This problem arose in the course of a theoretical physics project. We seek (complex) solutions of the imaginary time Benjamin--Ono equation

$$u_t-iu u_x-iu_{H,xx}=0$$

where $u_H(x,t)$ denotes the Hilbert transform with respect to the spatial ($x$) variable

$$u_H(x,t)=\frac{\text{P}}{\pi}\int_{-\infty}^{\infty}\frac{u(y,t)}{x-y}dy.$$

In particular, we are interested in similarity solutions of the form

$$u(x,t)=\frac{1}{\sqrt{t}}f\left(\frac{x}{\sqrt{t}}\right),$$

which gives rise to the equation

$$z f(z)+if(z)^2+2if_{H}'(z)=\text{const.}$$

Without the Hilbert transform in the last term things would be straightforward from this point, as we would have a Riccati equation. The same would go for solutions analytic in the upper or lower half plane, for which $f_H(z)=\mp i f(z)$. This would correspond to finding similarity solutions of the Burgers equation.

My question is: **are there `non-trivial' similarity solutions?** That is, solutions with $f(z)$ not analytic in the upper or lower half plane?

Any suggestions of how to proceed, and cope with the Hilbert transform, would be gratefully received!

**Update:** This problem (in real time) is discussed at the end of Section 3.5 of the book *Bilinear Transformation Method* by Yoshimasa Matsuno, with the comment "The analytical solution ... has not yet been obtained, therefore the numerical method may be applied to it".

**Another update:** Numerical solution is shown below, for the case of $\text{const.}=2i$. Blue and green lines are the real and imaginary parts respectively. At time zero this solution becomes $2i/(x+i\delta)$. Note that for $\text{const.}=-2i$, the only solution I can find is the trivial one $f(z)=-2i/(x+i\delta)$.