Lower bounds on the growth of Sobolev norms in some linear time dependent Schr\uf6dinger equations

In this paper we consider linear, time dependent Schrodinger equations of the form i partial derivative(t)psi = K-0 psi + V(t)psi, where K-0 is a positive selfadjoint operator with discrete spectrum and whose spectral gaps are asymptotically constant. We give a strategy to construct bounded perturbations V(t) such that the Hamiltonian K-0 + V(t) generates unbounded orbits. We apply our abstract construction to three cases: (i) the Har- monic oscillator on N, (ii) the half-wave equation on and (iii) the Dirac-Schrodinger equation on Zoll manifolds. In each case, V(t) is a smooth and periodic in time pseudodifferential operator and the Schrodinger equation has solutions fulfilling the optimal lower bound estimate parallel to psi(t)parallel to r greater than or similar to vertical bar t vertical bar as vertical bar t vertical bar >> 1

Traveling quasi-periodic water waves with constant vorticity

We prove the first bifurcation result of time quasi-periodic traveling waves solutions for space periodic water waves with vorticity. In particular we prove existence of small amplitude time quasi-periodic solutions of the gravity-capillary water waves equations with constant vorticity, for a bidimensional fluid over a flat bottom delimited by a space-periodic free interface. These quasi-periodic solutions exist for all the values of depth, gravity and vorticity, and restricting the surface tension to a Borel set of asymptotically full Lebesgue measure

Stokes waves at the critical depth are modulational unstable

This paper fully answers a long standing open question concerning the stability/instability of pure gravity periodic traveling water waves -- called Stokes waves -- at the critical Whitham-Benjamin depth $\mathtt{h}_{\scriptscriptstyle WB} = 1.363...$ and nearby values. We prove that Stokes waves of small amplitude $\mathcal{O}( \epsilon )$ are, at the critical depth $\mathtt{h}_{\scriptscriptstyle WB}$, linearly unstable under long wave perturbations. This is also true for slightly smaller values of the depth $\mathtt{h} > \mathtt{h}_{\scriptscriptstyle WB} - c \epsilon^2$, $c > 0$, depending on the amplitude of the wave. This problem was not rigorously solved in previous literature because the expansions degenerate at the critical depth. In order to resolve this degenerate case, and describe in a mathematically exhaustive way how the eigenvalues change their stable-to-unstable nature along this shallow-to-deep water transient, we Taylor expand the computations of arXiv:2204.00809v2 at a higher degree of accuracy, derived by the fourth order expansion of the Stokes waves. We prove that also in this transient regime a pair of unstable eigenvalues depict a closed figure "8", of smaller size than for $\mathtt{h} > \mathtt{h}_{\scriptscriptstyle WB}$, as the Floquet exponent varies.Comment: 52 pages, 6 figures, companion Mathematica code available at https://git-scm.sissa.it/amaspero/benjamin-feir-instability. arXiv admin note: text overlap with arXiv:2204.0080

Benjamin-Feir instability of Stokes waves in finite depth

Whitham and Benjamin predicted in 1967 that small-amplitude periodic traveling Stokes waves of the 2d-gravity water waves equations are linearly unstable with respect to long-wave perturbations, if the depth $\mathtt h$ is larger than a critical threshold $\mathtt{h}_{{WB}} \approx 1.363$. In this paper we completely describe, for any value of $\mathtt h >0$, the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent is turned on. We prove in particular the existence of a unique $\mathtt{h}_{WB}$, which coincides with the one in [9,38,2], such that, for any $0 < \mathtt h < {\mathtt{h}_{{WB}}}$ the eigenvalues close to zero remain purely imaginary and, for any $\mathtt h > {\mathtt{h}_{{WB}}}$, a pair of non-purely imaginary eigenvalues depicts a closed figure "8", parameterized by the Floquet exponent. As ${\mathtt h} \to \mathtt{h}_{WB}^+$ this figure "8" collapses to the origin of the complex plane. Our proof is based on a combination of Kato's similarity transformation theory and a "KAM" inspired block-diagonalization procedure. We exploit in crucial way the Hamiltonian and reversible structure of the water waves equations.Comment: arXiv admin note: substantial text overlap with arXiv:2109.1185