# “Two-speed” linear integro-differential equation

Working on a problem of many-electron dynamics in quantum dots I have arrived to an a following integro-differential equation: $$\frac{\partial}{\partial t} F(x,t)= - i (x+ v_1 t) F(x,t)-\alpha^2 g(x+v_2 t) \int g(y+v_2 t) F(y,t) dy$$ where $$g(z) = \sqrt{1-e^{-2 \beta z}} - e^{-\beta z}$$ and $v_1$,$v_2$, $\alpha$ and $\beta$ are real nonnegative constants with $v_2 \ge v_1$.

I'm looking for solution of the initial-value problem $F(x,0)=F_0(x)$.

1. For $v_1=v_2$, is it correct to reduce the equation to an ODE in $z=x+vt$ by seeking solutions in the form $F(x,t)=f(z)$? I hope the latter will be solvable by Wiener-Hopf technique.

2. Can you suggest an analytic strategy for the general case $v_1 \not = v_2$? Special cases of $v_1=0$ or $v_2=0$ are also of independent interest.

Any useful hints are welcome!

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