Small-time asymptotics and the emergence of complex singularities for the KdV equation

Abstract

While real-valued solutions of the Korteweg--de Vries (KdV) equation have been studied extensively over the past 50 years, much less attention has been devoted to solution behaviour in the complex plane. Here we consider the analytic continuation of real solutions of KdV and investigate the role that complex-plane singularities play in early-time solutions on the real line. We apply techniques of exponential asymptotics to derive the small-time behaviour for dispersive waves that propagate in one direction, and demonstrate how the amplitude, wavelength and speed of these waves depend on the strength and location of double-pole singularities of the initial condition in the complex plane. Using matched asymptotic expansions in the limit $t\rightarrow 0^+$, we show how complex singularities of the time-dependent solution of the KdV equation emerge from these double-pole singularities. Generically, their speed as they move from their initial position is of $\mathcal{O}(t^{-2/3})$, while the direction in which these singularities propagate initially is dictated by a Painlevé II (P$_{\mathrm{II}}$) problem with decreasing tritronquée solutions. The well-known $N$-soliton solutions of KdV correspond to rational solutions of P$_{\mathrm{II}}$ with a finite number of singularities; otherwise, we postulate that infinitely many complex-plane singularities of KdV solutions are born at each double-pole singularity of the initial condition. We also provide asymptotic results for some non-generic cases in which singularities propagate more slowly than in the generic case. Our study makes progress towards the goal of providing a complete description of KdV solutions in the complex plane and, in turn, of relating this behaviour to the solution on the real line.

Figures (13)

• Figure 1: Numerical solutions of the KdV model (\ref{['eq:kdv']}) on the real line using the $\mathrm{sech}^2$-type initial condition (\ref{['eq:ICsech']}), showing dispersive waves propagating to the left. Calculations are performed using the spin command Montanelli2020 in Chebfun Driscoll2014. For (a),(b) since $0< -A_0 <2$, there are no solitons moving to the right; (c) since $2<-A_0<6$, there is one soliton; (d) since $6<-A_0<12$, there are two solitons. Note, although hard to see on this scale, the dispersive waves are roughly the same size for each of these four examples, while the height of the initial hump $u_0(0)=-A_0$ of course increases as $-A_0$ increases.
• Figure 2: (a) [left panel] Numerical solution of the KdV model (\ref{['eq:kdv']}) on the real line, computed at $t=0.02$ using the $\mathrm{sech}^2$-type initial condition (\ref{['eq:ICsech']}) with $A_0=-1/4$; [middle panel] plots of $u_{\mathrm{num}}-(u_0+tu_1)$ (blue) and $U_{\hbox{\footnotesize dis}}$ (red circles) versus $x$ using same parameters as in left panel; [right panel] plots of $\left(u_{\mathrm{num}}-(u_0+tu_1)\right)/K$ (blue) and $U_{\hbox{\footnotesize scaled}}$, again with same parameters as left panel. (b) Same as (a), except that $A_0=-3/4$.
• Figure 3: A numerically computed amplitude of the scaled dispersive waves $\left(u_{\mathrm{num}}-(u_0+tu_1)\right)/K$ (blue dots) plotted for various values of the parameter $A_0$ at a fixed time $t=0.03$, compared with the asymptotic prediction $(2/3^{3/2}\pi^{1/2})\cos\left( \frac{\pi}{2}\sqrt{1-4A_0}\right)$ (red solid). (a) the $\mathrm{sech}^2$-type initial condition (\ref{['eq:ICsech']}); (b) the initial condition (\ref{['eq:ICxsquared']}); (c) the refined initial condition (\ref{['eq:newIC']}).
• Figure 4: (a) [left panel] Numerical solution of the KdV model (\ref{['eq:kdv']}) on the real line, computed at $t=0.02$ using the generic initial condition (\ref{['eq:ICxsquared']}) with $A_0=-1/4$; [middle panel] plots of $u_{\mathrm{num}}-(u_0+tu_1)$ (blue) and $U_{\hbox{\footnotesize dis}}$ (red circles) versus $x$ using same parameters as in left panel; [right panel] plots of $\left(u_{\mathrm{num}}-(u_0+tu_1)\right)/K$ (blue) and $U_{\hbox{\footnotesize scaled}}$, again with same parameters as left panel. (b) Same as (a), except that $A_0=-3/4$. (c) Same as (a), except that $A_0=-2$.
• Figure 5: (a) A snapshot of a numerical solution of (\ref{['eq:kdv']}) with the initial condition (\ref{['eq:ICxsquared']}), plotted for $t=0.3$ with the first nine wave crests indicated by red dots. (b)--(c) Numerically-determined crest locations as a function of time (solid red) together with the asymptotic prediction (\ref{['eq:crests']}) (blue dots) for $m=1,\ldots,9$. The slope of the hypotenuse of the triangle in the log-log plot indicates the scaling $|x_m|\sim \,\mathrm{constant}\, t^{1/3}$.