On linear degeneracy of integrable quasilinear systems in higher dimensions

Details On linear degeneracy of integrable quasilinear systems in higher dimensions On linear degeneracy of integrable quasilinear systems in higher dimensions

2009-09-30

arxiv.org/abs/0909.5685v2

Nonlinear Sciences

Exactly Solvable and Integrable Systems

36 pages, 19 figures. Added solutions via a Jacobi inversion problem on hyperellitic surfaces, humplike solutions and reductio

Scientific paper

We investigate $(d+1)$-dimensional quasilinear systems which are integrable by the method of hydrodynamic reductions. In the case $d\geq 3$ 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 $d\geq 6$. 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.

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