Mixed Superposition Rules in Lie Systems

• Mixed superposition rules are functional constructs that express the general solution of Lie systems by combining particular solutions of auxiliary systems with constant parameters.
• They are constructed using techniques such as imprimitive Lie algebras, semidirect sum decompositions, and coalgebra methods to ensure the integrability of complex differential equations.
• Their application spans various fields including quantum mechanics, thermodynamics, and geometric mechanics, offering a unified framework for solving nonlinear differential problems.

A mixed superposition rule is a functional construct that expresses the general solution of a Lie system—typically a time-dependent system of first-order ordinary or partial differential equations—in terms of particular solutions of auxiliary systems, which need not be identical to the original, and a set of constants. These rules generalize the classical superposition principle by allowing the auxiliary systems to be defined on different manifolds or to encode reductions associated with symmetry, fibration, or algebraic decompositions of the system’s Vessiot–Guldberg (VG) Lie algebra. Mixed superposition rules play a central role in unifying geometric, algebraic, and analytic approaches to solution generation across finite-dimensional Lie algebras, Hamiltonian systems on Dirac manifolds, and various classes of integrable and near-integrable systems, extending naturally to non-autonomous PDEs and physically relevant models.

1. Definition and Characterization

For a time-dependent differential equation system defined by a $t$-dependent vector field $X$ on a manifold $M$,

$\dot{x} = X(t, x),\quad x \in M,$

a mixed superposition rule is constructed via auxiliary systems $X_{(i)}$ on manifolds $M_i$, $i=1,\ldots, k$, and a smooth map

$\Phi: M_1 \times \cdots \times M_k \times M \to M$

such that any general solution $x(t)$ of $X$ can be written as

$$x(t) = \Phi\bigl(x_{(1)}(t), \ldots, x_{(k)}(t); p\bigr)$$

where each $x_{(i)}(t)$ is a particular solution of $X_{(i)}$ and $p$ is a constant determined by initial conditions.

The necessary and sufficient condition for the existence of a mixed superposition rule is that $X$ is a Lie system, i.e., its image is contained in a finite-dimensional real Lie algebra $V\subset\mathfrak X(M)$. This is a direct extension of the classical Lie–Scheffers theorem. Explicitly, the direct-product construction must yield a distribution whose projection onto the product of auxiliary manifolds is injective on an open dense subset, ensuring that the necessary $t$-independent first integrals exist and are non-degenerate in the relevant directions (Campoamor-Stursberg et al., 2 Nov 2025, Grabowski et al., 2012).

2. Construction Techniques

2.1 Imprimitive Lie Algebras

If the VG Lie algebra $V$ admits an invariant, integrable distribution $D\subset TM$, then $M$ is fibred over $M/D$. The system projects to a lower-dimensional base, and the mixed superposition rule may be obtained by combining solutions of the reduced system and the original. If $V\cap\Gamma(D)=0$, the map is constructed using $k$ generic solutions of the reduced system. If not, additional copies of $X$ are required, reflecting the dimension of $V\cap\Gamma(D)$, leading to mixed rules involving both the original and the reduced systems—e.g., for the $n$-dimensional Riccati system, solutions are built from one solution of the original and one of its Riccati projection (Campoamor-Stursberg et al., 2 Nov 2025).

2.2 Semidirect Sum Decomposition

If $V = V_1\overrightarrow{\oplus} V_2$ is a semidirect sum, $X$ decomposes as $X_1+X_2$ with each $X_i$ taking values in $V_i$. Mixed superposition rules then employ independent solutions from the $V_1$ and $V_2$ sectors: for the affine system $\dot x = A(t)x + b(t)$, $V_1\cong\mathfrak{gl}(n)$ and $V_2\cong\mathbb R^n$, leading to the classical affine superposition $x(t) = x_{(0)}(t) + \sum k_i x_{(i)}(t)$, with one solution from the full system and $n$ from the homogeneous part (Campoamor-Stursberg et al., 2 Nov 2025).

2.3 Coalgebra and Momentum Map Methods

Hamiltonian and Dirac–Lie systems exploit Poisson coalgebra and comodule structures. The addition map on the dual Lie algebra yields coalgebra morphisms, enabling the transfer of Casimir invariants via momentum maps to the configuration/manifold space. These maps, when combined with the direct-product or coaction, yield systems of $t$-independent first integrals whose joint level sets determine the mixed superposition rules. This approach is particularly systematic for systems with a compatible geometric structure (symplectic, Poisson, Dirac, or contact) (Campoamor-Stursberg et al., 2 Nov 2025, Ballesteros et al., 2013).

