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The Korteweg-de Vries equation with small dispersion: Higher order Lax- Levermore theory. (English) Zbl 0705.35125
The aim of the author is to give an asymptotic analysis of the solution to the initial value problem for the KdV equation: $u_ t=\sigma uu_ x+\epsilon^ 2u_{xxx}=0$ (with the small parameter $$\epsilon$$), $$\epsilon\to 0.$$
From this analysis, he deduces the nature of the oscillations, near shocks, of the equation $$u_ t+\sigma uu_ x=0$$ (i.e. without $$\epsilon$$).
Reviewer: M.Derridj

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 35B25 Singular perturbations in context of PDEs
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##### References:
 [1] Buslaev, Vestnik Leningrad Univ. 17 pp 56– (1962) [2] Cohen, Indiana U. Math. Jour. 34 pp 127– (1985) [3] Flaschka, Comm. Pure Appl. Math 33 pp 739– (1980) [4] Lax, Comm. Pure Appl. Math. 36 pp 253– (1983) [5] The hyperbolic nature of the zero dispersion KdV limit, Comm. P.D.E., 1988, pp. 495–514. · Zbl 0678.35081 [6] Venakides, Comm. Pure Appl. Math. 38 pp 125– (1985) [7] Venakides, Comm. Pure Appl. Math. 38 pp 883– (1985) [8] Venakides, AMS Trans. 301 pp 189– (1987) [9] Venakides, Comm. Pure Appl. Math. 42 pp 711– (1989) [10] and , Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, 1978. · Zbl 0417.34001
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