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!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.