Hi everybody! First post!(atleast in years and years).(adsbygoogle = window.adsbygoogle || []).push({});

The stationary KdV equation given by

$$ 6u(x)u_{x} - u_{xxx} = 0 $$.

It has a solution given by

$$ \bar{u}(x)=-2\sech^{2}(x) + \frac{2}{3} $$

This solution obeys the indentity

$$ \int_{0}^{z}\left(\bar{u}(y) - \frac{2}{3}\right)\int_{0}^{y}\left(\bar{u}(x) - \frac{2}{3}\right)dxdy = \bar{u}(z) + \frac{4}{3} $$

Is it possible to derive this kind of identity with out using the explicit form of the solution ##\bar{u}(x) ##? That is only by using the fact that ##\bar{u}(x) ## is a solution to (1). I tried differentiating (3) thrice on the left side but that generates a lot of different terms for which it is hard to use (1) to simplify. I really need some hints for this problem.

Thank you in advance :)

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# Derivation of an integral identity from the kdv equation.

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