Because this explosive entry is made up mainly of stock scale figures in double octaves which we will have played a thousand times if once along the way to getting to the point of learning this work, it's not the materials the entry has been made up from that's the problem but the context in which they're being deployed. Tchaikovsky was a competent enough player, but not a practising virtuoso of any kind, and in these kinds of situations in his concertos, it doesn't half show! :-)
The figuration itself being commonplace from étude literature, playing such double octaves at full power and high speed will hold no mysteries at this point. The difficulty rests elsewhere and for each of us the degree & nature of difficulty will vary according to our experience at the point of tackling this, so I'll list a number of them and their 'remedies' encountered in the field, in both myself and others, so you can select which resonate with you in your situation.
[A note in advance: there's absolutely *no* point in trying to learn this any other way than at speed from the outset. In the same way that we cannot learn to run by simply walking slowly and then speeding that up (the two sets of movements involved are completely different), playing this kind of bravura only exists at the speed it needs to exist, and it is *that* which we need to learn to assimilate & master.]
1. The Clock
It's handed to us by the strings, commonly at whole-note = +/- 70. We need to take it from them at bar 345. At bar 347, we then set our own clock fractionally, at just one or two bpm, *faster* as we prepare to enter. This ensures several things. Firstly, that in practice acoustically we will enter exactly on the timp-stroke underpinning the supporting tutti chord and, secondly, that in the auditorium we will sound exactly in time with the precedent the strings have just set. Without this tweak of the clock, we'll always acoustically sound fractionally slower than the strings, which is not good. (Because of the split seconds we lose in the upward leaps to come, it all irons out very quickly.)
2. Gesture
Because of the stock-in-trade nature of the actual double octave figure we already know all too well from étude literature, there is a temptation likewise to cluster these octaves, by analogy, into single, bar's-worth units separated by the major upward leaps. That will cause this whole explosive span to fragment. Instead, create 3, two-bar gesture clusters (bars [348+9] [350+1] [352+3]) followed by 2, three-bar gesture clusters ([354-6] + [357-9]) Once you do that, you'll instantly *feel* as well as see that this is, like most of these bravura arches, perfectly symmetrical in architecture, made up of two gesture 'super clusters' of 6 bars each.
What takes the most practice is a. to be able seamlessly to extend physical tension control to take on the entire length of each 'super-cluster' and b. do that 'as one' while successfully, simultaneously articulating the rapid gunfire octave clusters into their 2 and 3 based subsets within those two great spans.
3. Hazards
All carefully arrived at agreements reached in rehearsal notwithstanding, we will encounter the conductor who, subject to momentary 'inspiration', suddenly slows the last 4 eighths in the strings (bar 347) completely scuppering our own preparations to enter in the accelerated manner above. To be able to cope with that eventuality -- make yourself no illusions; it *will* happen one day..!
-- learn to negotiate the two-bar gesture clusters as though they start on beat 2 (alla breve), every other bar. (This means that the major leaps are now within the clusters, not linking them,.) Being able to negotiate the first clusters 'against the grain' like that can mean the difference between falling off the entire entry because of the conductor's 'moment of inspiration', or not.
4. The leaps
Train yourself to accelerate *in the air* as you negotiate the major leaps. The extra velocity will actually keep you acoustically in strict time, which otherwise the variable distances that have to be leaped across can cause to deteriorate. There is next to no space for any agogic inflection, never mind 'rubato', in these twelve bars, except where marked!
Hope this helps some of the way at least.
All the best,