Reduction of Multidimensional Wave Equations to Two-Dimensional Equations: Investigation of Possible Reduced Equations

We study possible Lie and non-classical reductions of multidimensional wave equations and the special classes of possible reduced equations - their symmetries and equivalence classes. Such investigation allows to find many new conditional and hidden symmetries of the original equations.

💡 Research Summary

The paper presents a comprehensive study of both Lie‑group and non‑classical (conditional) reductions of multidimensional wave equations, with a focus on the special classes of two‑dimensional equations that can be obtained through such reductions. Starting from the general wave operator (\Box u = F(u,\partial u,\dots )) in (n\ge 3) space‑time dimensions, the authors first perform a systematic Lie‑symmetry analysis. By writing the most general infinitesimal generator (X=\xi^{\mu}(x,u)\partial_{x^{\mu}}+\eta(x,u)\partial_{u}) and imposing the invariance condition, they derive the determining equations and solve them for arbitrary nonlinearities (F). The resulting symmetry algebra always contains the obvious translations in time and space, the rotations of the spatial (SO(n-1)) group, and, depending on the form of (F), possible scaling or conformal extensions. All admissible Lie symmetries are classified together with the constraints they impose on the nonlinearity.

The second major part introduces conditional (non‑classical) symmetries. Here the invariance requirement is relaxed: the generator (X) must leave the original PDE invariant only on the manifold defined by an auxiliary condition (Q(x,u,\partial u)=0). The authors derive the compatibility conditions between (X) and (Q) and incorporate them into the determining system. Solving this enlarged system yields many symmetry operators that are invisible to the pure Lie analysis. Typical examples include generators that become symmetries when the solution satisfies a characteristic relation such as (u_{t}+c,u_{x}=0) or a specific algebraic constraint on (u). These conditional symmetries generate new reduction ansätze and lead to families of exact solutions that are not obtainable by standard Lie reductions.

The core of the work is the reduction of the original (n)-dimensional equation to a two‑dimensional PDE. The authors employ equivalence transformations—point transformations that may involve both the independent variables and the dependent variable—to map ((x^{\mu},u)) to new variables ((\xi,\eta,v)). The transformation group is constructed from the previously identified Lie and conditional symmetries, guaranteeing that the reduced equation retains the invariant structure. After the transformation, the generic reduced form reads \