Dispersion Renormalization In One Kind Of FPU Chain

In this thesis, firstly,we indroduce one kind of wave turbulencemodel——generalized Fermi-Pasta-Ulam chain, and we collate langevinequation theory with numerical scheme result of FPU chain. We find thatthe dispersion is renormalized, actually, the new dispersive relation is thelinear dispersion multipled by a constant.Secondly, we study dynamic behavior of FPU chain with adoube-well potential energy, which occurs when, we note that thereseems no bare linear wave dynamics in this case. It’s amazing that we canalso find the renormalized dispersion in the wavenumber-frequencyspectral picture, this implies dispersive relation can be generated only bynonlinear term.Thirdly, we set one single particle in the double-well potential energyas an example and find linear dispersion of the double-well potential FPUchain, and also, we find out that the dynamic mechanism of double-wellpotential FPU chain is the same with the one in single-well potentialenergy.Fourthly, we obtain the renormalized dispersion via a self-consistencymean-field method, which is in the view of wave interaction. This time, weexplain the renormalized dispersion almost generated by trivial resonantfour wave interaction. Furthermore, we explain why renormalizeddispersion can be found in the double-well potential energy case.