TITLE: Solution transport equation
QUESTION [2 upvotes]: 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?

REPLY [0 votes]: 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!
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