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On the Cauchy problem for the Kadomtsev-Petviashvili equation. (English) Zbl 0787.35086
The author gives a global result for the Cauchy problem of the Kadomtsev- Petviashvili equation. The solution is globally well-posed for initial data in $$L^ 2(\mathbb{R}^ 2/\mathbb{Z}^ 2)$$. The result is obtained on extending a local result (obtained by applying a Picard fixed point argument) to a global one with the help of the $$L^ 2$$-conservation law.
The method is analogous to the one used for the Schrödinger and Korteweg-de Vries equations, and it is based on an analysis of multiple Fourier series.