The large-time development of the solution to an initial-value problem for the Korteweg-de Vries equation. II. Initial data has a discontinuous compressive step (Q2874616)

In the paper under review, the authors consider the initial-value problem for the Korteweg-de Vries equation NEWLINE\[NEWLINE u_t + uu_x + u_{xxx} = 0,\quad x\in \mathbb{R},\;t>0NEWLINE\]NEWLINE with \(u(x,0) = u_0 {\chi}_{\{x\leq 0\}}(x)\) and \( u(x,t) \sim u(x,0)\) as \(|x|\to \infty\) for every \(t>0\). Here, \(u_0>0\) and \(\chi_{\Omega}\) is the indicator function of a set \(\Omega\), that is, the initial profile is a step function. Using the methodology developed by the authors in [Matched asymptotic expansions in reaction-diffusion theory. London: Springer (2004; Zbl 1061.35002)], the detailed long-time asymptotic expansion of the solution to the initial-value problem is obtained. In particular, it is shown that the solution exhibits the formation of a dispersive shock wave.