Solution transport equation

Question

I have to solve the following equation $$\partial_t v+2A\cdot\nabla v-iB(x)\cdot Av=0$$ where $A$ is a constant vector and $B$ a smooth vector field. I can solve the transport equation $\partial_t v+2A\cdot\nabla v=0$ with characteristic method; how to treat the remaining term? Any suggestion?

Answer 1

Your equation, if expanded, reads:

$$ v_t + 2A_x v_x + 2A_y v_y + 2A_z v_z = i \, \mathbf{B}(x) \cdot \mathbf{A} \, v. $$

This is a linear 1st order PDE for which the method of characteristics reads:

$$ \frac{\mathrm{d}t}{1} = \frac{\mathrm{d}x}{2A_x} = \frac{\mathrm{d}y}{2A_y} = \frac{\mathrm{d}z}{2A_z} = \frac{\mathrm{d}u}{i \, \mathbf{B}(x) \cdot \mathbf{A} }. $$

Let me know if this helps.

Cheers!