Nonlocality and the Inverse Scattering Transform for the Pavlov Equation

As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form of integrable dispersionless multidimensional PDEs is nonlocal, and the proper choice of integration constants should be the one dictated by the associated inverse scattering transform (IST). Using the recently made rigorous IST for vector fields associated with the so-called Pavlov equation v xt + v yy + v x v x y − v y v x x = 0, in this paper we establish the following. 1. The nonlocal term ∂ x −1 arising from its evolutionary ́ form v t =∞ v x v y − ∂ x −1 ∂ y [v y + v x 2 ] corresponds to the asymmetric integral − x d x . 2. Smooth and well-localized initial data v(x, y, 0) evolve in ́ time developing, +∞ for t > 0, the constraint ∂ y M(y, t) ≡ 0, where M(y, t) = −∞ [v y (x, y, t) + (v x (x, y, t)) 2 ] d x. 3. Because no smooth and well-localized initial data can satisfy such constraint at t = 0, the initial (t = 0+) dynamics of the Pavlov equation cannot be smooth, although, because it was already established, small norm solutions remain regular for all positive times. We expect that the techniques developed in this paper to prove the above results should be successfully used in the study of the nonlocality of other basic examples of integrable dispersionless PDEs in multidimensions.