Newton's method and periodic solutions of nonlinear wave equations

Abstract

We prove the existence of periodic solutions of the nonlinear wave equationsatisfying either Dirichlet or periodic boundary conditions on the interval 10, rr]. The coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation. Using a version of Newton's method, we show that this equation has solutions provided the nonlinearity g(x, u ) satisfies certain generic conditions of nonresonance and genuine nonlinearity.

KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation

“…With further conditions on the nonlinearity-like reversibility or in the Hamiltonian case-the eigenvalues are purely imaginary, and the torus is linearly stable. The present situation is very different with respect to [18], [13]- [16], [8]- [9] and also [27]- [29], [2], where the lack of stability informations is due to the fact that the linearized equation has variable coefficients, and it is not reduced as in Theorem 1.4 below.…”

Quasi-periodic solutions for a nonlinear wave equation

“…where DU = j ω j U θ j , such that the straight windings θ(t) = ωt + θ 0 on the torus map into solutions of (3). Hence, in phase space they correspond to embedded invariant tori, on which in suitable coordinates the vector field is constant with linear flow.…”

“…But the condition given here is certainly not the sharpest. Also, we did not investigate the exceptional set for one point sets J given by (5) thoroughly because for the existence of Cantor discs of periodic solutions there are better results anyhow [3]. But it may well be that there are no exceptional points at all.…”

“…For the construction of periodic solutions, however, other approaches are more suitable. For example, in a pioneering paper, Craig and Wayne [3] extended the Lyapunov center theorem to the infinite dimensional hamiltonian system of the nonlinear wave equation. This, too, involves small divisor problems, but to a lesser extent requiring fewer nondegeneracy conditions.…”

Cantor families of periodic solutions of wave equations with C k nonlinearities

“…where Remark 4.1 One major difference with respect to the analytic case is making the sequence of finite dimensional truncations N n increase super-exponentially fast like in (4.3) (in [10]- [4] we had N n = N 0 2 n ). This is useful to prove the smallness of the remainder r n defined in (4.14), see remark 4.3.…”