# Manakov system

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[Maxwell's Equations](/source/Maxwell's_Equations), when converted to [cylindrical coordinates](/source/cylindrical_coordinates), and with the boundary conditions for an [optical fiber](/source/optical_fiber) while including [birefringence](/source/birefringence) as an effect taken into account, will yield the coupled [nonlinear Schrödinger equation](/source/nonlinear_Schr%C3%B6dinger_equation)s. After employing the [Inverse scattering transform](/source/Inverse_scattering_transform) (a procedure analogous to the [Fourier Transform](/source/Fourier_Transform) and [Laplace Transform](/source/Laplace_Transform)) on the resulting equations, the Manakov system is then obtained. The most general form of the Manakov system is as follows:

:<math>v_{1}'=-i\,\xi\,v_{1}+q_{1}\,v_{2}+q_{2}\,v_{3}</math>
:<math>v_{2}'=-q_{1}^{*}\,v_{1}+i\,\xi\,v_{2}</math>
:<math>v_{3}'=-q_{2}^{*}\,v_{1}+i\,\xi\,v_{3}.</math>

It is a coupled system of linear [ordinary differential equations](/source/ordinary_differential_equations). The functions <math>q_{1}, q_{2}</math> represent the envelope of the electromagnetic field as an initial condition.

For theoretical purposes, the [integral equation](/source/integral_equation) version is often very useful. It is as follows:

:<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>

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](/source/eigenvalue) parameter.  If a non-zero solution is desired, the imaginary part of the eigenvalue cannot change [sign](/source/Sign_(mathematics)); accordingly, most researchers take the imaginary part to be [positive](/source/Positive_number).

==References==
*{{cite journal | last=Menyuk | first=C. R. |title=Application of multiple-length-scale methods to the study of optical fiber transmission| journal=Journal of Engineering Mathematics | publisher=Springer Science and Business Media LLC | volume=36 | issue=1/2 | year=1999 | issn=0022-0833 | doi=10.1023/a:1017255407404 | pages=113–136| s2cid=9930111 }}
*{{cite journal | last1=Kaup | first1=D. J. | last2=Malomed | first2=B. A. | title=Soliton trapping and daughter waves in the Manakov model | journal=Physical Review A | publisher=American Physical Society (APS) | volume=48 | issue=1 | date=1993-07-01 | issn=1050-2947 | doi=10.1103/physreva.48.599 | pages=599–604| pmid=9909633 | bibcode=1993PhRvA..48..599K }}
*{{cite journal|first=S. V. |last=Manakov |title=[Remarks on the Integrals of the Euler Equations of the n-dimensional Heavy Top]|journal=Funktsional'nyĭ Analiz I Ego Prilozheniya|volume=10|issue=4|pages=93–94|year=1976|issn=0374-1990|language=ru}}

Category:Fiber optics
Category:Ordinary differential equations

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