Symmetries and reductions of integrable nonlocal partial differential equations

In this paper, symmetry analysis is extended to study nonlocal differential equations, in particular two integrable nonlocal equations, the nonlocal nonlinear Schr\"odinger equation and the nonlocal modified Korteweg--de Vries equation. Lie point symmetries are obtained based on a general theory and used to reduce these equations to nonlocal and local ordinary differential equations separately; namely one symmetry may allow reductions to both nonlocal and local equations depending on how the invariant variables are chosen. For the nonlocal modified Korteweg--de Vries equation, analogously to the local situation, all reduced local equations are integrable. At the end, we also define complex transformations to connect nonlocal differential equations and differential-difference equations.Comment: 10 page

A modified formal Lagrangian formulation for general differential equations

In this paper, we propose a modified formal Lagrangian formulation by introducing dummy dependent variables and prove the existence of such a formulation for any system of differential equations. The corresponding Euler--Lagrange equations, consisting of the original system and its adjoint system about the dummy variables, reduce to the original system via a simple substitution for the dummy variables. The formulation is applied to study conservation laws of differential equations through Noether's Theorem and in particular, a nontrivial conservation law of the Fornberg--Whitham equation is obtained by using its Lie point symmetries. Finally, a correspondence between conservation laws of the incompressible Euler equations and variational symmetries of the relevant modified formal Lagrangian is shown.Comment: 18 page

An automatic dynamic balancer in a rotating mechanism with time-varying angular velocity

We consider the system of a two ball automatic dynamic balancer attached to a rotating disc with nonconstant angular velocity. We directly compare the scenario of constant angular velocity with that when the acceleration of the rotor is taken into consideration. In doing so we show that there are cases where one must take the acceleration phase into consideration to obtain an accurate picture of the dynamics. Similarly we identify cases where the acceleration phase of the disc may be ignored. Finally, we briefly consider nonmonotonic variations of the angular velocity, with a view of maximising the basin of attraction of the desired solution, corresponding to damped vibrations.Comment: 15 pages, 5 figures, 7 table

Transformations, symmetries and Noether theorems for differential-difference equations

The first part of this paper develops a geometric setting for differential-difference equations that resolves an open question about the extent to which continuous symmetries can depend on discrete independent variables. For general mappings, differentiation and differencing fail to commute. We prove that there is no such failure for structure-preserving mappings, and identify a class of equations that allow greater freedom than is typical. For variational symmetries, the above results lead to a simple proof of the differential-difference version of Noether’s theorem. We state and prove the differential-difference version of Noether’s second theorem, together with a Noether-type theorem that spans the gap between the analogues of Noether’s two theorems. These results are applied to various equations from physics