Conditionally integrable PDEs, non-classical symmetries and applications

Some multi-dimensional nonlinear partial differential equations reduce to a linear equation in fewer dimensions after imposing one additional constraint. A large class of useful conditionally integrable reaction–diffusion equations follows from a single non-classical symmetry reduction, yielding an infinite-dimensional but incomplete solution space. This solution device extends further to other nonlinear equations such as higher-order reaction–diffusion, fractional-order diffusion and diffusion–convection. Applications are shown for population dynamics with or without weak Allee effects, speed-limited hyperbolic diffusion, material phase field dynamics, soil–water–plant dynamics and calcium transport on the curved surface of an oocyte. Beyond non-classical symmetry analysis, examples of other conditionally integrable equations are given; nonlinear diffusion in 1 + 2 dimensions and the Madelung–Burgers quantum fluid in 1 + 3 dimensions.