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I am wondering whether someone can help me with a non-linear PDE:

$$\frac{\partial^2\phi}{\partial t^2} = c\frac{\partial^2\phi}{\partial x^2} \left(\frac{\partial\phi}{\partial x} \right)^{n-1}$$

where, $\phi(x,t)$ is a function of $x$ and $t$, and $c$ and $n$ are constants ($c$ has dependency on $n$). This is not a homework problem and I am not a professional mathematician. I would appreciate any help! Even conjectures will help! thanks,

Newbee
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  • A trivial remark: linear functions are solution, i.e. $\phi(x,t)=A+Bx+Ct$ is a solution, where $A$, $B$, and $C$ are constants. – Paul Aug 29 '13 at 05:57
  • @newbee: great question for a first-timer, +1! Would love to hear some background information, where you got this equation, is it related to some physics problem, und so weiter! – Robert Lewis Aug 29 '13 at 05:58
  • @Paul, yes I have noticed that, but it is not very helpful for me. Thanks for the observation. – Newbee Aug 29 '13 at 08:04
  • @Robert Lewis It is actually related to a non-linear dynamical problem I am simulating. This is the Euler-Lagrange equation for that system. I have searched a lot, but couldn't find this equation anywhere. Thanks! – Newbee Aug 29 '13 at 08:05
  • @Newbee: very interesting! Could you be more specific? How does it arise as an Euler -Lagrange equation? From what? Can you provide any citings, references and/or links to relevant material. If you can, it would be much appreciated by yours truly. Thanks, Bob Lewis – Robert Lewis Aug 29 '13 at 08:30
  • @Newbee: could it be related to the porous medium equation? I dunno, looks sorta familiar/recognizable . . . – Robert Lewis Aug 29 '13 at 08:38
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    This might be a trivial addition to the above question, but a more compact form may strike a few more ideas. $$ \frac{\partial^{2}\phi}{\partial t^{2}};=;\frac{c}{n}\frac{\partial}{\partial x}\left(\frac{\partial\phi}{\partial x}\right)^{n} $$ I for one do not recognise this equation from any of the nonlinear pdes I have worked on personally. – Chinny84 Sep 01 '13 at 21:22

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