Nonlocal pseudosymmetries and Bäcklund transformations as $\mathcal{C}$-morphisms

This paper demonstrates that factorizing differential equations with respect to nonlocal pseudosymmetries yields Bäcklund transformations, which are interpreted as nonlocal $\mathcal{C}$-morphisms determined by the basic invariants of those symmetries.

The Gist

Imagine you are trying to solve a massive, tangled knot of string. This knot represents a complex differential equation—a mathematical rule that describes how things change in the real world, like waves in the ocean or the vibration of a guitar string.

For a long time, mathematicians have had a special trick called a Bäcklund transformation. Think of this trick as a "magic wand." If you have one solution to the knot (one way the string is arranged), the wand can instantly conjure up a new solution, often by solving a much simpler puzzle first. It's like having a recipe that lets you turn a plain loaf of bread into a fancy cake just by adding a secret ingredient.

However, finding these "magic wands" has always been difficult. Usually, mathematicians had to guess the right trick for each specific knot, one by one. It was like trying to find a key for every single lock in a giant castle without a master key.

This paper introduces a new "Master Key" system.

Here is the simple breakdown of what the authors, Diego and Tarcísio, discovered:

1. The Hidden Shadows (Nonlocal Pseudosymmetries)

Imagine your knot is sitting in a room. You can see the knot itself (the equation), but there are also "shadows" cast by the knot on the walls. These shadows aren't the knot itself, but they hold clues about its shape.

In math terms, these shadows are called nonlocal pseudosymmetries.

• Normal Symmetry: If you rotate a perfect circle, it looks the same. That's a normal symmetry.
• Pseudosymmetry: This is a "shadow symmetry." It's a hidden pattern that doesn't make the knot look the same immediately, but if you follow the pattern, it reveals how the knot can be stretched or twisted into a new shape.

The authors realized that these "shadows" are actually the blueprints for the magic wands (Bäcklund transformations).

2. The Blueprint (Factorization)

The paper shows a new way to use these shadows. Instead of guessing, you can take the "shadow" (the pseudosymmetry) and factorize it.

Think of factorization like taking a complex machine apart to see its gears.

• The authors found that if you take a specific type of shadow (one that comes from a "Riccati-type" structure, which is just a fancy way of saying a specific kind of curved relationship), you can break it down.
• When you break it down, the "basic parts" (invariants) of the shadow automatically tell you exactly how to build the magic wand.

It's like finding a hidden instruction manual inside the shadow itself. You don't need to guess the recipe; the shadow is the recipe.

3. The "C-Morphism" (The Bridge)

The paper describes these magic wands as C-morphisms.

• Imagine a bridge connecting two islands. One island is your original equation (the old solution). The other island is the new equation (the new solution).
• Usually, to cross this bridge, you have to integrate (solve) a system of equations, which is like walking across a rickety bridge.
• The authors show that these bridges are built directly from the "basic parts" of the shadows. Because the bridge is built from the fundamental structure of the shadow, it's a very sturdy, general way to cross over.

4. Why This Matters (The "One-Size-Fits-All" Approach)

Before this paper, finding a magic wand for a new equation was like trying to pick a lock with a different tool for every single door.

• The Old Way: "Okay, for this wave equation, I need to try this specific trick. For that heat equation, I need a different trick."
• The New Way: "I see a shadow pattern here. I will use this universal method to extract the magic wand. It works for almost any equation that has these shadows."

The authors tested this on several famous equations (like the KdV equation, which describes tsunamis, and the Sine-Gordon equation, which describes magnetic fields) and even discovered a brand new integrable equation (a new type of mathematical puzzle that can be solved perfectly).

The Big Picture Analogy

Imagine you are a chef trying to create new dishes.

• Before: You had to taste every single ingredient and guess how to mix them to make a new flavor. It took forever and was hit-or-miss.
• Now: The authors found that every ingredient has a "flavor shadow." If you look at the shadow, it tells you exactly which other ingredients to mix with it to create a perfect new dish. You don't need to guess anymore; you just follow the shadow's map.

In summary: This paper gives mathematicians a universal, structural method to generate new solutions for complex equations by looking at the "hidden shadows" (pseudosymmetries) of the equations, rather than guessing the solution for each case individually. It turns a chaotic search into a systematic, almost mechanical process.

Technical Summary

1. Problem Statement

The paper addresses the fundamental problem of determining Bäcklund transformations (BTs) for nonlinear partial differential equations (PDEs).

• Context: BTs are transformations that map solutions of a differential equation $E$ to solutions of another equation $E'$ (or to new solutions of $E$ itself, known as auto-Bäcklund transformations). They are crucial for generating new solutions and studying integrability.
• Challenge: Finding BTs is generally a difficult, open problem. Existing methods in the literature often rely on "case-by-case" analysis, particularly when the transformations affect independent variables (point transformations) or involve complex nonlocal dependencies.
• Gap: While pseudosymmetries (a generalization of symmetries introduced by Sokolov) have been linked to differential substitutions, their potential to systematically generate Bäcklund transformations—especially those involving nonlocal variables and independent variable changes—has remained underexplored.

2. Methodology

The authors propose a unified structural framework based on geometric theory of differential equations, specifically utilizing Jet spaces, C-morphisms, and nonlocal pseudosymmetries.

