Kivshar, Y. and Afanasjev, V.. Instability-induced dynamics ofdark solitons. Phys. Rev. E 54 (1996), 2015–32.CrossRefGoogle ScholarPubMed
28Sombra, A.. Bistable pulse collisions of the cubic-quintic nonlinear Schrödinger equation. Optics Comm. 94 (1992), 92–8.CrossRefGoogle Scholar
29Saarloos, W. Van and Hohenberg, P.. Fronts, pulses, sources, and sinks in the generalized complex Ginzburg–Landau equation. Phys. D 56 (1992), 303–67.CrossRefGoogle Scholar
30Soto-Crespo, J., Akhmediev, N. and Afanasjev, V.. Stability of the pulselike solutions of the quintic complex Ginzburg–Landau equation. J. Opt. Soc. Amer. B 13 (1996), 1439–49.CrossRefGoogle Scholar