Ermakov–Pinney and Emden–Fowler Equations: New Solutions from Novel Bäcklund Transformations
We study the class of nonlinear ordinary differential equations y″ y = F(z, y2), where F is a smooth function. Various ordinary differential equations with a well-known importance for applications belong to this class of nonlinear ordinary differential equations. Indeed, the Emden–Fowler equation, the Ermakov–Pinney equation, and the generalized Ermakov equations are among them. We construct Bäcklund transformations and auto-Bäcklund transformations: starting from a trivial solution, these last transformations induce the construction of a ladder of new solutions admitted by the given differential equations. Notably, the highly nonlinear structure of this class of nonlinear ordinary differential equations implies that numerical methods are very difficult to apply.
Keywordsnonlinear ordinary differential equation Bäcklund transformation Schwarzian derivative Ermakov–Pinney equation Emden–Fowler equation
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- 16.S. Chandrasekhar, An Introduction to the Study of Stellar Structures, Dover, New York (1958).Google Scholar
- 20.F. Major, V. N. Gheorghe, and G. Werth, Charged Particle Traps: Physics and Techniques of Charged Particle Field Confinement (Springer Ser. Atomic Optical Plasma Phys., Vol. 37), Springer, Berlin (2005).Google Scholar
- 30.S. Carillo and F. Zullo, “The Gross–Pitaevskii equation: Bäcklund transformations and admitted solutions,” Ricerche Mat. (2018 to appear); arXiv:1803.09228v2 [math-ph] (2018).Google Scholar