On the Integrability of theBenjamin‐OnoEquation on the Torus

Abstract

We prove that for any t ∈ R, the flow map S t of the Benjamin-Ono equation on the torus continuously extends to the Sobolev space H −s r,0 for any 0 < s < 1/2, but does not do so to H −s r,0for s > 1/2. Furthermore, we show that S t is sequentially weakly continuous on H −s r,0 for any 0 ≤ s < 1/2. Note that −1/2 is the critical Sobolev exponent of the Benjamin-Ono equation.

Exact Bohr-Sommerfeld Conditions for the Quantum Periodic Benjamin-Ono Equation

“…In § [9] we recall finite gap conditions for multi-phase solutions from [39]. In § [10] we derive formula (1.7) for the classical actions from results of Gérard-Kappeler [21], establish the Bohr-Sommerfeld conditions (1.8), and prove Theorem [1.3.1].…”

“…In § [4] we recall the classical Nazarov-Sklyanin hierarchy [40] and its presentation in terms of dispersive action profiles from [39]. In § [5] we identify the classical global action variables of Gérard-Kappeler [21] with the gaps in the dispersive action profiles from [39]. In § [6] we present the Hamiltonian and Lax operator of a preliminary geometric quantization of (1.1).…”

“…Classical Hamiltonian at Criticality. The Benjamin-Ono equation (1.1) is Hamiltonian: The classical Hamilton equations for (3.6) in Definition [3.2.1] are formal: M (a) is larger than the space L 2 (T) in which (1.1) is known to be well-posed [21,36]. From the perspective of dispersive equations, it is a coincidence that the symplectic space M (a) of (1.1) corresponds to its critical regularity s c = −1/2 as in Proposition [3.1.2].…”

“…of dispersive action profiles f (c|v; ε) from [39] for bounded v. We also state the characterization in Gérard-Kappeler [21] of their global action variables as integrals of the…”

“…. As a consequence, the eigenvalues C ↓ h (v; ε) of the embedded principal minor L + (v; ε) of the classical Lax operator L • (v; ε) can be calculated from those of L • (v; ε) by the shift relation 3), the definition of bands and gaps in [21] are…”