Solutions of quasi-geostrophic turbulence in multi-layered configurations

We consider quasi-geostrophic (Q-G) models in two- and three-layers that are useful in theoretical studies of planetary atmospheres and oceans. In these models, the streamfunctions are given by (1+2) partial differen- tial systems of evolution equations. A two-layer Q-G model, in a simpli- fied version, is dependent exclusively on the Rossby radius of deformation. However, the f-plane Q-G point vortex model contains factors such as the density, thickness of each layer, the Coriolis parameter, and the constant of gravitational acceleration, and this two-layered model admits a lesser number of Lie point symmetries, as compared to the simplified model. Finally, we study a three-layer oceanography Q-G model of special inter- est, which includes asymmetric wind curl forcing or Ekman pumping, that drives double-gyre ocean circulation. In three-layers, we obtain solutions pertaining to the wind-driven double-gyre ocean flow for a range of physi- cally relevant features, such as lateral friction and the analogue parameters of the f-plane Q-G model. Zero-order invariants are used to reduce the partial differential systems to ordinary differential systems. We determine conservation laws for these Q-G systems via multiplier methods.Comment: 14 pages, 6 figures, 1 tabl

An analysis of the invariance and conservation laws of some classes of nonlinear wave equations

We analyse nonlinear partial di erential equations arising from the modelling of wave phenomena. A large class of wave equations with dissipation and source terms are studied using a symmetry approach and the construction of conservation laws. Some previously unknown conservation laws and symmetries are obtained. We then proceed to use the multiplier (and homotopy) approach to construct conservation laws from which we obtain some surprisingly interesting higher-order variational symmetries. We also nd the corresponding conserved quantities for a large class of Gordon-type equations similar to those of the sine-Gordon equation and the relativistic Klein-Gordon equation. In particular, we direct our research and analysis towards a wave equation with non-constant coe cient terms, that is, coe cients dependent on time and space. Finally, we study a class of multi-dimensional wave equations

Nonlocal Representation of the sl(2,R)sl(2,R) Algebra for the Chazy equation

A demonstration of how the point symmetries of the Chazy Equation become nonlocal symmetries for the reduced equation is discussed. Moreover we construct an equivalent third-order differential equation which is related to the Chazy Equation under a generalized transformation, and find the point symmetries of the Chazy Equation are generalized symmetries for the new equation. With the use of singularity analysis and a simple coordinate transformation we construct a solution for the Chazy Equation which is given by a Right Painlev\'e Series. The singularity analysis is applied to the new third-order equation and we find that it admits two solutions, one given by a Left Painlev\'e Series and one given by a Right Painlev\'e Series where the leading-order behaviors and the resonances are explicitly those of the Chazy Equation.Comment: 6 pages, to appear in Quaestiones Mathematica

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

Approximate Lie Symmetry Conditions of Autoparallels and Geodesics

This paper is devoted to the study of approximate Lie point symmetries of general autoparallel systems. The significance of such systems is that they characterize the equations of motion of a Riemannian space under an affine parametrization. In particular, we formulate the first-order symmetry determining equations based on geometric requirements and stipulate that the underlying Riemannian space be approximate in nature. Lastly, we exemplify the results by application to some approximate wave-like manifolds