Quasi-Periodic Solutions for the Reversible Derivative Nonlinear Schrödinger Equations with Periodic Boundary Conditions

KAM theorem for degenerate infinite-dimensional reversible systems

“…On the other hand, KAM theory for infinite-dimensional Hamiltonian systems has also been extended to infinite-dimensional reversible systems [2,4,7,9,12,11,13,14,24]. However, to the best of our knowledge, these results are all based on classical non-degeneracy conditions, and there is still no works on KAM theory for degenerate infinite-dimensional reversible systems.…”

“…Proof. See[11] for the verification of the reversibility ofΦ * W . Since Φ * W = (DΦ) −1 W • Φ, we have Φ * W * ηr,U ≤ (DΦ) −1 * ηr,ηr,U W • Φ * ηr,U , By (3.8) and η 2 = E, 1 σ Φ − id * ηr,U , DΦ − Id * ηr,ηr,U ≤ 1, then by Lemma 6.10 it follows that W • Φ * ηr,U ≤ W * ηr,v .…”

Invariant tori of full dimension for higher-dimensional beam equations with almost-periodic forcing

“…For one-dimensional Hamiltonian systems, the existence of quasiperiodic solutions or almost-periodic solutions is also very significant in physics. It is well known that the infinite-dimensional KAM theory is powerful to obtain it (see [12][13][14][15][16][17][18][19][20][21]). However, the standard KAM theory fails to study higher-dimensional Hamiltonian PDEs because of the multiplicity of the eigenvalues.…”