Nonlocality and the inverse scattering transform for the Pavlov equation

TL;DR

This paper applies the inverse scattering transform to analyze the non-local structure of the Pavlov equation, revealing how initial data evolve and develop constraints, and highlighting the non-smooth nature of solutions at initial times.

Contribution

It establishes the proper non-local integral form for the Pavlov equation and describes the evolution constraints, advancing understanding of non-locality in integrable dispersionless PDEs.

Findings

The non-local term corresponds to an asymmetric integral from x to infinity.

Initial data evolve to satisfy a specific y-constraint for t>0.

Smooth initial data cannot satisfy the constraint at t=0, leading to non-smooth initial dynamics.

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 $v_{xt}+v_{yy}+v_xv_{xy}-v_yv_{xx}=0$, in this paper we establish the following. 1. The non-local term $\partial_x^{-1}$ arising from its evolutionary form $v_{t}= v_{x}v_{y}-\partial^{-1}_{x}\,\partial_{y}\,[v_{y}+v^2_{x}]$ corresponds to the asymmetric integral $-\int_x^{\infty}dx'$. 2. Smooth and well-localized initial data $v(x,y,0)$ evolve in time developing, for $t>0$, the constraint $\partial_y {\cal M}(y,t)\equiv 0$, where ${\cal M}(y,t)=\int_{-\infty}^{+\infty} \left[v_{y}(x,y,t) +(v_{x}(x,y,t))^2\right]\,dx$. 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.