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Consider generalized Hartree Equation

$ i \partial_t u + \Delta u + \left( \frac{1}{|x|^{\gamma}} \ast |u|^p \right)|u|^{p-2}u, \quad u(x, 0)=u_0(x)$ where $0<\gamma<d, p\geq 2, u(x,t)\in \mathbb C.$

When $p=2$ $-$ classical Hartree equation.

Scaling invariance: if $u(t,x)$ solves GHE, then so does $ u_{\lambda}(t,x)= \lambda^{\frac{d-\gamma +2}{2(p-1)}} u (\lambda^2 t, \lambda x)$ also.

My question is: What is the scaling invariance for fractional generalized Hartree equation: $ i \partial_t u - (- \Delta)^{\alpha/2} u + \left( \frac{1}{|x|^{\gamma}} \ast |u|^p \right)|u|^{p-2}u, \quad u(x, 0)=u_0(x)$

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  • $\begingroup$ Carlo already answered your question: but this is essentially just a plug-and-chug question. The linear terms determine the scaling relation between $t$ and $x$. And after that you just solve an algebraic equation for the scaling factor in front of the function. // Are you asking the question because you don't know how to do this computation (in which case Carlo's answer is unlikely to be too helpful), or because you cannot be bothered to do it yourself? $\endgroup$
    – Willie Wong
    Commented Jul 10, 2020 at 15:52

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substitution of $u_\mu(t,x)=\text{constant}\times u(\mu^\alpha t,\mu x)$ in the fractional GHE shows that this is a solution if $\text{constant}=\mu^{\frac{d-\gamma+\alpha}{2(p-1)}}$, so the scale invariance relation is $$u_\mu(t,x)=\mu^{\frac{d-\gamma+\alpha}{2(p-1)}}u(\mu^\alpha t,\mu x).$$

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