Linearisable Nonlinear Problems

• Linearisable nonlinear problems are defined as nonlinear systems that can be mapped to linear counterparts through change of variables or transformations.
• They employ methods like point and multipoint transformations, dynamic feedback, and iterative linearisation to satisfy compatibility and integrability conditions.
• These techniques enable explicit solutions and stability analysis in fields such as control theory, optimization, and spectral analysis.

A linearisable nonlinear problem is a nonlinear equation, system, or operator whose dynamics, solutions, or spectral properties can—under specific transformations or algorithmic procedures—be mapped to or encoded in an associated linear system. In the context of integrable systems, control theory, discrete mathematics, optimization, inverse problems, and spectral analysis, linearisability underpins both the classification and tractable solution of otherwise complex nonlinear entities. Techniques for establishing and exploiting linearisability vary, including point and multipoint transformations for difference or differential equations, systematic local or global linearization procedures for operator equations, dynamic or static feedback in control, and advanced numerical and symbolic algorithms to construct the linear mapping explicitly or indirectly. Nontrivial constraints and tests—often expressed in terms of algebraic, differential, or structural compatibility—govern when such mappings exist and whether they can be made algorithmic.

1. Definitions and General Principles

A nonlinear problem is called linearisable if there exists a change of variables, transformation, or lifting of the state and/or parameter space such that the original nonlinear system or equation is mapped to a linear one. This mapping may involve:

• Point transformations (functions depending only on the state at each site or time/space index),
• Multipoint transformations (involving neighboring values),
• Dynamic or static feedback (possibly involving state augmentation in control),
• Algebraic manipulations or parameter lifts (such as introducing auxiliary variables for eigenvector nonlinearities),
• Approximate linearization (e.g., via iterative local preconditioning).

Formally, in the setting of partial difference equations (as on a quad-graph), ordinary/partial differential equations, or nonlinear eigenproblems, one seeks transformations of the form

$\tilde{u} = \mathcal{T}(u, \partial u, \ldots)$

such that $\mathcal{A}(u, \ldots) = 0$ is reduced to a linear equation, possibly with constant or structured (e.g., periodic) coefficients.

Linearisability is subject to structural integrability or compatibility conditions, frequently encoded as Wronskian vanishing, Lie algebra dimensionality, specific algebraic constraints, rank requirements, or commutativity of parameter-dependent fields. These conditions can sometimes be organized into algorithmic checks, e.g., for PDEs, ODEs, or difference equations.

2. Linearization Techniques in Discrete and Continuous Equations

2.1. Discrete Quad-Graph Equations

For nonlinear quad-graph equations of the form

$$\mathcal{A}(u_{n,m}, u_{n+1,m}, u_{n,m+1}, u_{n+1,m+1}) = 0,$$

the primary tests for linearisability explored in (Levi et al., 2011) are:

• Point transformation: $\,\tilde{u}_{0,0} = f(u_{0,0})$. The functional equation derived by substitution yields necessary and sufficient conditions expressed via Wronskian vanishing, notably

$W_{(u_{0,1})}[F_{,u_{1,0}};F_{,u_{0,0}}] = 0,$

and further relations among relevant coefficients.

• Two-point transformation: $\,\tilde{u}_{0,0} = g(u_{0,0}, u_{0,1})$. The corresponding functional relations and their differentiations lead to compatibility equations, such as

$$\psi_{,u_{0,1}} - \frac{F_{,u_{0,1}}}{F_{,u_{0,0}}} \psi_{,u_{0,0}} = \mathcal{R}(u_{0,0}, u_{0,1}, u_{1,0}),$$

with cancellation of the $u_{1,0}$-dependence as a necessary criterion for existence.

• Generalized Hopf–Cole transformation: Transformations with multiplicative factors that absorb nonlinearity, e.g., for the discrete Burgers equation. These satisfy difference-differential compatibility conditions involving the transformation function and shifted arguments.

In each case, these algorithms translate the existence of a linearizing transformation into explicit (often overdetermined) systems of algebraic or differential equations, whose solution structure identifies both linearisable cases and the explicit transformation.

2.2. Ordinary Differential Equations

For ODEs rational in lower derivatives and solved for the highest order, the criteria for linearisability by point transformation are quantified via:

• Symmetry algebra dimensionality: For a second-order ODE, having an 8-dimensional Lie symmetry algebra is necessary and sufficient.
• Differential Thomas decomposition: Construction of a nonlinear PDE system whose unknowns determine the point transformation and the coefficients in the reduced linear equation (Lyakhov et al., 2017). This method explicitly generates all compatibility conditions and embedding relations required for linearization.

