Nonlocality and the inverse scattering transform for the Pavlov equation

Abstract

As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form of integrable dispersionless multidimensional PDEs is non-local, 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 $v_{xt}+v_{yy}+v_x{v}_{xy}-v_y{v}_{xx}=0$, in this paper we establish the following. 1. The non-local term ${\partial }_x^{-1}$ arising from its evolutionary form $v_t=v_x{v}_y-{\partial }_x^{-1}{\partial }_y[v_y+v_x^2]$ corresponds to the asymmetric integral $-{\int }_x^{\infty }d{x}^{\prime }$. 2. Smooth and well-localized initial data $v(x,y,0)$ evolve in time developing, for $t>0$, the constraint ${\partial }_y\mathcal{M}(y,t)\equiv 0$, where $\mathcal{M}(y,t)={\int }_{-\infty }^{+\infty }\left[v_y(x,y,t)+(v_x(x,y,t){)}^2\right]dx$. 3. Since no smooth and well-localized initial data can satisfy such constraint at $t=0$, the initial ($t=0+$) dynamics of the Pavlov equation can not be smooth, although, as 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 non-locality of other basic examples of integrable dispersionless PDEs in multidimensions.

arXiv comment19 pages

References in corpus (10)

• On integrability of (2+1)-dimensional quasilinear systems
• Integrable hydrodynamic chains
• A class of Einstein--Weyl spaces associated to an integrable system of hydrodynamic type
• The Cauchy Problem on the Plane for the Dispersionless Kadomtsev - Petviashvili Equation
• Inverse Scattering Problem for Vector Fields and the Cauchy Problem for the Heavenly Equation
• On the solutions of the dKP equation: nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking
• A hierarchy of integrable PDEs in 2+1 dimensions associated with 2 - dimensional vector fields
• Integrable dispersionless PDEs arising as commutation condition of pairs of vector fields
• Inverse Scattering Problem for Vector Fields and the Heavenly Equation
• The Cauchy problem for the Pavlov equation

Cited by in corpus (1)

• An integral geometry lemma and its applications: the nonlocality of the Pavlov equation and a tomographic problem with opaque parabolic objects