Lie Symmetry Analysis of PDEs

• Lie symmetry analysis of PDEs is a systematic method that identifies continuous transformations to simplify complex differential equations.
• It employs rigorous compatibility checks and differential syzygies to ensure valid symmetry reductions and integrability of the system.
• The framework enables reduction of PDEs to simpler forms, often converting them to ODEs for obtaining exact and closed-form solutions.

Lie symmetry analysis of partial differential equations (PDEs) is a comprehensive mathematical framework for uncovering and exploiting the continuous transformation groups (symmetries) admitted by a given PDE or system of PDEs. These symmetries provide a powerful lens for understanding the structure, integrability, and solution space of PDEs. Lie symmetry theory not only enables systematic reduction to simpler equations (often ODEs), facilitating explicit integration or classification of solutions, but also serves as a criterion for compatibility and integrability, especially in overdetermined or highly constrained systems.

1. Criteria for Lie Symmetries and Invariant Solutions

Lie symmetries are continuous, structure-preserving transformations of the independent and dependent variables of a PDE that map solutions to solutions. For a scalar PDE of the form $F[u] = 0$, a vector field $S$ is a Lie symmetry if its prolonged action annihilates $F$ on the solution manifold: $\operatorname{pr}^{(k)} S (F[u]) \big|_{F=0} = 0,$ where $k$ is the order of the equation. In the formal theory, the symmetry condition is expressed via the Lie (Jacobi) bracket or its higher-order analogues: $\{F, S\} = 0 \quad \big(\bmod \ J_{k+l-1}(E)\big),$ with $k = \operatorname{ord}(F)$, $l = \operatorname{ord}(S)$, and $J_{k+l-1}(E)$ denoting the ideal of differential consequences of order less than $k+l$. This encapsulates the requirement that the system and its symmetries are compatible up to a certain differential order (Kruglikov, 2011).

For a system $\mathcal{E}$ of $r$ PDEs $F_1[u] = \dots = F_r[u] = 0$, a Lie algebra generated by $S_1, \dots, S_k$ must satisfy

$$\{F_i, S_j\} = 0 \ \big(\bmod\ J_{k_i + l_j - 1}(E)\big),\quad i=1,\dots, r,\ j=1,\dots, k,$$

ensuring the invariance of the system under the entire algebra. The coupled system $F_1=0,\dots,F_r=0$ and $S_1=0,\dots, S_k=0$ must form a complete intersection for the existence of invariant (symmetry-reduced) solutions (Kruglikov, 2011).

2. Compatibility, Formal Integrability, and Differential Syzygies

The existence of Lie symmetries is deeply interwoven with compatibility and integrability properties of PDE systems. Overdetermined systems require the vanishing of all hidden or secondary compatibility conditions for existence of solutions—captured in the framework of differential syzygies and Spencer cohomology. A system $\mathcal{E}$ is formally integrable if, for every basis $\{s_j\}$ of algebraic syzygies among the symbols (the leading-order parts) of $F_\ell[u]$, the lifted differential syzygies $[S_j]$ vanish: $[S_j] = 0,$ implying that no further hidden compatibility conditions emerge and that the solution space has the expected dimension (Kruglikov, 2011). For scalar equations with jointly transversal characteristic varieties, the system is formally integrable if and only if all higher-order (Mayer) brackets vanish: $[F_i, F_j] = 0\quad \text{for all}\ i < j.$ Checking such compatibility conditions is essential before proceeding with symmetry-based reduction and solution construction.

3. Lie Algebras, Reduction, and Symmetry-Induced Simplification

Once a Lie algebra of symmetries is determined for a PDE or system, seeking solutions invariant under a subalgebra $G$ allows a systematic reduction in the number of independent variables. For a $k$-parameter Lie group, the resulting similarity (or symmetry-reduced) solution depends on $n-k$ variables ($n$ being the original number of independent variables), and the PDE is correspondingly reduced to a system of lower-dimensional PDEs or ODEs.

