Two sequences of fully-nonlinear evolution equations and their symmetry properties
Marianna Euler, Norbert Euler
TL;DR
This work analyzes two sequences of fully-nonlinear, odd-order evolution equations to map their symmetry structures and symmetry-integrability. The authors compute explicit Lie point symmetry algebras: Case 1 (with $n_k=k/(k+1)$) yields a $(2k+6)$-dimensional algebra, while Case 2 (with $n_k\neq k/(k+1)$) yields a $(2k+5)$-dimensional algebra, the difference arising from an extra symmetry generator $Z_5$. They introduce Conjecture 1, a set of order-dependent necessary conditions for Lie-Bäcklund symmetry–based integrability, and derive Corollaries shaping the integrable landscape: Case 1 contains at least two symmetry-integrable members (including the known $u_t=u_{3x}^{-1/2}$ and $u_t=u_{5x}^{-2/3}$), whereas Case 2 contains only one, $u_t=u_{3x}^{-2}$. A general form for fully-nonlinear symmetry-integrable 1+1D equations of order $n\ge 5$ is given as $u_t=f_1\big(u_{(2k+3)x}+f_2\big)^{- rac{k+1}{k+2}}+f_3$, with derivatives of $f_j$ allowed up to $u_{(2k+2)x}$. These results constrain possible integrable fully-nonlinear PDEs and guide future classification efforts.
Abstract
We obtain the complete Lie point symmetry algebras of two sequences of odd-order evolution equations. This includes equations that are fully-nonlinear, i.e. nonlinear in the highest derivative. Two of the equations in the sequences have recently been identified as symmetry-integrable, namely a 3rd-order equation and a 5th-order equation [Open Communications in Nonlinear Mathematical Physics, Special Issue in honour of George W Bluman, ocnmp:15938, 1--15, 2025]. These two examples provided the motivation for the current study. The Lie-Bäcklund symmetries and the consequent symmetry-integrability of the equations in the sequences are also discussed.