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Given the following nonlinear $PDE$: $$\partial_{tt} u(x,t)=ku(x,t)\partial_{xx}u(x,t)$$ with $$u(0,t)=u(L,t)=0$$ is it possible to solve it analytically? Could the solutions have singularities that can be interpreted as shock waves? Thanks in advance.

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Did you try that way? $$u(x,t)=P(x)T(t)$$ $$\partial_{tt} u(x,t)=ku(x,t)\partial_{xx}u(x,t)$$ $$P(x)T''(t)=kP(x)T(t)P''(x)T(t)$$ $$P(x)T''(t)=kP(x)P''(x)T^2(t)$$ $$\frac{T''(t)}{T^2(t)}=kP''(x)=c$$ c is a constant. –  Mathlover Apr 3 '12 at 12:24

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