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We introduce a deformation of the method of characteristics valid forHamiltonian perturbations of a scalar conservation law in the small dispersionlimit. Our method of analysis is based on the 'variational string equation', afunctional-differential relation originally introduced by Dubrovin in aparticular case, of which we lay the mathematical foundation.  Starting from first principles, we construct the string equation explicitlyup to the fourth order in perturbation theory, and we show that the solution tothe Cauchy problem of the Hamiltonian PDE satisfies the appropriate stringequation in the small dispersion limit. We apply our construction to explicitlycompute the first two perturbative corrections of the solution to the generalHamiltonian PDE. In the KdV case, we prove the existence of a quasi-trivialitytransformation at any order and for arbitrary initial data.

A deformation of the method of characteristics and the Cauchy problem for Hamiltonian PDEs in the small dispersion limit