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I have the following nonlinear dispersive PDEs

$$i \partial_t u- \partial_x^2 u =|\partial_x| |u|^2$$ where $f$ is some nice complex-valued function.

I am trying to use the ansatz $u(t,x) = e^{i \omega t}\varphi(x)$ to analyze the standing waves. However, I doubt that my calculations to the nonlinear term

\begin{align*} |\partial_x| |u|^2&=|\partial_x| \varphi^2\\ &=\mathcal{F}^{-1}(|\xi| \hat{\varphi^2})\\ &= \mathcal{F}^{-1}(|\xi|) * \varphi^2\\ \end{align*}

But I know that there is no point go the last part of the equation. Could you enlighten me please. Thanks in advance.

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    $\begingroup$ what does the absolute value $|\partial_x|$ of a differential mean? $\endgroup$
    – Carlo Beenakker
    Commented Sep 1, 2022 at 14:40
  • $\begingroup$ @CarloBeenakker, perhaps it is meant as a pseudodifferential operator. What I don't understand is where the function $f$ appears. $\endgroup$
    – Daniele Tampieri
    Commented Sep 1, 2022 at 14:46
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    $\begingroup$ Do you mean "$\lvert\partial\rvert$" (rather than just "$\lvert\partial$") in the title? $\endgroup$
    – LSpice
    Commented Sep 1, 2022 at 14:48
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    $\begingroup$ You can write $|\partial||u|^2=R\partial(u\overline{u})=2R\Re(\overline{u}\partial u)$, where $R$ is the Riesz transform. After this point, it depends on where you want to go $\endgroup$
    – Piero D'Ancona
    Commented Sep 1, 2022 at 18:39
  • $\begingroup$ @PieroD'Ancona I want to analyse the traveling wave solution of the ODE that will result from substituting the ansatz above. However, I do not know how to put the nonlinear term in terms of $varphi$ $\endgroup$
    – Mr. Proof
    Commented Sep 1, 2022 at 23:39

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