非定域性与Pavlov方程的逆散射变换
Nonlocality and the inverse scattering transform for the Pavlov equation
摘要：
与2+1维孤子偏微分方程一样，可积无色散多维偏微分方程的演化形式是非局部的，积分常数的正确选择应该由相关的逆散射变换（IST）决定。利用最近提出的与所谓的巴甫洛夫方程$v_{xt}+v_{yy}+v_xv_{xy}-v_yv_{xx}=0$相关联的向量场的严格IST，本文建立了以下结果。1.从其演化形式$v_{t}= v_{x}v_{y}-\partial^{-1}_{x}\,\partial_{y}\,[v_{y}+v^2_{x}]$产生的非局部项$\partial_x^{-1}$对应于非对称积分$-\int_x^{\infty}dx'$。2.平滑且局部化良好的初始数据$v(x,y,0)$随时间发展而演化，对于$t>0$，约束$\partial_y {\cal M}(y,t)\equiv 0$，其中${\cal M}(y,t)=\int_{-\infty}^{+\infty} \left[v_{y}(x,y,t) +(v_{x}(x,y,t))^2\right]\,dx$。3.由于没有光滑的和良好的本地化的初始数据可以满足这样的约束在$t=0$，初始（$t=0+$）的巴甫洛夫方程的动力学不可能是光滑的，虽然，因为它已经建立，小范数的解决方案保持正则的所有积极的时间。我们期望本文所发展的证明上述结果的方法，能成功地应用于其他基本的多维可积无色散偏微分方程的非定域性研究.
英文摘要：
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.
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