C. K. R. T. Jones (with J.Moeser and I.Gabitov)
To appear in Optics Letters
Mark J. Ablowitz, Gino Biondini, Sarbarish Charkravarty and Rudy L. Horne
Abstract: Four-wave mixing (FWM) in wavelength-division multiplexed systems with strong dispersion management and loss/amplification is comprehensively studied. The methods described apply to both solitons and quasi-linear return-to-zero systems. A linear model is introduced that describes the resonant growth and saturation of the FWM products. The model yields a resonance condition between the channel separation and the amplifier spacing, which in certain parameter regions, reproduces for strongly dispersion-managed systems the phase matching condition valid for classical solitons. As the dispersion map strength increases, the residual FWM decreases, but the FWM amplitude is found to increase inversely to the average dispersion in the system. A reduced linear model in also introduced which contains the basic features of FWM processes. Comparisons of both models with direct numerical simulations of the full nonlinear system demonstrate excellent agreement.
Accepted to JOSA B (Journal of the Optical Society of America B) in November 2002
T. Schäfer and C. E. Wayne
Abstract: We derive a partial differential equation that approximates solutions of Maxwell's equations describing the propagation of ultra-short optical pulses in nonlinear media and which extends the prior analysis of Alterman and Rauch. We discuss (non-rigorously) conditions under which this approximation should be valid, but the main contributions of this paper are: (1) an emphasis on the fact that the model equation for short pulse propagation may depend on the details of the optical susceptibility in the wavelength regime under consideration, (2) a numerical comparison of solutions of this model equation with solutions of the full nonlinear partial differential equation, (3) a local well-posedness result for the model equation and (4) a proof that in contrast to the nonlinear Schrödinger equation which models slowing varying wavetrains this equation has no pulse solutions which propagate with fixed shape and speed.
To be submitted
C. K. R. T. Jones (with R.Jackson and V.Zharnitsky)
To appear in Physica D
Rudy L. Horne
Abstract: This article introduces the reader to a certain class of nonlinear partial differential equations (pdes) which are characterized by a balancing between linear (usually dispersive) and nonlinear terms in the equation. The equations of interest usually allow for the existence of a special type of traveling wave solution: either solitary waves or solitons. This review article will summarize a few results arising from the study of both the Korteweg-deVries (KdV) and Nonlinear Schrodinger (NLS) equations. We begin our discussion with the KdV equation. We first give a historical perspective involving both experimental and mathematical evidence for the existence of solitary waves and solitons. We then show a mathematical derivation of the KdV equation using asymptotic techniques. Next, we prove that the KdV equation is Galilean invariant which allows us to connect solutions of the KdV equation to the Time Independent Schrodinger equation. We then give a brief description of the Inverse Scattering Transform which allows for a general approach to solving certain nonlinear pdes. Moving on the the NLS equation, we show that traveling wave solutions exits for this equation as well. We outline the Lax method concerning the solution of certain linear and nonlinear pdes. This method can be generalized and used to solve the NLS equation which we touch upon here. We also discuss a couple of applications of the NLS equation. We end the article by looking at a couple of the conserved quantities for both the KdV and NLS equations.
Submitted to the Conference for African-American Researchers in the Mathematical Sciences Proceedings, December 2002
T. Schäfer, R. O. Moore, and C. K. R. T. Jones
Abstract: We study the broadening of pulses in dispersion-managed fiber lines in which the dispersion has a random component. To model the dynamics of the pulse, we use the cubic nonlinear Schrödinger equation. For small distances, a straightforward perturbation expansion provides an analytical description of the pulse broadening. For longer distances over which the noise and nonlinearity interact, we approximate pulse evolution under the nonlinear Schrödinger equation by a system of coupled nonlinear ordinary differential equations. The validity of this approximation in the context of randomness is discussed. Finally, we exploit the fact that nonlinearity is small by averaging the evolution over the period of the dispersion map, thereby obtaining a new set of stochastic equations valid for a slow evolution over much longer scales.
To appear in Opt. Comm.
C. K. R. T. Jones (with R.Moore, B.Sandstede, W.Kath and J.Alexander)
Optical Communications, 195 (2001) 127-139
C. K. R. T. Jones (with P.Kevrekidis and T.Kapitula)
Phys Rev E, 63 (2001) 036602
C. K. R. T. Jones (with V.Zharnitsky, E.Grenier, J.Hesthaven and S.Turitsyn)
Phys Rev E 62 (2000) 7358
C. K. R. T. Jones (V.Zharnitsky, E.Grenier and S.Turitsyn)
Physica D 152-3 (2001)
C. K. R. T. Jones (with A.C.Yew and B.Sandstede)
Phys Rev E, 61 (2000) 5886-5892
C. K. R. T. Jones (with F.Geszetesy, Y.Latushkin and M.Stanislavova)
Indiana U. Math. Journal, 49 (2000) 221--243
C. K. R. T. Jones (with T.Küpper and K.Schaffner)
ZAMP, 52 (2001) 859-880
C. K. R. T. Jones (with P.Kevrekidis)
Phys Rev E, 61 (2000) 3114-3121
C. K. R. T. Jones (with S.Turitsyn, A.Aceves and V.Zharnitsky)
Phys Rev E, 59 (1999) 3843-3846
C. K. R. T. Jones (with A.Aceves, S.Turitsyn and V.Zharnitsky)
Phys Rev E, 58 (1998) 48-51
C. K. R. T. Jones (with J.Rubin)
Optics Communications, 140 (1997) 93-98