Vodová-Jahnová, Jiřina On symmetries and conservation laws of the Majda-Biello system. (English) Zbl 1326.35016 Nonlinear Anal., Real World Appl. 22, 148-154 (2015). Summary: In [Stud. Appl. Math. 112, No. 4, 341–390 (2004; Zbl 1141.86304)] J. A. Biello and A. J. Majda derived and studied the so-called reduced equations for equatorial baroclinic-barotropic waves, to which we refer as to the Majda-Biello system. The equations in question describe the nonlinear interaction of long-wavelength equatorial Rossby waves and barotropic Rossby waves with a significant midlatitude projection in the presence of suitable horizontally and vertically sheared zonal mean flows.Below we present a Hamiltonian structure for Majda-Biello system and describe all generalized symmetries and conservation laws for the latter. It turns out that there are only three symmetries corresponding to \(x\)-translations, \(t\)-translations and to a scaling of \(t\), \(x\), \(u\) and \(v\), and four conservation laws, one of which is associated with the conservation of energy, the second conserved quantity is just the Hamiltonian functional and the other two are Casimir functionals of the Hamiltonian operator admitted by our system. Our result provides inter alia a rigorous proof of the fact that the Majda-Biello system has just the conservation laws mentioned in the paper by Majda and Biello. Cited in 3 Documents MSC: 35B06 Symmetries, invariants, etc. in context of PDEs 35Q86 PDEs in connection with geophysics 37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) Keywords:Hamiltonian operators; equatorial baroclinic-barotropic waves; long-wavelength equatorial Rossby waves; barotropic Rossby waves; Casimir functionals Citations:Zbl 1141.86304 PDFBibTeX XMLCite \textit{J. 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