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I have to solve the following nonlinear PDE: $$\partial_t u(x,t)=ku(x,t)^2 \partial_{xx}u(x,t)$$ where $k$ is a constant with $k>0$.

Is it possible to find some symmetry in this equation which could help to solve it? Thanks in advance.

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2 Answers 2

up vote 4 down vote accepted

There are at least 3 types of special solutions connected to some notion of symmetry that you might consider.

1) Solutions in separated variables: $u(x,t)=X(x)T(t)$. This leads to $$ k\,XX''=\frac{T'}{T^3}=\lambda\quad\text{constant.} $$ The resulting ODE's can be solved and you obtain a family of solutions.

2) Traveling wave solutions: $u(x,t)=\phi(x-c\,t)$, $c\in\mathbb{R}$. The resulting equation in $\phi$ is again solvable: $$ -c\,\phi'=k\,\phi^2\phi''. $$

3) Self-similar solutions of the form $u(x,t)=t^{\alpha/2} v(x\,t^{-(\alpha+1)/2})$. Then $v=v(\xi)$ satisfies the ODE $$ \frac{\alpha}{2}\,v-\frac{\alpha+1}{2}\,\xi\,v'=k\,v^2v''. $$

As for the possibility of obtaining a general solution, I am not very optimistic.

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This is a well studied PDE with respect to the symmetries analysis method. You can find a lot of articles in google scholar about it. For your convenience here they are its Lie point symmetries:

\begin{array}{l} \mathfrak X_1 = \partial _x \\ \mathfrak X_2 = \partial _t \\ \mathfrak X_3 = u\partial _u+x\partial _x \\ \mathfrak X_4 = 2 t\partial _t-u\partial _u \\ \end{array} To make a connection with the reductions/ansatzes proposed by Julián Aguirre, the third is connected with the symmetry $(\alpha+1)\mathfrak X_3+\mathfrak X_4$, the second with the symmetry $c\mathfrak X_1+\mathfrak X_2$ and as for the first one it's the $-2c\mathfrak X_2+\mathfrak X_4$ when $\lambda\ne0$ and the $\mathfrak X_2$ when $\lambda=0$.

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