Existence and stability of N -pulses on optical fibers with phase-sensitive amplifiers

Abstract The propagation of pulses in optical communication systems in which attenuation is compensated by phase-sensitive amplifiers is investigated. A central issue is whether optical fibers are capable of carrying several pieces of information at the same time. In this paper, multiple pulses are shown to exist for a fourth-order nonlinear diffusion model due to Kutz and co-workers (1994). Moreover, criteria are derived for determining which of these pulses are stable. The pulses arise in a reversible orbit-flip, a homoclinic bifurcation investigated here for the first time. Numerical simulations are used to study multiple pulses far away from the actual bifurcation point. They confirm that properties of the multiple pulses including their stability are surprisingly well-predicted by the analysis carried out near the bifurcation.

[1]  Björn Sandstede,et al.  Stability of pulses on optical fibers with phase-sensitive amplifiers , 1997 .

[2]  Bjorn Sandstede.  Instability of localized buckling modes in a one-dimensional strut model , 1997, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[3]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[4]  W. Kath,et al.  Pulse propagation in nonlinear optical fiber lines that employ phase-sensitive parametric amplifiers , 1994 .

[5]  Bernold Fiedler,et al.  Homoclinic period blow-up in reversible and conservative systems , 1992 .

[6]  J. Nathan Kutz,et al.  Stability of Pulses in Nonlinear Optical Fibers Using Phase-Sensitive Amplifiers , 1996, SIAM J. Appl. Math..

[7]  Björn Sandstede,et al.  Stability of multiple-pulse solutions , 1998 .

[8]  Björn Sandstede,et al.  A numerical toolbox for homoclinic bifurcation analysis , 1996 .

[9]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[10]  M. Shub.  Global Stability of Dynamical Systems , 1986 .

[11]  Bernold Fiedler,et al.  Discretization of homoclinic orbits, rapid forcing, and "invisible" chaos , 1996 .

[12]  A. Hasegawa,et al.  Guiding-center soliton in optical fibers. , 1990, Optics letters.

[13]  J. Alexander,et al.  A topological invariant arising in the stability analysis of travelling waves. , 1990 .

[14]  Eiji Yanagida,et al.  Branching of double pulse solutions from single pulse solutions in nerve axon equations , 1987 .

[15]  R. Remmert,et al.  Theory of Complex Functions , 1990 .

[16]  A. Vanderbauwhede.  Subharmonic branching in reversible systems , 1990 .

[17]  Alan R. Champneys,et al.  Subsidiary homoclinic orbits to a saddle focus for reversible systems , 1994 .

[18]  Linn F. Mollenauer,et al.  Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber , 1991 .