## Kurtosis, 4th order diffusion, and wave speed

Here's a very simple problem students can do on their own to investigate how higher-order effects in linear parabolic equations can alter the wave-speed properties.

Consider a linear partial differential equation exhibiting non-zero 4th derivative. This is applicable to local expansion of kernels in integral-differential equations. Murray talks about this some.

$\frac{\partial N}{\partial t} = RN + D\frac{\partial^{2}N}{\partial x^{2}} + K \frac{\partial^{4}N}{\partial x^{4}}.$ $\dot{u} = u + u'' + K u''''.$

This is important in the limiting expansion of integro-differential equations. One can look for exponential traveling wave with intensity $\theta$ so that $u = \bar{C} e^{\theta(x-ct)},$ and this gives $0 = \bar{C} e^{\theta(x-ct)} \left( K \theta^{4} + \theta^{2} + c\theta + 1 \right).$ Applying elimination theory with criticality condition $\partial c / \partial \theta = 0$, $4K\theta^{3}+2\theta+c_{min}=0$, so the minimum wave speed $c_{min}$ must satisfy $(9Kc_{min}^{2}-8K+2)\theta +c_{min}(1+12K)=0.$ This leads to nasty 8th degree polynomial in $c_{min}$ and $K.$ However, using groebner basis theory, one finds a rotated hyperbola and an ellipse/hyperbola govern the solution: \begin{eqnarray} 2 \theta^{2} + 3 \theta c_{min} + 4 &=& 0 \label{eq:1}\\ 9 c_{min}^{2} - 4(12K+1) \theta^{2} &=& 32 \label{eq:2} \end{eqnarray} These allow the problem to be reduced to simple 1-d root-finding.

The question is whether there exists a minimum speed for these waves. Well, for given $c$, it is straight forward to show when $K>0$ there exist $0$ or $2$ invasion speeds for given wave intensities $\theta$ by inspecting $$K \theta^{4} + \theta^{2} + 1 = - c \theta.$$ When $K<0$ there are will be $2$ or $4$ solutions for $\theta$, but inspecting $c = - \frac{ K \theta^{4} + \theta^{2} + 1 }{ \theta },$ we see that for every possible $c$, there exists at least one $\theta > 0$ -- there is no band-gap in allowed speeds. Thus, we can not define any wave speeds for $K < 0$ without changing model fundamentals.

In this case ($K\geq 0$), $\frac{\partial c_{min}}{\partial K}>0$, such that $c_{min}$ grows without bound. For $K>0, c_{min}>2$ there is the very nice asymptotic approximation that $$K \sim \frac{27}{256}c_{min}^{4} - \frac{9}{16}c_{min}^{2} + \frac{1}{2}$$ which isn't too bad even when $K\sim 0$. For small $K$, $c_{min} = 2 + K + O(K^2)$, which recovers the standard diffusion behavior.