A deformation of the method of characteristics and the cauchy problem for hamiltonian PDEs in the small dispersion limit

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 partial differential equation (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 Korteweg-de Vries (KdV) case, we prove the existence of a quasitriviality transformation at any order and for arbitrary initial data.