It is a coupled system of linear ordinary differential equations. The functions represent the envelope of the electromagnetic field as an initial condition.
For theoretical purposes, the integral equation version is often very useful. It is as follows:
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 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; accordingly, most researchers take the imaginary part to be positive.
Menyuk, C. R. (1999). "Application of multiple-length-scale methods to the study of optical fiber transmission". Journal of Engineering Mathematics. 36 (1/2). Springer Science and Business Media LLC: 113–136. doi:10.1023/a:1017255407404. ISSN0022-0833. S2CID9930111.
Manakov, S. V. (1976). "[Remarks on the Integrals of the Euler Equations of the n-dimensional Heavy Top]". Funktsional'nyĭ Analiz I Ego Prilozheniya (in Russian). 10 (4): 93–94. ISSN0374-1990.