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Nonlinear dispersion equations: Smooth deformations, compactions, and extensions to higher orders
V. A. Galaktionov Department of Mathematical Sciences, University of Bath, Math, BA2 7AY, UK
Abstract:
The third-order nonlinear dispersion PDE, as the key model,
\begin{equation}
u_t=(uu_x)_{xx}\quad\text{in}\quad\mathbb R\times\mathbb R_+.
\label{1}
\end{equation}
is studied. Two Riemann's problems for (1) with the initial data
$S_{\mp}(x)=\mp\operatorname{sign}{x}$ create shock (
$u(x,t)\equiv S_{-}(x)$) 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 (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
$x\in\mathbb R^N$) 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
$$
u_t=(u^2u_x)_{xx},\quad u_{tt}=(uu_x)_{xx},\quad u_{tt}=uu_x,\quad
u_{ttt}=(uu_x)_{xx},\quad u_t=(uu_x)_{xxxxxx},\quad \text{ets.}
$$
Key words:
Odd-order quasi-linear PDE, shock and rarefaction waves, entropy solutions, self-similar patterns.
UDC:
519.63 Received: 24.04.2008
Fulltext:
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