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$$u_t+yu_x=-y-\mathbb{H}(y_x)$$

where y satisfies the inviscid Burgers equation

$$y_t+yy_x=0$$

and the Hilbert Transform is defined as

$$\mathbb{H}(f) = PV \int_{-\infty}^{\infty} \frac{f(x')}{x-x'} \ dx',$$

where PV means principal value.

The solutions for y are determined by the initial conditions, so that if y(t=0,x)=F(x) then y=F(x-yt).

Now, I know how to solve the homogeneous part of the equation, as well as the particular solution for the term -y on the right hand side of the governing equation. However, I cannot find a solution to

$$u_t+yu_x=-\mathbb{H}(y_x).$$

Any tips on how to find solutions for a system like this are appreciated.