Fourth-Order Pattern Forming PDEs: Partial and Approximate Symmetries

This paper considers pattern forming nonlinear models arising in the study of thermal convection and continuous media. A primary method for the derivation of symmetries and conservation laws is Noether's theorem. However, in the absence of a Lagrangian for the equations investigated, we propose the use of partial Lagrangians within the framework of calculating conservation laws. Additionally, a nonlinear Kuramoto-Sivashinsky equation is recast into an equation possessing a perturbation term. To achieve this, the knowledge of approximate transformations on the admissible coefficient parameters is required. A perturbation parameter is suitably chosen to allow for the construction of nontrivial approximate symmetries. It is demonstrated that this selection provides approximate solutions.