A conditionally integrable non-reciprocal wave equation with diode properties

Abstract

A known class of conditionally integrable partial differential equations is extended to include those that can be reduced by a non-classical symmetry to a linear Kirchhoff equation. From any steady solution to that linear equation, there follows an exact time-dependent solution to a nonlinear hyperbolic equation. An example solution is constructed in two space dimensions and one time dimension. By a change of variable, in one space dimension these nonlinear partial differential equations are equivalent to a nonlinear wave equation with diode-like properties that break reciprocity. These properties are illustrated by an exact solution in one dimension.