关于第二天堂方程和巴甫洛夫方程的解
On the solutions of the second heavenly and Pavlov equations
摘要：
最近我们解决了单参数向量场族的逆散射问题，并利用这个结果构造了一类与多维向量场对易有关的可积非线性偏微分方程（如Plebanski的天堂方程）的Cauchy问题的形式解，无色散的Kadomtsev-Petviashvili（dKP）方程和二维无色散的Toda（2ddT）方程，以及一维矢量场的对易，如Pavlov方程。我们还表明，相关的Riemann-Hilbert反问题是强大的工具，以建立柯西问题的解决方案，如果在有限时间内打破，以构建其长期的行为和特征类的隐式解决方案。在本文中，我们使用上述理论，集中在天堂和巴甫洛夫方程，i）建立他们的本地化的解决方案发展没有打破，不同的情况下，dKP和2ddT; ii）构造长期行为的解决方案的柯西问题; iii）特征的一个特殊类的隐式解决方案的天堂方程。
英文摘要：
We have recently solved the inverse scattering problem for one parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations connected with the commutation of multidimensional vector fields, like the heavenly equation of Plebanski, the dispersionless Kadomtsev - Petviashvili (dKP) equation and the two-dimensional dispersionless Toda (2ddT) equation, as well as with the commutation of one dimensional vector fields, like the Pavlov equation. We also showed that the associated Riemann-Hilbert inverse problems are powerfull tools to establish if the solutions of the Cauchy problem break at finite time,to construct their longtime behaviour and characterize classes of implicit solutions. In this paper, using the above theory, we concentrate on the heavenly and Pavlov equations, i) establishing that their localized solutions evolve without breaking, unlike the cases of dKP and 2ddT; ii) constructing the longtime behaviour of the solutions of their Cauchy problems; iii) characterizing a distinguished class of implicit solutions of the heavenly equation.
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