The cubic Szegő equation on the real line: explicit formula and well-posedness on the Hardy class (Q6568762)

The authors propose an explicit formula for the solution of the cubic Szegő equation on the real line based on the Lax pair structure of the equation. The Lax pair involves Hankel operators on the Hardy class \(H^2(\mathbb{R})\). The proof of the formula is essentially a calculation based on the Lax pair formulation of the cubic Szegő equation, which relies on the commutation properties of Hankel and Toeplitz operators with the infinitesimal generator of the Lax-Beurling semigroup in \(H^2(\mathbb{R})\). Since the notion of Hankel operators can make sense for arbitrary symbols in \(H^2(\mathbb{R})\), the authors are next able to extend the flow of solution to the whole Hardy space \(H^2(\mathbb{R})\). To this end, they make use of spectral theory of dissipative operators and semigroup theory.