Nijenhuis geometry IV: conservation laws, symmetries and integration of certain non-diagonalisable systems of hydrodynamic type in quadratures
Bolsinov A.V. Konyaev A.Y. Matveev V.S.
1 October 2024Institute of Physics
Nonlinearity
2024#37Issue 10
The paper contains two lines of results: the first one is a study of symmetries and conservation laws of gl -regular Nijenhuis operators. We prove the splitting theorem for symmetries and conservation laws of Nijenhuis operators, show that the space of symmetries of a gl -regular Nijenhuis operator forms a commutative algebra with respect to (pointwise) matrix multiplication. Moreover, all the elements of this algebra are strong symmetries of each other. We establish a natural relationship between symmetries and conservation laws of a gl -regular Nijenhuis operator and systems of the first and second companion coordinates. Moreover, we show that the space of conservation laws is naturally related to the space of symmetries in the sense that any conservation laws can be obtained from a single conservation law by multiplication with an appropriate symmetry. In particular, we provide an explicit description of all symmetries and conservation laws for gl -regular operators at algebraically generic points. The second line of results contains an application of the theoretical part to a certain system of partial differential equations of hydrodynamic type, which was previously studied by different authors, but mainly in the diagonalisable case. We show that this system is integrable in quadratures, i.e. its solutions can be found for almost all initial curves by integrating closed 1-forms and solving some systems of functional equations. The system is not diagonalisable in general, and construction and integration of such systems is an actively studied and explicitly stated problem in the literature.
37K05 , 37K06 , 37K10 , 37K25 , 37K50 , 53A20 , 53B10 , 53B20 , 53B30 , 53B50 , 53B99 , 53D17 , conservation law , integrable systems , Nijenhuis operator , quasilinear system , symmetries
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School of Mathematics, Loughborough University, Loughborough, LE11 3TU, United Kingdom
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russian Federation
Moscow Center for Fundamental and Applied Mathematics, Moscow, 119992, Russian Federation
Institut für Mathematik, Friedrich Schiller Universität Jena, Jena, 07737, Germany
La Trobe University, Melbourne, Australia
School of Mathematics
Institute of Mathematics and Mathematical Modeling
Faculty of Mechanics and Mathematics
Moscow Center for Fundamental and Applied Mathematics
Institut für Mathematik
La Trobe University
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