Abstract
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 \({\mathbb {R}}^3\) 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.
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The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.
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Funding
The authors were supported by the National Natural Science Foundation of China (Grant No. 11601232) and the fundamental research funds for the Central Universities, China (Grant Nos. KJQN201717 and KYZ201537). The first author was also partially supported by the National Natural Science Foundation of China (Grant No. 11775116).
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Ren, X., Zhao, S. Reducibility for a class of quasi-periodic linear Schrödinger equations and its application. Nonlinear Dyn 111, 21207–21239 (2023). https://doi.org/10.1007/s11071-023-08925-6
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DOI: https://doi.org/10.1007/s11071-023-08925-6