How does variable thrust solid rocket booster work? What’s the mechanism for controlling how fast solid fuel burns?
I think to understand that you must first ask, what is the purpose of the variable thrust requirement? Depending on what you are trying to achieve, some options either present themselves or remove themselves.
The most obvious reason for a BVR missile to want vary thrust is to increase total time under controlled thrust to boost range and terminal energy state. Being able to reduce thrust when turning can also help with agility, but would come with significantly higher technical challenges and costs, so not sure that juice would be worth the squeeze, especially for a BVRAAM.
So, with the above assumption in place, the most obvious and technically feasible solution would be to simply build upon and expand the dual pulse principle of the PL15 and add in more burn stages. Going from dual pulse to triple or even quadruple pulse would be relatively straightforwards, but just how many stages you can do will ultimately come down to how light and compact you can make your stop/start stage boundaries and how rapidly and reliably you can trigger subsequent stages after the previous one burns out.
At one end of the spectrum, you can have say 4-6 one second burn pulses in your first stage section to be used during cruise phase where each pulse will burn out, and the missile will cruise on inertia for a predetermined time before triggering the next pulse and vastly increase cruise range. Then the final stage kicks in as one continuous burn for terminal engagement.
Ideally you will want to be able to jettison the first stage both to boost terminal performance to minimise the dead weight and aerodynamic drag you are hauling, but also to allow you to engage targets at much close ranges without massively nerfing your missile’s performance by limiting its burn characteristics to those of the first stage on-and-off burn mode.
At the other end of the spectrum, if you can make the pulse separation mechanism light and compact enough to give you the ability to pack in enough such pulses to control burn time to 0.25-0.5 seconds (which isn’t actually that many pulses, since the AIM120C5 has a totally burn time of between 7-9s, so if we just use a nice round 10s total burn time estimate for the PL16, with 4-6s for the first stage, that leaves 4-6s for the second stage, so that’s 8-24 pulses, which while a lot, isn’t stupidly high either). The key make or break tech would be how fast re-ignition could happen and how reliable they can make it, since the number of pulses will multiply the failure rate, so they need the new re-ignition mechanism to have a reliability of 4-12 times of that used on the PL15 to have the same failure rate on re-ignition for example.
Personally, I think in terms of tech feasibility, having a 6s first stage broken into 4-6 pulses of 1-1.5s duration each and a 4s second stage full burn for terminal engagement seems the most likely approach. With the micro-pulse terminal stage design more likely as a later block version or a totally new missile design.