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What is the necessary conditions we need to transform second order PDE to a system of ODE? e.g. If I have $$a^2*u_{tt}- u_{xx}+ u*u_{x}=0 $$ what conditions I needed to transform it to a system of ODE?

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If we use a travelling wave solution we have $z=x-ct$ $$ c^{2}a^{2}\frac{\mathrm{d}^{2}\Phi}{\mathrm{dz}^{2}}-\frac{\mathrm{d}^{2}\Phi}{\mathrm{dz}^{2}}+\Phi\frac{\mathrm{d}\Phi}{\mathrm{dz}}=0 $$ or $$ \lambda\frac{\mathrm{d}^{2}\Phi}{\mathrm{dz}^{2}}+\Phi\frac{\mathrm{d}\Phi}{\mathrm{dz}}=0 $$ where $\lambda=a^{2}c^{2}-1$.

we can change of variables to reduce to a first order ode

$$ \lambda p\frac{\mathrm{d}p}{\mathrm{d\Phi}}+\Phi p=0\\ p=\frac{\mathrm{d}\Phi}{\mathrm{dz}} $$ This is separable and can be solved to yield $$ \frac{\mathrm{d}\Phi}{\mathrm{dz}} = -\frac{\Phi^{2}}{2\lambda}+D. $$ Then solving this equation for $\Phi$ involves computing the following integral $$ \int\frac{1}{\Phi^{2}-2\lambda D}\mathrm{d}\Phi = -\frac{t}{2\lambda}+C_{1}. $$ now we have to split the cases for the integration constant $D$ ( I leave this for you to do: But the hint is consider $2\lambda D = \pm k^{2}$ and zero.).

The above is assuming we can have a constant reference frame where the evolution can be considered to have a constant $c$. Also, this is not solved with any conditions on boundaries or initial value problem. This is just a travelling wave solution but the end solutions should solve the equation as required.

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