Workshop: Global rational solutions of the one dimensional Half-Wave Maps equation, Patrick Gérard

Date: 2023-09-12

Time: 09:00 - 09:50

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Patrick Gérard, University of Paris-Saclay


The one dimensional Half-Wave Maps equation is the following quasi-linear hyperbolic system, \(\partial_t S=S \times |D_x| S \) where \(x\) belongs to the real line and \(S=S(t,x)\) is valued into the two dimensional sphere. This equation is known to be locally wellposed for Sobolev perturbations of constant states, and to admit a Lax pair, but no conservation law controlling a Sobolev regularity \(H^s\) for \(s\) \(>\) \(1/2\) is known. If the initial datum is a rational perturbation of a constant state, the smooth solution is known to remain a rational perturbation of this state as long as it exists. We prove that such a solution is always global in time. The main ingredient is an explicit formula for the smooth solution, involving Toeplitz operators on the Hardy space, and which is inherited from a spin version of the Benjamin-Ono equation.

This is a joint work with Enno Lenzmann.