This reduction relies on constructing canonical variables (group invariants) by solving the characteristic system associated with the infinitesimal generators of the Lie algebra: $$\frac{dx_1}{\xi_1} = \cdots = \frac{dx_n}{\xi_n} = \frac{du}{\eta},$$ where $\xi_i$, $\eta$ are the coefficients of the infinitesimal generator. In the case when the symmetry algebra is 'maximal' for the equation (i.e., of largest possible dimension given the number of variables and the structure of the system), the reduction can sometimes lead to complete integration in closed form, often via successive application of symmetry reductions or through contact/Lie–Bäcklund symmetries (Kruglikov, 2011).

It is critical to observe that if the symmetry algebra is too large relative to the number of variables—especially for linear or degenerate systems—invariant solutions may be trivial. Thus, the richness of the symmetry must be balanced against the geometry and analytical features of the PDE to avoid over-reduction.

4. Models with Large Symmetry Algebras and Exact Integrability

A central theme is that PDEs possessing large or even 'maximal' symmetry algebras often exhibit exact integrability properties. This is particularly evident for classical models such as certain Monge equations, the Korteweg-de Vries (KdV) equation under specific symmetry subalgebras, and the Kadomtsev–Pogutse system (Kruglikov, 2011).

In these situations, the joint system—the PDE plus its symmetry constraints—has a characteristic variety of maximal codimension, meaning that all possible compatibility (syzygy) conditions are saturated by the symmetry. No additional constraints obstruct integration, so closed form (Darboux integrable) or explicitly parametrized solutions are available. In such cases, the process essentially reduces the original PDE to an ODE or to a system for which integration methods (such as the method of characteristics, first integrals, or quadratures) can be employed systematically.

The paper provides concrete formulas for these scenarios. For a scalar equation $F[u]=0$ of order $k$ and a symmetry $S$ of order $l$: $$\{ F, S \} = 0 \ \left(\bmod \, J_{k+l-1}(E)\right),$$ which guarantees S-invariant solutions exist, provided the system is nondegenerate and the dimensions are compatible.

5. Role of Formal Theory: Brackets, Symbols, and Cohomology

Formal integrability and compatibility analysis of PDEs via differential brackets and the symbolic module is essential for a rigorous understanding of the solution space. The compatibility checks based on brackets—such as Mayer or higher–multibrackets—provide necessary and sufficient conditions for prolongation closure and supply bridges between algebraic syzygies among the symbols of differential operators and their differential counterparts on the solution level (Kruglikov, 2011).

This is recast in the language of Spencer cohomology and symbolic modules, which allow for the quantification and classification of possible obstructions. Quantization-type maps (the 'arrow $q$' in the paper) systematically connect algebraic relations at the level of the symbols to differential syzygies, providing a practical computational route to verifying integrability and identifying hidden constraints.

6. Representative Formulas and Computational Criteria

The methodology is highlighted by several core formulas:

• Symmetry Invariance Condition (scalar case):

$$\{ F, S \} = 0\,\,(\bmod\, J_{k+l-1}(E)),\quad k = \operatorname{ord}(F), \; l = \operatorname{ord}(S)$$

• Invariance for system:

$$\{ F_i, S_j \} = 0\,\,(\bmod\, J_{k_i + l_j - 1}(E)), \quad i = 1,\dots, r;\; j = 1,\dots, k$$

Given the appropriate dimensionality (i.e., combined codimension equal to the sum of equations and symmetries), these ensure the compatibility and integrability of the symmetry-reduced (invariant) system (Kruglikov, 2011).

7. Significance and Applications

Lie symmetry analysis of PDEs is not a purely formal exercise; it is tightly coupled to the existence and explicit construction of invariant and exact solutions, integrability criteria, and the deeper geometric structure of differential systems. The approach bridges the classical geometric theory—emphasizing compatibility, syzygies, and prolongation— with practical tools for reduction and integration. When maximal or sufficiently rich symmetry algebras are present, exact, closed form solutions can often be constructed by systematic symmetry reduction (Kruglikov, 2011).

The framework is applicable in diverse settings: from the rigorous analysis of compatibility in overdetermined systems, through explicit construction of exact solutions and reductions in mathematical physics, to providing structural understanding in geometric and applied PDEs. The interplay of Lie symmetry, differential syzygies, and integrability ensures that the analysis encompasses both the algebraic and analytic aspects of PDE theory.