I'm tryng to solve the following: \begin{equation} u_{tt}(t,x)-u_{xx}(t,x)=K\cdot\text{Exp}\left (u(t,x)\right) \end{equation} with $K \in \mathbb{R}\setminus\{0\}$. For now I don't care about boundaries conditions, then you can set them as you want.
2026-03-29 17:33:37.1774805617
Solution for a particular nonlinear wave equation (in 2D)
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In fact this belongs to a PDE of the form http://eqworld.ipmnet.ru/en/solutions/npde/npde2103.pdf.
The general solution is $u(t,x)=f(x-t)+g(x+t)-2\ln\left(n\int^{x-t}e^{f(r)}~dr-\dfrac{K}{8n}\int^{x+t}e^{g(s)}~ds\right)~.$