A. Theoretical Framework

1. C-morphisms (Lie-Bäcklund maps): The authors treat BTs as regular C-morphisms. A C-morphism is a map between infinite jet spaces that preserves the Cartan distribution (contact structure). This allows BTs to be viewed as maps that transform integral manifolds (solutions) of one equation to another.
2. Differentiable Coverings: The approach utilizes differentiable coverings (extensions of the equation space by nonlocal variables). Specifically, the authors focus on Riccati-type differentiable coverings derived from Zero-Curvature Representations (ZCRs).
   • Given a ZCR (a flat connection) for an equation $E$, one can construct a linear system whose integrability condition yields $E$.
   • By reducing the dimension of the auxiliary vector space in the ZCR (via gauge transformations), one obtains a system of Riccati-type equations defining a covering $Y(\infty) \to E(\infty)$.
3. Nonlocal Pseudosymmetries: The core innovation is the use of pseudosymmetries defined on these differentiable coverings.
   • A pseudosymmetry $Y$ of a distribution $D$ satisfies $[Y, X] \in \langle Y \rangle + D$ for generators $X$ of $D$.
   • Unlike standard symmetries, pseudosymmetries do not necessarily map solutions to solutions directly but allow for factorization.
4. Factorization Procedure:
   • The authors demonstrate that if a differentiable extension $V$ of an equation $E$ admits a pseudosymmetry (or $r$-pseudosymmetry), one can compute the invariants of this pseudosymmetry.
   • These invariants define a new system of variables $(x', u')$.
   • The relations between the original variables and these invariants constitute a Bäcklund transformation. The target equation $E'$ is effectively the "quotient" or "factorization" of the extended system by the pseudosymmetry.

B. Algorithmic Steps

1. Start with a PDE $E$ admitting a ZCR.
2. Construct the associated Riccati-type differentiable covering $Y$ (introducing nonlocal variables $\rho$).
3. Identify a nonlocal conservation law $\mu$ associated with the ZCR.
4. Search for pseudosymmetries of the covering $Y$ that are pseudoprolongations of vector fields relative to $\mu$.
5. Compute the basic system of invariants of these pseudosymmetries.
6. Express the new dependent variables (and potentially independent variables) in terms of these invariants to derive the BT.

3. Key Contributions

• Unification of BTs and Pseudosymmetries: The paper establishes a rigorous link showing that Bäcklund transformations are determined by the basic invariants of nonlocal pseudosymmetries. This shifts the problem of finding BTs to the problem of finding pseudosymmetries in a covering space.
• General Structural Approach: Unlike previous case-by-case methods, this framework provides a general structural algorithm applicable to any equation admitting a ZCR.
• Handling Independent Variables: The method naturally handles cases where the BT affects the independent variables (e.g., $x' = x + f(\rho)$), which is often difficult in traditional symmetry-based approaches.
• Generalization to $r$-pseudosymmetries: The authors extend the theory to $r$-pseudosymmetries (systems of vector fields), demonstrating their necessity for higher-dimensional coverings (e.g., $sl_3(\mathbb{R})$ ZCRs).
• Discovery of New Integrable Equations: The framework is applied to derive BTs for a new integrable modification of the Harry-Dym equation, demonstrating the method's ability to uncover integrability structures in previously unstudied equations.

4. Results and Examples

The authors validate the framework through numerous representative examples, recovering known transformations and deriving new ones:

• KdV Equation:
  • Recovered the standard auto-Bäcklund transformation and the Darboux transformation (via a 2-pseudosymmetry on a doubled Riccati system).
  • Showed that the Darboux transformation arises naturally from the invariants of a pseudosymmetry on a 2-dimensional covering.
• Sine-Gordon Equation:
  • Derived the classic auto-Bäcklund transformation ($z' = z + 4 \arctan \rho$) from the invariants of a nonlocal pseudosymmetry.
• Short-Pulse Equation:
  • Successfully derived an auto-Bäcklund transformation that modifies both the dependent variable and the independent variable $x$.
• Camassa-Holm Equation:
  • Recovered known BTs and derived a Darboux-type transformation using a 2-pseudosymmetry.
• Harry-Dym and Modified Harry-Dym (mHD):
  • Derived BTs for the standard Harry-Dym equation.
  • Novel Result: Applied the method to a new modification of the Harry-Dym equation ($z_t = z^3 z_{xxx} + \dots$), successfully deriving its auto-Bäcklund transformation and proving its integrability.
• Tzitzeica Equation:
  • Demonstrated the power of 2-pseudosymmetries. For the Tzitzeica equation (associated with an $sl_3(\mathbb{R})$ ZCR), a single pseudosymmetry was insufficient, but a 2-pseudosymmetry yielded the known auto-Bäcklund transformation.
• Coupled Schrödinger System:
  • Derived an auto-Bäcklund transformation for a coupled system, further illustrating the method's applicability to systems of PDEs.

5. Significance

• Systematic Discovery: The paper moves the field from ad-hoc discovery of BTs to a systematic, algorithmic procedure based on the geometry of coverings and pseudosymmetries.
• Geometric Insight: It clarifies the geometric nature of Bäcklund transformations as C-morphisms arising from the factorization of differential coverings.
• Integrability Criterion: The existence of such pseudosymmetries in a Riccati-type covering serves as a strong indicator of integrability. The method provides a tool to test the integrability of new equations (like the mHD example).
• Handling Nonlocality: By explicitly working with nonlocal variables and differentiable coverings, the method resolves complexities associated with nonlocal transformations that often hinder traditional Lie symmetry analysis.

In conclusion, the paper provides a powerful, general geometric machinery for constructing Bäcklund transformations, unifying various known results and opening new avenues for discovering integrable structures in nonlinear PDEs.