Noether-Like Operators and First Integrals for Generalized Systems of Lane-Emden Equations

Abstract

Coupled systems of Lane–Emden equations are of considerable interest as they model several physical phenomena, for instance population evolution, pattern formation, and chemical reactions. Assuming a complex variational structure, we classify the generalized system of Lane–Emden type equations in relation to Noether-like operators and associated first integrals. Various forms of functions appearing in the considered system are taken, and it is observed that the Noether-like operators form an Abelian algebra for the corresponding Euler–Lagrange-type systems. Interestingly, we find that in many cases, the Noether-like operators satisfy the classical Noether symmetry condition and become the Noether symmetries. Moreover, we observe that the classical Noetherian integrals and the first integrals we determine using the complex Lagrangian approach turn out to be the same for the underlying system of Lane–Emden equations.

FMNEICS: fractional Meyer neuro-evolution-based intelligent computing solver for doubly singular multi-fractional order Lane–Emden system

“…This historical Lane-Emden model exists in astrophysics, quantum mechanics, and spherical cloud of gas, but considered stiff to solve due to the singular point at the origin. Many existing deterministic schemes have been applied to solve Lane-Emden model (Wazwaz 2001;He and Ji 2019;Singh et al 2019a,b;Asadpour et al 2019;Khalifa and Hassan 2019;Cȃruntu et al 2019) and its general form is written as follows (Farooq 2019;Hadian-Rasanan et al 2020;Sabir 2020):…”

First Integrals of Two-Dimensional Dynamical Systems via Complex Lagrangian Approach

“…Similarly, for the above stated forms of P(x), F and G along with few more, ample literature on the classification of the above system with respect to Lie symmetries, Noether symmetries and production of associated first integrals is available. In this regard, the interested reader is referred to [18][19][20][21] (and references in these papers for more details). Seeking the symmetry properties and conservation laws of LE-system (2) with respect to different cases which we present here is novel.…”

“…Seeking the symmetry properties and conservation laws of LE-system (2) with respect to different cases which we present here is novel. It is essential as it generates further interest in researchers who can apply it to a more specific scenario as, e.g., in [18] for P(x) = 1. Moreover, unless this more general approach is presented in the literature, the application of (2) in specific cases can likely be overlooked due to its intrinsic difficult nature.…”