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On Korteweg-de Vries, Kadomtsev-Petviashvili and Boussinesq equations. On modulation theory. (Russian) Zbl 0724.35096

For the Kadomtsev-Petviashvili equation, evolution equations of the Whitham’s type for slow evolution of parameters of a periodic wave are derived in a general form, when the parameters depend on all the variables x,y, and t. The derivation does not make use of an explicit form of the underlying periodic wave; instead, it is based on the averaging technique (the Bogolyubov’s method). The particular case of the one-dimensional (y-independent) system is analyzed in more detail. It is demonstrated that this third-order system (which is more general than the system of the Whitham’s equations for the Korteweg-de Vries equation) admits three Riemann’s invariants. For this system, the equation for the characteristic velocities is investigated as well. Finally, the same is done for the Boussinesq equation. The corresponding third-order system governing the slow modulations of the periodic wave is also demonstrated to admit three Riemann invariants.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
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