Dispersive shock wave interactions and asymptotics

Dispersive shock waves (DSWs) are physically important phenomena that occur in systems dominated by weak dispersion and weak nonlinearity. The Korteweg–de Vries (KdV) equation is the universal model for systems with weak dispersion and weak, quadratic nonlinearity. Here we show that the long-time-asymptotic solution of the KdV equation for general, steplike data is a single-phase DSW; this DSW is the “largest” possible DSW based on the boundary data. We find this asymptotic solution using the inverse scattering transform and matched-asymptotic expansions. So while multistep data evolve to have multiphase dynamics at intermediate times, these interacting DSWs eventually merge to form a single-phase DSW at large time.

Figure 1

Numerical simulations (using the scheme in Ref. [14]) of three well-separated steps (at $t=0$, with ${\varepsilon }^2=0.1$ and $c=1$). Here we see (a) three single-phase DSWs at $t=1$; (b) and (c) strong interaction and multiphase dynamics; and (d) eventual merging to form a single-phase DSW.

Figure 2

Numerically computed solutions of the KdV equation for the initial conditions shown in gray. (a) Nonvanishing boundary conditions, $c\ne 0$. This solution has three basic regions: rapid decay in region A, right of the DSW; strong nonlinearity of width $O\left(t\right)$ in region B; and an oscillating tail in region C, left of the DSW. (b) Vanishing boundary conditions, $c=0$. Here, the solution has four basic regions (see Ref. [19]). Region III has strong nonlinearity with height $O\left[{\left(\log t\right)}^{1/2}{t}^{-2/3}\right]$ and width $O\left[t^{1/3}{\left(\log t\right)}^{2/3}\right]$.

Figure 3

The value of $r_2\left(\chi \right)$ found numerically for $0<\chi <10c^2$, where $\chi \equiv \zeta /t$. For comparison, we include $-r_2\left(\chi \right)$ as a dashed line and a numerical simulation of $u(x,t)$ in gray (inside the envelope of $r_2$ and $-r_2$) for a single step at (a) $t/\varepsilon =10$ and (b) $t/\varepsilon =200$. Note that $\chi =0$ corresponds to $x\sim -2c^{2}t$ and $\chi =10c^2$ to $x\sim -12c^{2}t$.