Dmitry Treschev

We discuss algebraic and combinatorial aspects of the Hamiltonian normal form theory. The main objective is to describe the normal form near a singular point by an explicit formula in terms of the original Hamiltonian, avoiding the normalization procedure. In the case of one degree of freedom we compute the normal form as an explicit nonlinear functional, applied to the original Hamiltonian. In arbitrary dimension the corresponding formulas are more complicated but still explicit.

We consider the problem of spectral stability of traveling wave solutions $u=\gamma(x-Wt)$ for a system of viscous conservation laws $\partial_t u + \partial_x F(u) = \partial^2_x u$. Such solutions correspond to heteroclinic trajectories $\gamma$ of a system of ODEs. In general conditions of stability can be obtained only numerically. We propose a model class of piecewise linear (discontinuous) vector fields $F$ for which the stability problem is reduced to a linear algebra problem. We show that the stability problem makes sense in such low regularity and construct several examples of stability loss. Every such example can be smoothed to provide a smooth example of the same phenomenon.