Manakov system

If the scientist begins with Maxwell's Equations, converts to cylindrical coordinates, and uses the boundary conditions for a fiber optic cable while including birefringence as an effect, he would arrive at the coupled nonlinear Schrödinger equations. He could then employ the Inverse Scattering Transform, a procedure analogous to the Fourier Transform and Laplace Transform, to obtain the Manakov system. The most general form of the Manakov system is as follows:

$v_{1}'=-i\,\xi \,v_{1}+q_{1}\,v_{2}+q_{2}\,v_{3}$

$v_{2}'=-q_{1}^{*}\,v_{1}+i\,\xi \,v_{2}$

$v_{3}'=-q_{2}^{*}\,v_{1}+i\,\xi \,v_{3}.$

It is a coupled system of linear ordinary differential equations. The functions $q_{1},q_{2}$ represent the envelope of the electromagnetic field as an initial condition.

For theoretical purposes, the integral equation version can be very useful at times. It is as follows:

$\lim _{t\to a}e^{i\xi t}v_{1}-\lim _{t\to b}e^{i\xi t}v_{1}=\int _{a}^{b}[e^{i\xi t}\,q_{1}\,v_{2}+e^{i\xi t}\,q_{2}\,v_{3}]\,dt$

$\lim _{t\to a}e^{-i\xi t}v_{2}-\lim _{t\to b}e^{-i\xi t}v_{2}=-\int _{a}^{b}e^{-i\xi t}\,q_{1}^{*}\,v_{1}\,dt$

$\lim _{t\to a}e^{-i\xi t}v_{3}-\lim _{t\to b}e^{-i\xi t}v_{3}=-\int _{a}^{b}e^{-i\xi t}\,q_{2}^{*}\,v_{1}\,dt$

The reader may make further substitutions and simplifications depending on the limits used and the assumptions about boundary or initial conditions. One important concept is that $\xi$ is complex. The reader must make some assumptions about this eigenvalue parameter. If you assume the solutions are relatively "nice", and you do not want to have the zero solution, then the imaginary part of the eigenvalue cannot change sign. Most researchers take the imaginary part to be positive.