Abstract
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.
Highlights
The family of nonlinear fourth-order partial differential equations (PDEs) ut + λuux + αuxx + βuxxx + γuxxxx + ζ(u) = 0, α, β, γ, λ − constants, describes several fundamental phenomena in physical processes
Over a wide range of parameter values, this family of equations possesses solutions that take the form of patterns such as pulses [15] and spatio-temporal pattern formation in extended systems
Our interest lies in a subclass of these equations called the Swift-Hohenberg (SH) [28] and Kuramoto-Sivashinsky (KS) equations [17,27] that exhibit interesting pattern formations
Summary
Our interest lies in a subclass of these equations called the Swift-Hohenberg (SH) [28] and Kuramoto-Sivashinsky (KS) equations [17,27] that exhibit interesting pattern formations The former was introduced as a simple model for the Rayleigh-Benard instability of roll waves, but has since appeared in connection with Taylor-Couette flow [25] and in the study of lasers [18]. The latter PDE, the KS equation is one of the simplest physically interesting nonlinear systems and is frequently encountered in the study of continuous media which exhibits a chaotic behavior In the literature, it has offered insight on the turbulence in full-fledged Navier-Stokes boundary shear flows [7], descriptions of stability of flame fronts, reaction diffusion systems, long waves on the interface between two viscous fluids and unstable drift waves in plasmas.
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