Taiwanese Journal of Mathematics

Construction of Periodic Solutions for Nonlinear Wave Equations by a Para-differential Method

Bochao Chen, Yixian Gao, and Yong Li

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This paper is concerned with the existence of families of time-periodic solutions for the nonlinear wave equations with Hamiltonian perturbations on one-dimensional tori. We obtain the result by a new method: a para-differential conjugation together with a classical iteration scheme, which have been used for the nonlinear Schrödinger equation in [22]. Avoiding the use of KAM theorem and Nash-Moser iteration method, though a para-differential conjugation, an equivalent form of the investigated nonlinear wave equations can be obtained, while the frequencies are fixed in a Cantor-like set whose complement has small measure. Applying the non-resonant conditions on each finite-dimensional subspaces, solutions can be constructed to the block diagonal equation on the finite subspace by a classical iteration scheme.

Article information

Taiwanese J. Math., Volume 21, Number 5 (2017), 1057-1097.

Received: 31 October 2016
Revised: 14 December 2016
Accepted: 15 December 2016
First available in Project Euclid: 1 August 2017

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Zentralblatt MATH identifier

Primary: 35L05: Wave equation 35S50: Paradifferential operators

periodic solutions para-differential conjugation iteration scheme


Chen, Bochao; Gao, Yixian; Li, Yong. Construction of Periodic Solutions for Nonlinear Wave Equations by a Para-differential Method. Taiwanese J. Math. 21 (2017), no. 5, 1057--1097. doi:10.11650/tjm/7914. https://projecteuclid.org/euclid.twjm/1501599183

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