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.
Nonlinear d'Alembert equation
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pde
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2Did 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. – 2012-04-03