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Symmetry analysis and conservation laws for the Hunter-Saxton equation. (English) Zbl 1264.35100
Summary: In this paper, the problem of determining the most general Lie point symmetries group and conservation laws of a well known nonlinear hyperbolic PDE in mathematical physics called the Hunter-Saxton equation (HSE) is analyzed. By applying the basic Lie symmetry method for the HSE, the classical Lie point symmetry operators are obtained. Also, the algebraic structure of the Lie algebra of symmetries is discussed and an optimal system of one-dimensional subalgebras of the HSE symmetry algebra which creates the preliminary classification of group invariant solutions is constructed. Particularly, the Lie invariants as well as similarity reduced equations corresponding to infinitesimal symmetries are obtained. Mainly, the conservation laws of the HSE are computed via three different methods including Boyer’s generalization of Noether’s theorem, first homotopy method and second homotopy method.

35G50Nonlinear higher-order systems of PDE
35B06Symmetries, invariants, etc. (PDE)
37K05Hamiltonian structures, symmetries, variational principles, conservation laws
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
37K15Integration of completely integrable systems by inverse spectral and scattering methods
65H20Global numerical methods for nonlinear algebraic equations, including homotopy approaches
82D30Random media, disordered materials (statistical mechanics)
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