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 investigates both Lie and non‑classical (Q‑conditional) reductions of the multidimensional nonlinear wave equation, commonly written as the d’Alembert operator ∎ u = F(u) with ∎ = ∂₀² – ∂₁² – … – ∂ₙ². The author’s main goal is to understand how this high‑dimensional PDE can be systematically reduced to a two‑dimensional equation while preserving or revealing new symmetry structures.

The study begins by recalling the classical Lie reduction method (direct method) and its limitations for equations containing second‑order derivatives in many independent variables. To overcome these difficulties, the author introduces an ansatz u = φ(y, z) where y(x) and z(x) are two new scalar functions of the original coordinates. Substituting this ansatz into the original wave equation leads to a set of compatibility conditions that can be written as a d’Alembert–Hamilton system: ∎ y = R(y, z), ∎ z = S(y, z), together with three scalar relations y_μ y^μ = r(y, z), y_μ z^μ = q(y, z), z_μ z^μ = s(y, z).

The crucial observation is that the discriminant Δ = r s – q² determines the type of the reduced equation. Four distinct cases arise:

1. Elliptic case (Δ > 0). The reduced system is compatible iff the functions V and Φ satisfy V = h(v, v*)∂{v*}Φ/Φ, where h = 1/R{vv*} and (h∂_{v*})^{n+1}Φ = 0. The resulting two‑dimensional equation has the form h(v, v*)