Nonresonant Mode Coupling for Classes of Korteweg-de Vries Equations

The method of mode coupling in third order is applied to generalized Korteweg-de Vries equations. These equations contain terms which are each a product of a function of the dependent variable u times one space derivative of u, up to a derivative of the fifth order. The resulting amplitude and phase equations are then specialized to three different classes of equations. For equations of the first class the amplitudes do not change, although all modes are coupled for the determination of the phases. For equations of the second class, all amplitude equations are truly coupled and the phases are constant. Finally, for equations of the third class, which includes the KdV equation, each mode behaves as if it were the only one present, so that only self-modulation is possible.