7 research outputs found
Cheng Equation: A Revisit Through Symmetry Analysis
The symmetry analysis of the Cheng Equation is performed. The Cheng Equation
is reduced to a first-order equation of either Abel's Equations, the analytic
solution of which is given in terms of special functions. Moreover, for a
particular symmetry the system is reduced to the Riccati Equation or to the
linear nonhomogeneous equation of Euler type. Henceforth, the general solution
of the Cheng Equation with the use of the Lie theory is discussed, as also the
application of Lie symmetries in a generalized Cheng equation.Comment: 10 pages. Accepted for publication in Quaestiones Mathematicae
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Similarity solutions and Conservation laws for the Bogoyavlensky-Konopelchenko Equation by Lie point symmetries
The 1 + 2 dimensional Bogoyavlensky-Konopelchenko Equation is investigated
for its solution and conservation laws using the Lie point symmetry analysis.
In the recent past, certain work has been done describing the Lie point
symmetries for the equation and this work seems to be incomplete (Ray S (2017)
Compt. Math. Appl. 74, 1157). We obtained certain new symmetries and
corresponding conservation laws. The travelling-wave solution and some other
similarity solutions are studied.Comment: 12 pages. Accepted for publication in Quaestiones Mathematica
Noether’s Theorem and Symmetry
In Noether’s original presentation of her celebrated theorem of 1918, allowance was made for the dependence of the coefficient functions of the differential operator, which generated the infinitesimal transformation of the action integral upon the derivatives of the dependent variable(s), the so-called generalized, or dynamical, symmetries. A similar allowance is to be found in the variables of the boundary function, often termed a gauge function by those who have not read the original paper. This generality was lost after texts such as those of Courant and Hilbert or Lovelock and Rund confined attention to point transformations only. In recent decades, this diminution of the power of Noether’s theorem has been partly countered, in particular in the review of Sarlet and Cantrijn. In this Special Issue, we emphasize the generality of Noether’s theorem in its original form and explore the applicability of even more general coefficient functions by allowing for nonlocal terms. We also look for the application of these more general symmetries to problems in which parameters or parametric functions have a more general dependence on the independent variables