Double waves in multi-dimensional systems of hydrodynamic type: a necessary condition for integrability

2004/12/24 by E. V. Ferapontov, Ferapontov, E. V., К. Р. Хуснутдинова +2
Earth and Planetary Sciences · Physics and Astronomy · #Aquatic and Environmental Studies #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems #nlin.SI

paper · pdf · doi:10.48550/arxiv.nlin/0412064

27 pages, Latex, to appear in Proc. Royal Soc. A. The formulation of the main result (Theorem 2) is refined and a full proof is given

openalex publication_date 2004/12/24 · arxiv created 2005/11/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

An invariant differential-geometric approach to the integrability of (2+1)-dimensional systems of hydrodynamic type u_t+A(u)u_x+B(u)u_y=0 is developed. It is proved that the existence of special solutions known as `double waves' is equivalent to the diagonalizability of an arbitrary matrix of the two-parameter family (kE+A)^-1(lE+B). Since the diagonalizability can be effectively verified by differential-geometric means, this provides a simple necessary condition for integrability.

Citations

• Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory

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