Solve analytically coupled nonlinear PDE?

Question

I want to solve this coupled nonlinear set of equations:

$$ \frac{\partial f}{\partial t}=\frac{\partial^2 f}{\partial x^2}+ag^n $$ $$ \frac{\partial g}{\partial t}=\frac{\partial^2 g}{\partial x^2}+b\frac{\partial f}{\partial x} $$ When $n = 1$, this is a linear system of equations and the wave ansatz $e^{\lambda t - ikx}$ is a solution. I thus expect waves to remain a solution, but a pretty different one. I am particularly interested in the cases $n=2$ and $n=3$. When it concerns PDEs I am very rusty, let alone nonlinear ones. Is there a recipe to solve this analytically? Note that if the solution involves an infinite sum of wave modes, I am only interested in the first mode. Small values of $a$ are also ok.

EDIT: I am also potentially interested in the solution of this coupled nonlinear set of equations: $$ \frac{\partial f}{\partial t}=ag^n $$ $$ \frac{\partial g}{\partial t}=b\frac{\partial f}{\partial x} $$ The second-order derivatives likely have an important impact on the solution, but it is not impossible for the behavior I am interested in to still occur in these simplified equations. In this case, we can differentiate the equation for $g$ with respect to $t$ and substitute the equation for $f$ and get an equation solely for the variable $g$: $$ \frac{\partial^2 g}{\partial t^2}=ab\frac{\partial g^n(x,t^\prime)}{\partial x}=nabg^{n-1}\frac{\partial g(x,t^\prime)}{\partial x}. $$ How do I solve this?

Answer 1

HINT: This is not a complete solution as requested by OP.
We can find closed solutions for some special cases $n=2, b=0$ (Maple CAS):

$$f\! \left(x,t\right) = -\frac{a c_{2}^{2} c_{3}^{2} {\mathrm e}^{2 k_{1}^{2} t-2 k_{1} x}}{2 k_{1}^{2}}-\frac{a c_{1}^{2} c_{3}^{2} {\mathrm e}^{2 k_{1}^{2} t+2 k_{1} x}}{2 k_{1}^{2}}+\frac{a c_{1} c_{2} c_{3}^{2} {\mathrm e}^{2 k_{1}^{2} t}}{k_{1}^{2}}-\frac{a c_{5} c_{3}^{2} {\mathrm e}^{-\sqrt{2}\, k_{1} x+2 k_{1}^{2} t}}{2 c_{6} k_{1}^{2}}+c_{6} c_{4} {\mathrm e}^{k_{2}^{2} t+k_{2} x}+c_{6} c_{5} {\mathrm e}^{k_{2}^{2} t-k_{2} x}-\frac{a c_{4} c_{3}^{2} {\mathrm e}^{2 k_{1}^{2} t+\sqrt{2}\, k_{1} x}}{2 c_{6} k_{1}^{2}}$$

$$g\! \left(x,t\right)=c_{1} c_{3} {\mathrm e}^{k_{1}^{2} t+k_{1} x}+c_{2} c_{3} {\mathrm e}^{k_{1}^{2} t-k_{1} x}$$

Special case $n=3, b=0$:

$$f\! \left(x,t\right) = -\frac{a c_{1}^{3} c_{3}^{3} {\mathrm e}^{3 k_{1}^{2} t+3 k_{1} x}}{6 k_{1}^{2}}-\frac{a c_{2}^{3} c_{3}^{3} {\mathrm e}^{3 k_{1}^{2} t-3 k_{1} x}}{6 k_{1}^{2}}+\frac{3 a c_{2} c_{1}^{2} c_{3}^{3} {\mathrm e}^{3 k_{1}^{2} t+k_{1} x}}{2 k_{1}^{2}}+\frac{3 a c_{1} c_{2}^{2} c_{3}^{3} {\mathrm e}^{3 k_{1}^{2} t-k_{1} x}}{2 k_{1}^{2}}+\frac{a c_{4} c_{3}^{3} {\mathrm e}^{3 k_{1}^{2} t+\sqrt{3}\, k_{1} x}}{6 c_{6} k_{1}^{2}}+\frac{a c_{5} c_{3}^{3} {\mathrm e}^{-\sqrt{3}\, k_{1} x+3 k_{1}^{2} t}}{6 c_{6} k_{1}^{2}}+c_{6} c_{4} {\mathrm e}^{k_{2}^{2} t+k_{2} x}+c_{6} c_{5} {\mathrm e}^{k_{2}^{2} t-k_{2} x}$$

$$g\! \left(x,t\right)=c_{1} c_{3} {\mathrm e}^{k_{1}^{2} t+k_{1} x}+c_{2} c_{3} {\mathrm e}^{k_{1}^{2} t-k_{1} x}$$

But don't know if there are closed solutions for $b\neq 0$ or general $n\in \mathbb{Z}$.

Visualization $[f(x,t),g(x,t)]$ with example values $n=2, a=1, b=0, k_1=i, k_2=2 i, c_1=c_2=c_3=c_4=c_5=c_6=1$:

Addendum

For case $n=1, b\neq 0$ we'll get expressions like this in the solution:

$${\mathrm e}^{\frac{\left(\mathit{RootOf}\left(\textit{_}Z^{3} c_{1}^{6}+4 c_{2} \textit{_}Z^{2} c_{1}^{4}+5 c_{2}^{2} \textit{_}Z c_{1}^{2}-c_{1}^{3} a b+2 c_{2}^{3}, \mathit{index}=1\right) c_{1}^{2}+2 c_{2}\right) \left(c_{1} x+c_{2} t+c_{3}\right)}{c_{1}^{2}}}$$

It means that we have to determine the roots of a third order polynomial in $\textit{_}Z$.
In general these polynomials must be solved numerically!