A Solving forced PDE with method of Characteristics

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1. Apr 26, 2016

nickthequick

I'm trying to solve the following PDE:
$$u_t+yu_x=-y-\mathbb{H}(y_x)$$
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.

2. May 2, 2016