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Keywords = Abel’s ODE of the second kind

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18 pages, 376 KB  
Article
Traveling-Wave Solutions of Several Nonlinear Mathematical Physics Equations
by Petar Popivanov and Angela Slavova
Mathematics 2025, 13(6), 901; https://doi.org/10.3390/math13060901 - 7 Mar 2025
Viewed by 851
Abstract
This paper deals with several nonlinear partial differential equations (PDEs) of mathematical physics such as the concatenation model (perturbed concatenation model) from nonlinear fiber optics, the plane hydrodynamic jet theory, the Kadomtsev–Petviashvili PDE from hydrodynamic (soliton theory) and others. For the equation of [...] Read more.
This paper deals with several nonlinear partial differential equations (PDEs) of mathematical physics such as the concatenation model (perturbed concatenation model) from nonlinear fiber optics, the plane hydrodynamic jet theory, the Kadomtsev–Petviashvili PDE from hydrodynamic (soliton theory) and others. For the equation of nonlinear optics, we look for solutions of the form amplitude Q multiplied by eiΦ, Φ being linear. Then, Q is expressed as a quadratic polynomial of some elliptic function. Such types of solutions exist if some nonlinear algebraic system possesses a nontrivial solution. In the other five cases, the solution is a traveling wave. It satisfies Abel-type ODE of the second kind, the first order ODE of the elliptic functions (the Weierstrass or Jacobi functions), the Airy equation, the Emden–Fawler equation, etc. At the end of the paper a short survey on the Jacobi elliptic and Weierstrass functions is included. Full article
(This article belongs to the Section C1: Difference and Differential Equations)
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13 pages, 448 KB  
Article
Solutions of Magnetohydrodynamics Equation through Symmetries
by Rangasamy Sinuvasan, Amlan K. Halder, Rajeswari Seshadri, Andronikos Paliathanasis and Peter G. L. Leach
Symmetry 2023, 15(10), 1908; https://doi.org/10.3390/sym15101908 - 12 Oct 2023
Viewed by 1792
Abstract
The magnetohydrodynamics (1 + 1) dimension equation, with a force and force-free term, is analysed with respect to its point symmetries. Interestingly, it reduces to an Abel’s Equation of the second kind and, under certain conditions, to equations specified in Gambier’s family. The [...] Read more.
The magnetohydrodynamics (1 + 1) dimension equation, with a force and force-free term, is analysed with respect to its point symmetries. Interestingly, it reduces to an Abel’s Equation of the second kind and, under certain conditions, to equations specified in Gambier’s family. The symmetry analysis for the force-free term leads to Euler’s Equation and to a system of reduced second-order odes for which singularity analysis is performed to determine their integrability. Full article
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