This talk will explain how a similar hidden symmetry structure exists for all central force systems in $n$ dimensions when local constants of motion are considered. In particular, every such system possesses $2n-1$ local constants of motion, including a generalized LRL vector, plus an integral of motion that depend explicitly on time. The latter quantity will be shown to lead to an enlarged symmetry group structure.A key tool is a version of Noether's theorem holding in the space of configuration variables extended to include time.

Mathematical and physical properties of the generalized LRL vector and the additional integral of motion, along with the associated symmetry group, will be presented.The $n$-dimensional Kepler potential and isotropic oscillator are used as examples.

Two situations arise for the admitted trivial Lie point symmetries of a linear PDE. A third situation arises for an admitted nontrivial Lie point symmetry of a PDE.

The method is algorithmic and has been implemented in Mathematica. The software is being used to compute conservation laws of nonlinear PDEs occurring in the applied sciences and engineering. The software package will be demonstrated for PDEs that model, e.g., shallow water waves, ion-acoustic waves in plasmas, sound waves in nonlinear media, transonic gas flow, and stress and displacement in elastic materials. Examples include the Korteweg-de Vries and Zakharov-Kuznetsov equations, and a constitutive equation arising in elasticity.

In this talk, we explore alternative approaches to deriving approximate yet analytically tractable PDEs from nonlinear perturbed systems of the form $u_{tt} - u_{xx} + \epsilon N(u) = 0$, where $\epsilon$ is a small parameter and $N$ is nonlinear. Using methods such as multiple scales, we obtain reduced models that allow for a more manageable analysis of complex nonlinear phenomena, including shock formation and wave breaking. Numerical experiments, including error analysis and conservation testing, are presented for the dynamics governed by the PDE $u_{tt} = (1 + \epsilon u_x^2)u_{xx}$, where we can determine the break times, correct the multivalued solution, and track the shockwave. Similar tests are also performed after the addition of a nonlinear viscosity term: $u_{tt} = (1 + \epsilon u_x^2)u_{xx} + \eta(u_x^2u_{xt})_x$

Abstract:

The Lie-point symmetry method is used to find some closed-form solutions for a constitutive equation modeling stress in elastic materials and a technology diffusion model.

The Lie algebra for the governing PDE system for a constitutive equation modeling stress in elastic materials is five-dimensional. Using the optimal system of one-dimensional subalgebras, closed-form solutions for the model are obtained. Based on the scaling symmetry of the PDE and using Euler and homotopy operators, several conservation laws are computed with symbolic software.

A critical component of economic growth is growth in productivity which is dependent on technology adoption. While most technologies are created in developed economies, they diffuse to developing economies through various channels such as trade, migration and knowledge spillovers. The first model that integrates compartmental models with diffusion is developed to analyze technology adoption within a framework of a system of second-order non-linear partial differential equations. A three-dimensional Lie algebra is established for a technology diffusion model. The combinations of Lie symmetries are used to obtain reductions and establish closed-form solution for the technology diffusion model. The closed-form solutions allow for graphical representations of the technology diffusion process over an effective distance and time and show the commonly observed S-curve path of technology diffusion. Furthermore, a sensitivity analysis is performed to develop policy insights into the factors influencing the diffusion of technology.

Many simplified models have been derived from Euler and Navier-Stokes equations to describe specific phenomena, such as surface and internal waves. These models aim to reduce the complexity of the original equations while preserving essential features of the phenomena and offering physical insight and computational accuracy. Examples include dimension and symmetry reductions, linearizations, and more general approximations based on asymptotic relationships. Fundamental nonlinear partial differential equations of mathematical physics, including Burgers', Korteweg-de Vries, nonlinear Schrödinger, and Kadomtsev-Petviashvili equations, arise in this context. Reduced models often reveal rich mathematical structures; their exact solutions can closely describe physical phenomena.

I will discuss a model of nonlinear internal waves in a stratified system of two non-mixing fluids with different densities, contained in a horizontal channel. This model, developed by Miyata and then Choi and Camassa, was derived through layer-averaging under the "shallow water" assumption, which assumes a small ratio of channel depth to wavelength, without requiring wave amplitudes to be small. I will introduce a transformation that simplifies the Choi-Camassa model, reducing it to a simpler dimensionless form, and demonstrate that the model admits simple physical exact solutions, including traveling waves, cnoidal waves, and kinks.