We introduce a deformation of the method of characteristics valid for Hamiltonian perturbations of a scalar conservation law in the small dispersion limit. Our method of analysis is based on the 'variational string equation', a functional-differential relation originally introduced by Dubrovin in a particular case, of which we lay the mathematical foundation.
Starting from first principles, we construct the string equation explicitly up to the fourth order in perturbation theory, and we show that the solution to the Cauchy problem of the Hamiltonian PDE satisfies the appropriate string equation in the small dispersion limit. We apply our construction to explicitly compute the first two perturbative corrections of the solution to the general Hamiltonian PDE. In the KdV case, we prove the existence of a quasi-triviality transformation at any order and for arbitrary initial data.
Carleman estimates for the regularization of ill-posed Cauchy problems
这是一个调查，这是作者以前关于Carleman估计在逆问题中的应用的调查的延续，J. Inverse and Ill-Posed Problems，21，477-560，2013。本文证明了某些线性偏微分方程不适定Cauchy问题的Tikhonov泛函可以由这些偏微分方程的无界线性算子生成。这些是那些运营商的Carleman估计是有效的，例如椭圆，抛物和双曲运营商的二阶。最小化的收敛速度建立使用Carleman估计。对非线性反问题，如障碍物重构问题和系数反问题的推广也进行了讨论。
Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations
具有$ε^2$阶小色散的Korteweg de Vries（KdV）方程的Cauchy问题的特征在于出现了$ε$阶波长的快速调制振荡区。这些振荡近似地由KdV的椭圆解描述，其中振幅、波数和频率不是常数，而是根据Whitham方程演化。本文定量分析了KdV方程在小色散极限下的数值解与$ε$值介于$10^{-1}$和$10^{-3}$之间时的近似解之间的差异。数值结果是兼容的差异的顺序$ε$内的“内部”的Whitham振荡区，顺序$ε^{1/3}$在左边界外的Whitham区和顺序$ε^{1/2}$在右边界外的Whitham区。
