Modulation parameters

When a solution is close to a soliton at each time t, the position, amplitude, phase, etc. of the soliton that best approximates the solution are collectively known as the modulation parameters. One has some freedom regarding how to define the "best" approximation, but once one does, these parameters will obey some ordinary differential equations (coupled with a PDE governing the error term between the solution and the soliton), known as the modulation equations.

The presence of modulation parameters is closely tied to the symmetries of the underlying equation.