Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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.

share|cite|improve this answer

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 $\lambda=0$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.