Reducibility for a class of quasi-periodic linear Schrödinger equations and its application

By modifying a Kuksin’s estimate, the coefficients of a one-dimensional quasi-periodic linear Schrödinger equation can be reduced to constants, and thus, the existence of the quasi-periodic solutions is obtained. Moreover, for the reduced system, the boundedness, the blowup and the specific form of the quasi-periodic solutions are analyzed in detail, via the depiction of the phase portrait with respect to the pseudo-time in R 3 and the growth of solutions from the spectrum theory of the associated lattice Schrödinger operator numerically. The result is based on infinite-dimensional KAM theory and bifurcation theory, which is original. The reduction techniques can also be used to the Ginzburg–Landau equation, which can be applied to traffic flow hydrodynamics. The solitary wavelike quasi-periodic solutions obtained can be further refined and applied to optical soliton communication.