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| [[Maxwell's Equations]], when converted to [[cylindrical coordinates]], and with the boundary conditions for an [[optical fiber]] while including [[birefringence]] as an effect taken into account, will yield the coupled [[nonlinear Schrödinger equation]]s. After employing the [[Inverse scattering transform]] (a procedure analogous to the [[Fourier Transform]] and [[Laplace Transform]]) on the resulting equations, the Manakov system is then obtained. The most general form of the Manakov system is as follows:
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| :<math>v_{1}'=-i\,\xi\,v_{1}+q_{1}\,v_{2}+q_{2}\,v_{3}</math>
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| :<math>v_{2}'=-q_{1}^{*}\,v_{1}+i\,\xi\,v_{2}</math>
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| :<math>v_{3}'=-q_{2}^{*}\,v_{1}+i\,\xi\,v_{3}.</math>
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| It is a coupled system of linear [[ordinary differential equations]]. The functions <math>q_{1}, q_{2}</math> represent the envelope of the electromagnetic field as an initial condition.
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| For theoretical purposes, the [[integral equation]] version is often very useful. It is as follows:
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| :<math>\lim_{x\to a}e^{i\xi x}v_{1}-\lim_{x\to b}e^{i\xi x}v_{1}=\int_{a}^{b}[e^{i\xi x}\,q_{1}\,v_{2}+e^{i\xi x}\,q_{2}\,v_{3}]\,dx</math> | |
| :<math>\lim_{x\to a}e^{-i\xi x}v_{2}-\lim_{x\to b}e^{-i\xi x}v_{2}=-\int_{a}^{b}e^{-i\xi x}\,q_{1}^{*}\,v_{1}\,dx</math> | |
| :<math>\lim_{x\to a}e^{-i\xi x}v_{3}-\lim_{x\to b}e^{-i\xi x}v_{3}=-\int_{a}^{b}e^{-i\xi x}\,q_{2}^{*}\,v_{1}\,dx</math>
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| One may make further substitutions and simplifications, depending on the limits used and the assumptions about boundary or initial conditions. One important concept is that <math>\xi</math> is complex; assumptions must be made about this [[eigenvalue]] parameter. If a non-zero solution is desired, the imaginary part of the eigenvalue cannot change [[Sign (mathematics)|sign]]; accordingly, most researchers take the imaginary part to be [[Positive number|positive]].
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| ==References== | |
| *C. Menyuk, ''Application of multiple-length-scale methods to the study of optical fiber transmission'', Journal of Engineering Mathematics 36: 113-136, 1999, Kluwer Academic Publishers, Netherlands.
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| *D. Kaup, B. Malomed, ''Soliton Trapping and Daughter Waves in the Manakov Model'', Physical Review A, Vol. 48, No. 1, July 1993.
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| *S. V. Manakov, ''Remarks on the Integrals of the Euler Equations of the n-dimensional Heavy Top'', Functional Anal. Appl., Vol. 10, pp. 93–94, 1976.
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| [[Category:Fiber optics]]
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| [[Category:Ordinary differential equations]]
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