I am trying to solve the following PDE

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

where $$\mathbb{H}(g)=P.V. \int_{-\infty}^{\infty} \frac{g(x')}{x-x'} dx'$$

is the Hilbert Transform, P.V. means principal value, while $y=f(x-y t)$ is an implicit function given by the initial conditions, and is a solution to the inviscid Burgers equation, i.e. it satisfies $$y_t+yy_x=0.$$

Furthermore, the initial condition is that at $t=0$, $u=0$.

$\bf{Attempt\ at\ a\ solution:}$

The Lagrange equations for this system are $$dt=\frac{dx}{y}=-\frac{du}{y+\mathbb{H}(y_x)}.$$

Equating the second and third relation, we have $$u = -\left(x+\int \frac{1}{y}\mathbb{H}(y_x) \ dx\right)+c_1$$ for a constant $c_1$. Furthermore, the first relation implies

$$\frac{dx}{dt}= y=f(x-yt).$$

As $y$ satisfies Burgers equation it is constant along the characteristics and this becomes a trivial integration, yielding $$x-y t=c_2.$$

This relation defines the characteristics of the system. This should implies an arbitrary function (based on the initial conditions) $$h (x-y t)$$ should also satisfy the governing equation, but when I substitute this back into the original PDE I am finding it doesn't satisfy the equation.

Ignoring this for now, and equating the first and third terms in the Lagrange equations, I find $$u=-\int y+\mathbb{H}(y_x) \ dt+c_3.$$

The solution clearly has the form

$$u=-x +h(x-yt) +S$$

where $S$ is the particular solution satisfying the term related to the Hilbert Transform. I cannot figure out how to find this term though, and this is where I am asking for help.


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