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 $f(z)=-2i/(x+i\delta)$.
(source: cam.ac.uk)