3. Explicit Mixed Superposition Examples

Schrödinger Lie Systems

For inhomogeneous linear systems with VG algebra the Schrödinger group $\mathcal S(1)$, mixed rules are established via explicit algebraic relations among four generic solutions: $$\begin{aligned} x_{(1)} &= x_{(2)} + (k_2 x_{(3)} - k_1 x_{(4)}), \ y_{(1)} &= y_{(2)} + (k_2 y_{(3)} - k_1 y_{(4)}). \end{aligned}$$ Regularity of the integrals ensures the map covers the full solution space (Campoamor-Stursberg et al., 2 Nov 2025).

Riccati-Type and Higher-Order Systems

For higher-order Riccati equations arising from projective reductions of linear ODEs,

$y^{(s-1)} = F(t, y, y', ..., y^{(s-2)}),$

the mixed rule defines $y(t)$ in terms of $s$ independent solutions $x^{(a)}(t)$ of the underlying linear problem: $$y(t) = \frac{\sum_{a=1}^s k_a x^{(a)\,(s-1)}(t)}{\sum_{a=1}^s k_a x^{(a)}(t)}.$$ This generalizes the well-known Riccati superposition and systematically reduces solving nonlinear equations to integrating linear ones (Grabowski et al., 2012).

Dirac–Lie, Calogero–Moser, and Thermodynamic Systems

In Dirac–Lie, contact, or thermodynamic systems, mixed superposition rules utilize the semidirect algebraic structure or the imprimitive fibration. For time-dependent Calogero–Moser models, they express the solution in terms of both the full system and its reduced or decoupled subsystems. For thermodynamics in contact geometry, $(T, P)$ variables are reconstructed from solutions of subsystems governing $(U, S, \nu)$ and their projections (Campoamor-Stursberg et al., 2 Nov 2025).

4. Role in Hamiltonian, Dirac, and Lie–Hamilton Systems

In Hamiltonian contexts, mixed superposition rules cohere naturally with coalgebra and momentum-map frameworks:

• The Lie–Hamilton system is described by a finite-dimensional algebra of Hamiltonians, with the Poisson coalgebra structure enabling explicit computation of invariants and superposition rules.
• When the Lie algebra splits into direct or semidirect sums, mixed superposition rules emerge, expressing one system’s general solution in terms of particular solutions to both subsystems.
• In Dirac geometry, the generalized coalgebra method produces constants of motion compatible with the underlying geometric structure, ensuring the applicability of the method to contact and presymplectic settings (Campoamor-Stursberg et al., 2 Nov 2025, Ballesteros et al., 2013).

5. Applications Beyond ODEs: PDEs and Operator Superposition

Mixed superposition rules generalize to Lie systems of PDEs. For non-autonomous PDEs

$\partial_{t^i}u = F_i(t, u), \quad i=1, \ldots, s,$

the method provides a recursive hierarchy by expanding around approximate solutions (e.g., $u(t) = u_0(t) + \epsilon v(t) + \epsilon^2 w(t)$). The resulting systems in $(v, w)$ admit VG algebras with semidirect or imprimitive structure, permitting mixed superposition rules, as in the Tzitzéica equation (Campoamor-Stursberg et al., 2 Nov 2025).

In operator theory, such as the superposition of $(s, p)$-fractional Laplacians studied by Di Pierro et al., the mixed operator is defined via measure-theoretic summation: $L(u)(x) = \int_0^1 (-\Delta_p)^s u(x)\,d\mu(s),$ with existence and multiplicity results under dominating positive measure on higher exponents and suitably bounded negative parts (Dipierro et al., 2023).

6. Broader Impact and Outlook

Mixed superposition rules unify geometric and algebraic solution approaches for broad classes of Lie systems, including those with symmetry reductions, inhomogeneities, and compatible Hamiltonian or Dirac structures. They reduce the integration of complex, nonlinear systems to algebraic manipulations involving particular solutions of simpler or linear subsystems, as illustrated in the Riccati hierarchy, quantum Hamiltonians, and PDE reduction frameworks.

Current research extends the methodology to quantum deformations, higher-order perturbations, and classes of non-integrable systems by exploiting the flexibility in the choice of auxiliary systems and the algebraic machinery of coalgebras, comodules, and momentum maps. The approach also addresses topological subtleties—e.g., tensor products in nuclear Fréchet categories—while providing explicit solution formulas in physically and mathematically significant models (Campoamor-Stursberg et al., 2 Nov 2025).

The systematic comparison of imprimitive geometric constructions, semidirect sum algebraic decompositions, and coalgebra methods yields a versatile toolkit for constructing and applying mixed superposition rules across differential geometry, mathematical physics, and applied analysis.