The general theory of Whitham for slowly-varying non-linear wavetrains is extended to the case where some of the defining partial differential equations cannot be put into conservation form. Typical examples are consideredin plasma dynamics and water waves in which thelack of a conservation form is due to dissipation; anadditional non-conservative element, the presence of anexternal force, is treated for the plasma dynamics example. Certain numerical solutions of the water wavesproblem (the Korteweg-de Vries equation with dissipation)are considered and compared with perturbation expansionsabout the linearized solution; it is found that the firstcorrection term in the perturbation expansion is anexcellent qualitative indicator of the deviation of thedissipative decay rate from linearity.
A method for deriving necessary and sufficient conditionsfor the existence of a general uniform wavetrainsolution is presented and illustrated in the plasmadynamics problem. Peaking of the plasma wave is demonstrated, and it is shown that the necessary and sufficient existence conditions are essentially equivalent to the statement that no wave may have an amplitude larger than the peaked wave.
A new type of fully non-linear stability criterion is developed for the plasma uniform wavetrain. It isshown explicitly that this wavetrain is stable in thenear-linear limit. The nature of this new type ofstability is discussed.
Steady shock solutions are also considered. By aquite general method, it is demonstrated that the plasmaequations studied here have no steady shock solutionswhatsoever. A special type of steady shock is proposed,in which a uniform wavetrain joins across a jump discontinuity to a constant state. Such shocks may indeedexist for the Korteweg-de Vries equation, but are barredfrom the plasma problem because entropy would decreaseacross the shock front.
Finally, a way of including the Landau dampingmechanism in the plasma equations is given. It involvesputting in a dissipation term of convolution integralform, and parallels a similar approach of Whitham inwater wave theory. An important application of thiswould be towards resolving long-standing difficultiesabout the "collisionless" shock.
