On the basis of the condition for nuclear burning wave existence in the neutron-multiplicating media (U-Pu and Th-U cycles) we show the possibility of surmounting the so-called dpaparameter problem, and suggest an algorithm of the optimal nuclear burning wave mode adjustment, which is supposed to yield the wave parameters (fluence/neutron flux, width and speed of nuclear burning wave) that satisfy the dpa-condition associated with the tolerable level of the reactor materials radioactive stability, in particular that of the cladding materials.It is shown for the first time that the capture and fission cross-sections of 238 U and 239 Pu increase with temperature within 1000-3000 K range, which under certain conditions may lead to a global loss of the nuclear burning wave stability. Some variants of the possible stability loss due to the so-called blow-up modes (anomalous nuclear fuel temperature and neutron flow evolution) are discussed and are found to possibly become a reason for a trivial violation of the traveling wave reactor internal safety.