Recent progress on nonlinear wave equations via KAM theory∗†

In this note, the present author’s recent works on nonlinear wave equations via KAM theory are introduced and reviewed. The existence of solutions, periodic in time, for non-linear wave (NLW) equations has been studied by many authors. A wide variety of methods such as bifurcation theory and variational techniques have been brought on this problem. See [11] and the references therein, for example. There are, however, relatively less methods to find the quasi-periodic solutions of NLW or other PDE’s. The KAM theory is a very powerful tool in order to construct families of quasi-periodic solutions, which are on an invariant manifold, for some nearly integrable Hamiltonian systems of finite many degrees of freedom. In the 1980’s,the celebrated KAM theory has been successfully extended to infinitely dimensional Hamiltonian systems of short range so as to deal with certain class of Hamiltonian networks of weakly coupled oscillators. Vittot & Bellissard [27], Frohlich, Spencer & Wayne [15] showed that there are plenty of almost periodic solutions for some weakly coupled oscillators of short range. In [30], it was also shown that there are plenty of quasi-periodic solutions for some weakly coupled oscillators of short range. Because of the restrict of short range, those results obtained in [27, 15] does not apply to PDE’s. In the 1980-90’s, the KAM theory has been significantly generalized, by Kuksin[17, 18, 19], to infinitely dimensional Hamiltonian systems without being of short range so as to show that there is quasi-periodic solution for some class of partial differential equations. Also see Pöschel[24]. Let us focus our attention to the following nonlinear wave equation utt − uxx + V (x)u + u + h.o.t. = 0, (1) subject to Dirichlet and periodic boundary conditions on the space variable x. 1. Dirichlet boundary condition. In 1990, Wayne[28] obtained the time-quasiperiodic solutions of (1), when the potential V is lying on the outside of the set of some “bad” potentials. In [28], the set of all potentials is given some Gaussian measure and then the set of “bad” potentials is of small measure. Kuksin[17] assumed the potential V depends on n-parameters, namely, V = V (x; a1, ..., an), and showed that there are many quasi-periodic solutions of (1) for “most” (in the sense of Lebesgue measure) parameters a’s. However, their results exclude the constant-value potential V (x) ≡ m ∈ R, in particular, V (x) ≡ 0. When the potential V is constant, the parameters required can be extracted from the ∗In memory of Prof Xunjing LI †Supported by NNSFC ‡School of mathematics, Fudan University, Shanghai 200433, China, Email: [email protected]