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LANL, 02/05/03. Shedding and Interaction of solitons in imperfect medium. Misha Chertkov (Theoretical Division, LANL). In collaboration with Ildar Gabitov (LANL) Igor Kolokolov (Budker Inst.) Vladimir Lebedev (Landau Inst.) Yeo-Jin Chung (LANL) Sasha Dyachenko (Landau Inst.).

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LANL, 02/05/03

Shedding and Interaction of solitons in imperfect medium

Misha Chertkov(Theoretical Division, LANL)

In collaboration with

Ildar Gabitov (LANL)

Igor Kolokolov (Budker Inst.)

Vladimir Lebedev (Landau Inst.)

Yeo-Jin Chung (LANL)

Sasha Dyachenko (Landau Inst.)

``Statistical Physics of

Fiber Optics


Zoltan Toroczkai (LANL)

Pavel Lushnikov (LANL)

Jamison Moeser (Brown U)

Tobias Schaefer (Brown U)

Avner Peleg (LANL)

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  • What was the idea? Fiber Optics. Statistics.

  • What we did first(Pinning method of pulse confinement in a fiber with fluctuating dispersion)

  • What we did recently(Shedding and interaction of solitons in imperfect medium)

  • Other activities and plans(Polarization Mode Dispersion, Wave-length Division Mulitplexing, Dispersion Management, etc)

  • Suggestions forExtensive DNS, Experiment, Field Trials

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Fiber Electrodynamics

  • Monomode

  • Weak nonlinearity,

  • slow in z


averaging over amplifiers

NLS in the envelope approximation

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- dispersion length

- pulse width

- nonlinearity length

- pulse amplitude

Soliton solution

Dispersion balances nonlinearity

Integrability (Zakharov & Shabat ‘72)

Model A

Nonlinear Schrodinger Equation

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Gabitov, Turitsyn ‘96

Smith,Knox,Doran,Blow,Binnion ‘96

Turitsyn et al/Optics Comm

163 (1999) 122

Model B

Dispersion management

Lin, Kogelnik, Cohen ‘80

  • dispersion compensation aims to prevent

  • broadening of the pulse (in linear regime)

  • four wave mixing (nonlinearity) is suppressed

  • effect of additive noise is suppressed

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Stochastic model

(unrestricted noise)

Noise is conservative

No jitter

Abdullaev and co-authors ‘96-’00

Noise in dispersion. Statistical Description.

DSF, Gripp, Mollenauer Opt. Lett. 23, 1603, 1998

Optical-time-domain-reflection method.

Measurements from only one end of fiber by

phase mismatch at the Stokes frequency

Mollenauer, Mamyshev, Neubelt ‘96

Questions:Does an initially localized pulse

survive propagation?

Are probability distribution functions

of various pulse parameters getting


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Unrestricted noise

Describes slow evolution of the original field

if nonlinearity is weak

Nonlinearity dies (as z increases)

== Pulse degradation

correlation length

Noise is strong

D >> 1

Question:Is there a constraint that one can impose on the random chromatic

dispersion to reduce pulse broadening?

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The restricted model



Random uniformly

distributed ]-.5,.5[

Pinning method

Constraint prescription:

the accumulated dispersion should be pinned to zero

periodically or quasi-periodically

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Weak nonlinearity


noise average

The nonlinear kernel

does not decay (with z) !!!

noise and pinning period


The averaged equation

does have a steady (soliton like)

solution in the restricted case

Restricted (pinned) noise

Model A

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The averaged equation

does have a steady solution

Restricted (pinned) noise. DM case.

Weak nonlinearity

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Numerical Simulations

Fourier split-step scheme

Fourier modes

Model A

Model B

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Practical recommendations for improving fiber system performance

that is limited by randomness in chromatic dispersion.

The limitation originates from the accumulation of the integral

dispersion. The distance between naturally occurring nearest zeros

grows with fiber length. This growth causes pulse degradation.

We have shown that the signal can be stabilized by periodic or

quasi-periodic pinning of the accumulated dispersion.

M.Chertkov,I. Gabitov,J.Moeser

US patent+PNAS 98, 14211 (2001)

M.C.,I.G., P. Lushnikov, Z.Toroczkai, JOSAB (2002)

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Shedding and Interaction of solitons

in imperfect medium

M.Chertkov,I. Gabitov,

I.Kolokolov, V.Lebedev,

JETP Lett. 10/01

MC, Y. Chung, A. Dyachenko, IG,IK,VL PRE Feb 2003


Second order adiabatic pert. theory

(Kaup ’90) +

Statistical averaging



*What statistics does describe the radiation emitted due to disorder

by a single soliton, pattern of solitons?

How far do the radiation wings extend from the peak of the soliton(s)?

What is the structure of the wings?

*How strong is the radiation mediating interaction between the solitons?

How is the interaction modified if we vary the soliton positions

and phases within a pattern of solitons?

Model A

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Single Soliton Story

“Asymptotic freedom”:

soliton is distinguishable from

the radiation at any z


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Radiation tail + forerunner

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Soliton phase mismatch

Soliton position shift is

Gaussian zero mean

random variable

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Multi-soliton case

The z-dependence is similar

to the one described by

Elgin-Gordon-Haus jitter

Infinite pattern(continuous flow of information)

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Bare case

Pinned case

Single soliton decay

Two-soliton interaction

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Statistical Physics of Fiber Communications

We planned to addressed:

  • Single pulse dynamics

  • Fluctuating dispersion

  • Dispersion Management and Fluctuations

  • Raman term +noise

  • Polarization Mode Dispersion

  • Additive (Elgin-Gordon-Hauss) noise optimization

  • Joint effect of the additive and multiplicative noises

  • Mutual equilibrium of a pulse and radiation closed in a box

  • (wave turbulence on a top of a pulse) driven by a noise

  • Many-pulse, -channel interaction

  • Statistics of the noise driven by the interaction

  • Suppression of the four-wave mixing (ghost pulses) by the pinning?

  • Dynamics in a channel under the WDM

  • Multi mode fibers

  • noise induced enhancement of the information flow

  • ...