Nonlinear Dispersion Equations: Smooth Deformations, Compactions, and Extensions to Higher Orders

Reference:

Galaktionov, V. A., 2008. Nonlinear Dispersion Equations: Smooth Deformations, Compactions, and Extensions to Higher Orders. Computational Mathematics and Mathematical Physics, 48 (10), pp. 1823-1856.

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Official URL:

http://dx.doi.org/10.1134/S0965542508100084

Abstract

The third-order nonlinear dispersion PDE, as the key model, u(t) = (uu(x))(xx) in R x R+ is studied. Two Riemann's problems for (0.1) with the initial data S--/+ (x) = -/+ sgnx create shock (u (x, t) equivalent to S-(x)) and smooth rarefaction (for the data S+) waves (see [16]). The concept of "delta-entropy" solutions and others are developed for establishing the existence and uniqueness for (0.1) by using stable smooth delta-deformations of shock-type solutions. These are analogous to entropy theory for scalar conservation laws such as u(t) + uu(x) = 0, which were developed by Oleinik and Kruzhkov (in x is an element of R-N) in the 1950s-1960s. The Rosenau-Hyman K (2, 2) (compacton) equation u(t) = (uu(x))(xx) + 4uu(x), which has a special importance for applications, is studied. Compactons as compactly supported travelling wave solutions are shown to be delta-entropy. Shock and rarefaction waves are discussed for other NDEs such as u(t) = (u(2)u(x))(xx), u(tt) = (uu(x))(xx), u(tt) = uu(x), u(ttt) = (uu(x))(xx), u(t) = (uu(x))(xxxxx), etc.

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