Abstract
The Frobenius KDV equation and the Frobenius KP equation are introduced, and the Frobenius Kompaneets equation, Frobenius Burgers equation and Frobenius Harry Dym equation are constructed by taking values in a commutative subalgebra Z2ε in the paper. The five equations are selected as examples to help us study the self-adjointness of Frobenius type equations, and we show that the first two equations are quasi self-adjoint and the last three equations are nonlinear self-adjointness. It follows that we give the symmetries of the Frobenius KDV and the Frobenius KP equation in order to construct the corresponding conservation laws.
Highlights
In the field of mathematical and physical science, nonlinear partial differential equations (NLPDE) have wide applications, including nonlinear optics [1], fluid flows [2], plasma physics [3], excitable media [4], and so on [5,6]
In order to study the quasi self-adjointness of the Frobenius type equation, we introduce the Frobenius KDV equation and Frobenius KP equation
The Frobenius KDV equation and Frobenius KP equation are introduced as two examples in order to study the quasi self-adjointness of the Frobenius type equation
Summary
In the field of mathematical and physical science, nonlinear partial differential equations (NLPDE) have wide applications, including nonlinear optics [1], fluid flows [2], plasma physics [3], excitable media [4], and so on [5,6]. Frobenius KDV equation, Frobenius Camassa-Holm equation, Frobenius Hunter-Saxton equation, and so on were obtained based on the Frobenius-Virasoro algebra in [20] It follows that Li considered the Hirota quadratic equation of the commutative version of extended multi-component Toda hierarchy in [21], and the gauge transformation and symmetries of the Zn -BKP hierarchy were investigated in [22]. Dym equation are constructed for investigating the nonlinear self-adjointness of the Frobenius type equation. We choose these five Frobenius equations because the corresponding single-component equations, incuding the KP equation, the KDV equation, the Kompaneets equation, the Burgers equation and the Harry Dym equation are very representative in mathematical physics and related fields. After that the conservation laws of Frobenius type equations are found by means of symmetries
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