Nonlinear dispersion equations: Smooth deformations, compactions, and extensions to higher orders

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
$$ u_t=(u^2u_x)_{xx},\quad u_{tt}=(uu_x)_{xx},\quad u_{tt}=uu_x,\quad u_{ttt}=(uu_x)_{xx},\quad u_t=(uu_x)_{xxxxxx},\quad \text{ets.} $$