I was wondering if there is a general (theoretical, not experimental) method for finding the dispersion relation for waves in a medium, say given the equation governing purturbations in the medium? For linear, homogeneous equations (the only type I've come across so far as a second-year physics undergraduate), it is clear you can simply substitute in a general sinusoid/exponential form to obtain this, however it is not clear to me how to approach this for non-linear and/or inhomogeneous equations.
I think the exact methods depend on the type of system. The simplest system is probably gases with one free electron. Here you can find how the dipole moment of the polarizability of the neucleus+electron changes when light is shined on this gas-light system. The you can see how this varies when changing the frequency of light (giving you a dispersion relation).
Brief summary of scientific direction: Using complex values of velocity and coordinates when solving nonlinear partial differential equations
Just as the square equation has complex roots, the nonlinear partial differential equations have complex solutions. It turns out that the complex solution is probabilistic. The physical meaning of the real part is the average value of the solution, and the imaginary part means the standard deviation. The nonlinear Navier-Stokes equation is reduced to an infinite system of ordinary differential equations of the first order. The complex coordinates of the equilibrium position describe the turbulent solution. Problems arise when recalculating the imaginary part of a complex solution into a real solution. But in the attached articles, for which the abstract describes the solution to these problems. For different types of roughness, the solution to these problems is different.
- YAKUBOVSKIY, EG. "STUDY OF NAVIER-STOKES EQUATION SOLUTION I. THE GENERAL SOLUTION OF NONLINEAR ORDINARY DIFFERENTIAL EQUATION." EUROPEAN JOURNAL OF NATURAL HISTORY 3 (2016): 60-66. https://world-science.ru/pdf/2016/3/14.pdf
- YAKUBOVSKIY, EG. "STUDY OF NAVIER-STOKES EQUATION SOLUTION II. THE USE OF LAMINAR SOLUTIONS." EUROPEAN JOURNAL OF NATURAL HISTORY 3 (2016): 67-83.https://world-science.ru/pdf/2016/3/15.pdf
- YAKUBOVSKIY, E. G. "STUDY OF NAVIER–STOKES EQUATION SOLUTION III. THE PHYSICAL SENSE OF THE COMPLEX VELOCITY AND CONCLUSIONS." EUROPEAN JOURNAL OF NATURAL HISTORY 3 (2016): 84-87. https://www.world-science.ru/pdf/2016/3/16.pdf