Geodesic
Dispersive rarefaction wave with a large initial gradient
Ural mathematical journal, Tome 3 (2017) no. 1, pp. 33-43

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Consider the Cauchy problem for the Korteweg-de Vries equation with a small parameter at the highest derivative and a large gradient of the initial function. Numerical and analytical methods show that the obtained using renormalization formal asymptotics, corresponding to rarefaction waves, is an asymptotic solution of the KdV equation. The graphs of the asymptotic solutions are represented, including the case of non-monotonic initial data.
Keywords: The Korteweg-de Vries, Cauchy problem, Asymptotic behavior, Rarefaction wave. 
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Alexander E. Elbert; Sergey V. Zakharov. Dispersive rarefaction wave with a large initial gradient. Ural mathematical journal, Tome 3 (2017) no. 1, pp. 33-43. http://geodesic.mathdoc.fr/item/UMJ_2017_3_1_a2/