On Linear Degeneracy of Integrable Quasilinear Systems in Higher Dimensions

We investigate (d + 1)-dimensional quasilinear systems which are integrable by the method of hydrodynamic reductions. In the case d ?? 3 we formulate a conjecture that any such system with an irreducible dispersion relation must be linearly degenerate. We prove this conjecture in the 2-component case, providing a complete classification of multi- dimensional integrable systems in question. In particular, our results imply the non- existence of 2-component integrable systems of hydrodynamic type for d ?? 6. In the second half of the paper we discuss a numerical and analytical evidence for the impossibility of the breakdown of smooth initial data for linearly degenerate systems in 2 + 1 dimensions.