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Self-adjointness, group classification and conservation laws of an extended Camassa-Holm equation. (English) Zbl 1371.35253

Summary: In this paper, we prove that equation \(E \equiv u_1-u_{_x2_t}+u_xf(u)-au_xu_{}x^2-buu_{x^3}=0\) is self-adjoint and quasi self-adjoint, then we construct conservation laws for this equation using its symmetries. We investigate a symmetry classification of this nonlinear third order partial differential equation, where \(f\) is smooth function on \(u\) and \(a\), \(b\) are arbitrary constants. We find three special cases of this equation, using the Lie group method.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A30 Geometric theory, characteristics, transformations in context of PDEs
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
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Full Text: Euclid