I have been spending time with the Tzitzeica equation, the travelling-wave solution of which is troubling me (in particular the ansatz given in Equation $(5)$):
For $u=u(x, t)$ \begin{equation} u_{xt} = e^{-u}+e^{-2u} \tag{1} \end{equation}
Under the transformation $v(x, t)=e^{-u}$ one obtains \begin{align} v = e^{-u} &\iff - \ln v = u \\ \implies u_t &= -\frac{v_t}{v} \\ \tag{2} \text{and}\, u_{tx} &= \frac{-v_{tx}v+v_{t}v_{x}}{v^{2}} \end{align} Inserting $(2)$ into $(1)$ we obtain \begin{align} -v_{tx}v - v_t v_x &= v^{2} \left( e^{-u}-e^{-2u}\right) \\ \iff 0 &= vv_{xt}-v_t v_x+v^{3}+v^{4} \tag{3} \end{align}
For so-called travelling-wave solutions set (into $(3)$) $$v=v(x, t), \qquad \xi=x-wt, \qquad v(x,t) = v(\xi)$$ to obtain \begin{equation} -wvv''+w(v')^2 + v^3 +v^4=0 \tag{4} \end{equation}
Now, the travelling-wave transformation above allows us to write solutions in the form $$v(\xi) = \sum_{k=0}^{N} A_k \left( \frac{\psi'(\xi)}{\psi( \xi)}\right) \tag{5}$$ Where the $A_k\, (k = 0,1,2,3,\ldots)$ are arbitrary constants to be determined and $\psi$ is an as yet undetermined function.
It is at this point that I cannot seem to progress other that inputting $(5)$ into $(4)$ and trying to determine the values of the $A_k$.
I owuld have thought that I could determine $N$ from the ansatz in $(5)$ to get a simpler solution.
Am I correct in thinking this?
Any help is greatly appreciated.
There seems to be a typo. The Tzitzeica-Bullough-Dodd equation is $u_{xt} = e^{u} - e^{-2u}$, according to this reference (eqn 2.9)
https://arxiv.org/pdf/1209.5517.pdf
Travelling wave solutions of the equation in your post may be found explicitly. Look for $u(x,t) = U(x-ct)$. Then $$ -cU'' = e^{-U} + e^{-2U} $$ and the general solutions are
$$ \begin{align} U_1(\xi) &= \log \left(\frac{e^{-\sqrt{A} \xi } \left(-A c+2 e^{\sqrt{A} \xi }-e^{2 \sqrt{A} \xi }-1\right)}{2 A c}\right)\\ U_2(\xi) &= \log \left(\frac{-A c e^{\sqrt{A} \xi }-e^{-\sqrt{A} \xi }-e^{\sqrt{A} \xi }+2}{2 A c}\right) \end{align} $$ where $A$ is a constant and $c$ is the wave speed.