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Given the nonlinear PDE $$ \partial^2\phi+\phi^3=0 $$ I consider the characteristics $\xi=\kappa\cdot x$. Then I look for a solution in the form $$ \phi(x)=a\cdot\chi(\xi). $$ Then, provided $\kappa^2=\frac{a}{2}$, I get the ODE defining of the Jacobi ${\rm sn}$ function and the final solution takes the form $$ \phi(x)=a\cdot{\rm sn}(\xi+b,-1). $$ This solution has two arbitrary integration constants $a$ and $b$ that yield a family of solutions of the PDE we started from.

My question is this: Could I claim this set of solutions to be unique in some sense? Otherwise, under what conditions this claim could be supported? It is generally known that such a nonlinear equation could provide rich families of solutions so, this question appears to me not that easy to address.

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    $\begingroup$ With the necessary pinch of salt, WolframAlpha seems to corroborate your statement. $\endgroup$
    – Hachino
    Commented Jan 23, 2015 at 14:14
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    $\begingroup$ Why did you put this on hold? What is wrong with this formulation of the question? Please, check these papers arxiv.org/abs/1412.1955 arxiv.org/abs/1409.2351 arxiv.org/abs/1306.6530 arxiv.org/abs/0907.4053 Why should not this question fit your criteria for research level? $\endgroup$
    – Jon
    Commented Jan 23, 2015 at 23:28
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    $\begingroup$ The question I tried to ask is how general are these solutions. Please, could you suggest a better rewording? $\endgroup$
    – Jon
    Commented Jan 27, 2015 at 15:27

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