Q:

Solving PDE with traveling wave Ansatz

I have the following PDE (where $a,b,c$ are constants)
$$\frac{\partial h}{\partial t}(x,t) =a\frac{\partial}{\partial x}\bigg(h^3(x,t)\frac{\partial h}{\partial x}(x,t)-bh^3(x,t)\bigg), $$
and I want to solve with a traveling wave in the form
$$h(x,t)=F(\xi),\quad \xi = x-ct,\quad F(0)=0,\quad \lim_{\xi\rightarrow -\infty}F(\xi) = h_0.$$
I start by writing
$$\frac{\partial h}{\partial t} = -cF'(\xi),\quad \frac{\partial h}{\partial x} = F'(\xi), \quad \frac{\partial^2 h}{\partial x^2} = F''(\xi).$$
The equation I want to solve then becomes
$$-cF'-aF^2\bigg(3(F')^2+FF''-bF'\bigg) = 0.$$
I'm stuck on proceeding from here. I should be looking to integrate. I would appreciate some help. I am looking to solve for the shape of the wavefront, I'm guessing that will be given implicitly, given the nature of the problem. Thanks.

A:

I wouldn't expand the r.h.s using differentiation rules as done in OP. Instead, let's do the following.
Injecting the traveling wave Ansatz $h(x,t)=F(\xi)$ with $\xi=x-ct$ in the PDE gives the differential equation
$$
-c F' = a\left(F^3 F' - b F^3\right)' .
$$
Integrating once, we have
$$
-c F = aF^3 (F' - b) +k_1 ,
$$
where $k_1$ is an arbitrary integration constant. Evaluating the previous equation at $\xi=0$ leads to $k_1=0$. If we assume that $F'$ vanishes as $\xi\to {-\infty}$, then we have the constraint
$$
{h_0} =\pm\sqrt{\frac{c}{ab}} \, .
$$
Finally, non-trivial solutions satisfy a Chini equation of the form
$$
F' = b- \frac{c}a F^{-2}
$$
for $\xi\in \Bbb R^*$.

