KDV limit of the hydromagnetic waves in cold plasma

In this paper, we study the long wavelength limit for the hydromagnetic waves propagating across a magnetic field in cold plasma. Based on the Gardner–Morikawa transform and the reductive perturbation method, it is demonstrated that as $$\varepsilon \rightarrow 0$$ε→0, the solutions of such hydromagnetic waves converge to the solution of the Korteweg–de Vries equation on an $$O(\varepsilon ^{-3/2})$$O(ε-3/2) time interval.

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