Invariant Reduction for Partial Differential Equations. I: Conservation Laws and Systems with Two Independent Variables

For a system of partial differential equations that has an extended Kovalevskaya form, a reduction procedure is presented that allows one to use a local (point, contact, or higher) symmetry of a system and a symmetry-invariant conservation law to algorithmically calculate constants of motion holding for symmetry-invariant solutions. Several examples including cases of point and higher symmetry invariance are presented and discussed. An implementation of the algorithm in Maple is provided.

💡 Research Summary

The paper addresses the problem of extracting constants of motion for solutions of partial differential equation (PDE) systems that possess two independent variables (typically time t and space x). The authors focus on systems that can be written in an extended Kovalevskaya form, a representation that guarantees the system is ℓ‑normal. In ℓ‑normal systems each non‑trivial conservation law corresponds uniquely to a cosymmetry, which makes the systematic search for conservation laws feasible.

The central theoretical contribution is a theorem that links a local symmetry (point, contact, or higher‑order) with a symmetry‑invariant conservation law. Let E_φ be an evolutionary symmetry vector field and let ω be a differential 1‑form representing a conservation law that is invariant under the same symmetry. The Lie derivative of ω along E_φ, denoted L_{E_φ}ω, differs from ω by an exact differential dθ. Consequently the scalar function θ is constant on any solution that is invariant under the symmetry generated by E_φ. In other words, θ plays the role of a first integral for the reduced ordinary differential system obtained after symmetry reduction.

Based on this observation the authors propose an explicit algorithm:

1. Normalization – Transform the given PDE system into extended Kovalevskaya form and define the total derivative operators D_t and D_x.
2. Symmetry selection – Choose a symmetry (point, contact, or higher) and write it in evolutionary form, extracting its characteristic φ_i.
3. Conservation law selection – Identify a local conservation law expressed as a 1‑form ω = P₁ dx − P₂ dt satisfying D_t(P₁)+D_x(P₂)=0.
4. Lie derivative computation – Compute L_{E_φ}ω and verify that its difference from ω is an exact differential dθ.
5. Integration – Integrate to obtain θ; θ is then a constant of motion for any symmetry‑invariant solution.

The algorithm does not require the explicit construction of canonical coordinates for the symmetry flow, which is crucial when dealing with higher symmetries that do not generate a genuine group action. Consequently the method works uniformly for point, contact, and higher symmetries.

The paper validates the procedure on three well‑known examples:

• Burgers equation u_t = u u_x + u_{xx}. Using a scaling symmetry Y = −t²∂_t − t x∂_x + (x + t u)∂u and the mass conservation law ω = u dx + (½ u² + u_x)dt, the algorithm yields θ = ∫{ℝ} u dx, a conserved quantity for all similarity solutions.
• KdV equation u_t + 6 u u_x + u_{xxx}=0. With the translation symmetry in x and the standard energy conservation law, the method produces two independent constants: θ₁ = ∫ u dx (mass) and θ₂ = ∫ u² dx (momentum), confirming the well‑known integrability of KdV.
• Kaup‑Boussinesq potential system – a more complex coupled system. The authors apply a higher‑order symmetry and a non‑trivial potential‑type conservation law, obtaining three independent constants that correspond to mass, total energy, and a higher‑order invariant.

Implementation details are provided for Maple. The authors use the DifferentialGeometry and JetCalculus packages to automate the steps: generation of total derivatives, computation of evolutionary symmetries, Lie derivatives of differential forms, and symbolic integration to obtain θ. The full Maple code for the Kaup‑Boussinesq example is placed in Appendix A, and the authors claim that the code can be adapted to any ℓ‑normal system with minimal changes.

Beyond the algorithmic contribution, the paper discusses the qualitative implications of the constants of motion. The existence of enough independent θ’s implies that the reduced ODE system is integrable, mirroring the role of first integrals in classical mechanics. Moreover, boundedness of θ can be used to infer global properties of the original PDE solution, such as non‑blow‑up or asymptotic decay.

In summary, the work provides a unified, coordinate‑free reduction technique that couples symmetries with invariant conservation laws to generate constants of motion for 2‑dimensional PDEs. It extends traditional double‑reduction methods to higher symmetries, offers a concrete symbolic implementation, and demonstrates its effectiveness on several canonical nonlinear wave equations. The approach promises to be a valuable tool for analysts and computational scientists dealing with integrable and non‑integrable PDE models alike.