3. Linearisation in Optimization, Inverse Problems, and Applied Algorithms

3.1. Dynamic ODE-Based Methods in Optimization

Transforming a nonlinear optimization problem (especially NLPs with constraints) into a linearisable ODE system (the Dynamic Optimization Equation, DOE) leverages Lyapunov theory. The DOE is constructed as

$$\frac{d\theta}{dt} = -K_0 \left( \nabla f + h_\theta^T \lambda_e + g_\theta^T \lambda_k \right)$$

with explicit pseudo-inverse-based multiplier expressions accommodating possible singular or redundant constraints (Zhang et al., 2018). This guarantees global convergence to the KKT point under mild regularity and obviates the need for step length selection.

3.2. Iterative Linearisation and Model Correction

For general operator equations $F(u)=0$ (possibly in infinite dimensions), prominent approaches include fixed-point local linearisation,

$A(u^n)(u^{n+1} - u^n) = -F(u^n),$

with suitable preconditioning, and the iterative-linearization Galerkin (ILG) methodology for discretized PDEs (Heid et al., 2018), offering separate a posteriori control over linearisation and discretization errors.

In inverse problems, the sequential model correction approach (Arjas et al., 2023) recursively solves a convexified (linearized) problem, updating the approximation error at each step—a strategy subsuming Gauss–Newton in the least-squares case.

3.3. Control and Discretization Preservation

Dynamic and static feedback linearization, including discrete-time feedback and the effect of discretization, is addressed in (Jindal et al., 3 Jun 2024), where geometrically motivated discretization schemes (using retraction maps) preserve feedback linearizability under sampling, allowing retention of linear control law applicability.

4. Structural and Algorithmic Compatibility Tests

Linearisability is only generically possible for special structures or under restrictive compatibility conditions. For example:

• In quad-graph equations, explicit expressions for Wronskians or other compatibility invariants must vanish (Levi et al., 2011).
• For time-optimal control, the existence of a linearizing transformation $z = F(t, x)$ for $\dot{x} = a(t, x) + b(t, x)u$ is regulated by analytic commutativity of the iterated vector fields, full-rank conditions, and a combinatorial indicial equation with prescribed integer roots (Sklyar et al., 2022).
• In nonlinear eigenvalue problems (with eigenvector nonlinearities), exact linearizability requires that the nonlinear scalar (e.g., $(r^T x)/(s^T x)$) is isolated and lifted into an extended multiparameter problem, with explicit block-structured formulations and operator determinants reflecting the linear structure (Claes et al., 2021).

Such criteria usually manifest as overdetermined systems, permitting only selected families of nonlinear problems to admit explicit linearization.

5. Applications and Concrete Examples

• Nontrivial discrete equations: The discrete Liouville, $\mathcal{Q}_+$ and Hietarinta equations, as well as QRT-type equations, provide fertile testbeds for the classification and explicit construction of linearizing transformations (Levi et al., 2011).
• Laurent property recurrences: Certain families of nonlinear recurrences, though not cluster-algebra mutations, exhibit twofold linearisability: they simultaneously satisfy a linear relation with constant coefficients and another with periodic coefficients. Associated monodromy matrices yield first integrals and connect the nonlinear problem to spectral theory (e.g., the dressing chain of Schrödinger operators) (Hone et al., 2013).
• Nonlinear spectral problems: Universal computational approaches using nonlinear injection moduli and the Solvability Complexity Index (SCI) hierarchy achieve provably convergent, optimal algorithms for the spectrum and pseudospectrum computation in nonlinear operator pencils, regardless of reduction to linearity by transformation (Colbrook et al., 23 Apr 2025).

6. Integrability, Invariance, and Broader Impact

The existence of a linearizing transformation is often coincident with deeper notions of integrability (C-integrability in the discrete sense), invariance under symmetry groups, or the possession of conserved quantities (monodromy invariants). Algorithmic frameworks for testing and constructing linearisations thus not only delineate the border between integrability and non-integrability but also generate explicit transformations applicable to explicit solution, classification, and analysis—spanning from discrete systems to quantum computing contexts. These developments underpin the robust and general utility of linearisability for advancing both fundamental and applied research in nonlinear analysis, control, and computation.