New origins of hidden symmetries for partial differential equations

Hidden symmetries of differential equations are point symmetries that arise unexpectedly in the increase (equivalently decrease) of order, in the case of ordinary differential equations, and variables, in the case of partial differential equations. The origins of Type II hidden symmetries (obtained via reduction) for ordinary differential equations are understood to be either contact or nonlocal symmetries of the original equation while the origin for Type I hidden symmetries (obtained via increase of order) is understood to be nonlocal symmetries of the original equation. Thus far, it has been shown that the origin of hidden symmetries for partial differential equations is point symmetries of another partial differential equation of the same order as the original equation. Here we show that hidden symmetries can arise from contact and nonlocal/potential symmetries of the original equation, similar to the situation for ordinary differential equations.     

Model for ion-initiated trench etching

We model the plasma etching of trenches by Langmuir kinetics for neutral molecules and bombarding ions. The parallel combination of an isotropic etch rate for the neutrals and an anisotropic etch rate for the ions gives an effective etch rate. The ion etch rate is proportional to the normal surface component of the ion energy flux. An approximate analytical expression for the composite etch rate offers a new approach to the computation of etch profiles for these mixed systems. Etch profiles are displayed for three cases: the nearly ion flux-limited regime, an intermediate case, and the nearly neutral-flux limited regime for the trench bottom. The numerical calculation of the etch profiles follows from the integration of three characteristic strip equations which are nonlinear first-order ordinary differential equations (ODE's)     

Perturbation Expansion of Potential of Charged Dielectric Sphere in Asymmetrical Electrolytes

A perturbation expansion for the electrostatic potential of a uniformly charged, dielectric sphere is extended to include asymmetrical electrolytes in the large radius limit. The potential, valid for q?/?T > 1 and small ?D/a near the sphere, is solved in a similar manner to that for 1-1 electrolytes.     

Master partial differential equations for a Type II hidden symmetry

An approach for determining a class of master partial differential equations from which Type II hidden point symmetries are inherited is presented. As an example a model nonlinear partial differential equation (PDE) reduced to a target PDE by a Lie symmetry gains a Lie point symmetry that is not inherited (hidden) from the original PDE. On the other hand this Type II hidden symmetry is inherited from one or more of the class of master PDEs. The class of master PDEs is determined by the hidden symmetry reverse method. The reverse method is extended to determine symmetries of the master PDEs that are not inherited. We indicate why such methods are necessary to determine the genesis of Type II symmetries of PDEs as opposed to those that arise in ordinary differential equations (ODEs).     

Modification of plasma-etched profiles by sputtering

Two-dimensional etch profiles are modeled for plasma etching. The etch rate dependence on the angle of incidence of the bombarding ions on the etched surface has a sputtering-type yield. The etch profile is advanced in time by an evolution equation for an etch rate proportional to the modified ion energy flux. Approximate analytical expressions for the etch rates are derived as a product of the etch rates in the absence of the sputtering-type yield and a weighting factor that depends on the angle the ion drift velocity makes with the normal to the wafer surface. The weighting factor is determined from experimental measurements of the angular dependence of ion beam etching by sputtering. These etch rates are valid when the ratio of the ion drift speed to the ion thermal speed is large compared to one. The etching is modeled in the ion flux-limited regime for simplicity. The modifications of the shape of etch profiles of a long rectangular trench and a waveguide structure or strip are treated     

Type II hidden symmetries of the second heavenly equation

A Type II hidden symmetry of the non-linear second heavenly equation in gravitational physics is identified. Its provenance from other partial differential equations is studied. Two reductions of the second heavenly equation produce the Monge–Ampère equation in similarity variables and new analytic solutions are possible.