4
$\begingroup$

Consider the initial-value problems in $d=1$ $$\begin{cases} i\partial_tu = \Delta^2 u \\ u(x,0)=u_0 \end{cases}$$ and $$\begin{cases} i\partial_t u= \Delta u \\ u(x,0)=u_0, \end{cases}$$ Solutions to these equations obey the dispersive estimates $$\|e^{-it\Delta^2}u_0\|_{L^\infty} \lesssim t^{-1/4} \|u_0\|_{L^1} \\ \|e^{-it\Delta}u_0\|_{L^\infty} \lesssim t^{-1/2} \|u_0\|_{L^1}.$$ By conservation of the $L^2$ norm, we thus expect for long times, a localized initial data will in the first case spread like $\Delta x\gtrsim t^{1/2} $ and in the second case, $\Delta x \gtrsim t$ (indeed for the latter, taking a Gaussian as our initial data, we have that $\Delta x \sim t$ for large $t$). Hence, the latter seems to be more dispersive since it can delocalize faster.

However, we also know that for the former, the ratio of the group velocity to the phase velocity is 2 times as large compared to the latter. So from this perspective, we would predict the former to be more dispersive.

What gives?

EDIT: The second dispersive estimate follows from computing $\mathcal{F}^{-1}(e^{-i|\xi|^2t})(x)= C t^{-1/2} e^{-i|x|^2/(4t)}$ while the first follow from this paper. The phase and group velocities for the first equation are $\omega(k)/k= k^3$ and $\partial_k k^4=4k^3$. For the second, they are $\omega(k)/k= k$ and $\partial_k k^2=2k$.

Cross-posted to MathOverflow

$\endgroup$
5
  • $\begingroup$ Attaching some of your calculations would help a lot here. How did you obtain these "dispersive estimates", and what do they mean? What does $\Vert \mathrm e^{\mathrm it\Delta} u_0 \Vert_\infty$ have to do with dispersion? How did you compute the phase and group velocities? $\endgroup$
    – K.defaoite
    Commented May 15 at 20:00
  • $\begingroup$ See my edits. They are dispersive because since the $L^\infty$ norm decays and the $L^2$ norm remains constant, the solution must spread out as time increases. $\endgroup$
    – Diffusion
    Commented May 16 at 3:39
  • $\begingroup$ Thanks, but I still don't understand the significance of $\mathrm e^{\mathrm it~\mathscr L}$ where $\mathscr L$ is either one of the operators. $\endgroup$
    – K.defaoite
    Commented May 16 at 15:05
  • 1
    $\begingroup$ They are solutions of $i\partial_t u = \mathscr{L}u$. $\endgroup$
    – Diffusion
    Commented May 16 at 16:35
  • 1
    $\begingroup$ Possibly one way to think about it: The speed (e.g. group velocity) at which low frequencies travel is more relevant to dispersion here since you’re measuring dispersion against an $L^1$ norm. Near $k=0$, the group velocity for the first equation is much smaller than for the usual Schrödinger equation, which could explain the slower decay. $\endgroup$ Commented May 18 at 0:16

0

You must log in to answer this question.