Invariant Reduction for Partial Differential Equations. I: Conservation Laws and Systems with Two Independent Variables

This paper presents an algorithmic reduction procedure for systems of partial differential equations in extended Kovalevskaya form that utilizes local symmetries and symmetry-invariant conservation laws to calculate constants of motion for symmetry-invariant solutions, accompanied by illustrative examples and a Maple implementation.

The Gist

Imagine you are trying to solve a massive, chaotic puzzle. This puzzle represents the laws of physics governing a system, like water flowing in a river or waves crashing on a shore. In the world of mathematics, these laws are written as Partial Differential Equations (PDEs). They are notoriously difficult to solve because they describe how things change in multiple directions at once (like time and space).

This paper introduces a clever "shortcut" or a special tool to solve these puzzles. Here is the breakdown using simple analogies:

1. The Problem: The Chaotic Ocean

Think of a PDE system as a stormy ocean. It's wild, with waves crashing everywhere.

• Conservation Laws are like the "rules of the ocean." For example, the total amount of water (mass) or energy in a specific area never disappears; it just moves around. Mathematically, these are equations that say, "What goes in must come out."
• Symmetries are like the "patterns" in the storm. Maybe the waves look the same if you shift them slightly to the right, or if you speed up time. These patterns are called symmetries.

Usually, finding a specific solution (a calm spot in the storm) is incredibly hard. But what if you only care about solutions that follow a specific pattern (symmetry)?

2. The Old Way: Changing the Map

Traditionally, to find these pattern-following solutions, mathematicians would try to "rotate the map." They would invent new coordinates (like changing from a grid to a spiral) to make the math simpler.

• The Flaw: This is like trying to navigate a city by redrawing the streets every time you turn a corner. It works for simple patterns (like moving straight), but if the pattern is complex or abstract (called "higher symmetries"), you can't even draw the new map. The math gets stuck.

3. The New Method: The "Motion Tracker"

The authors of this paper propose a new way that doesn't require redrawing the map. They use a concept called Invariant Reduction.

Imagine you are watching a dancer (the solution) who is moving to a specific rhythm (the symmetry).

• The Insight: If the dancer is perfectly in sync with the rhythm, and you also know a rule about their energy (a conservation law), you can predict exactly how much energy they have at any moment without tracking every single step.
• The "Constant of Motion": In the paper, they calculate a specific number (a "constant") that stays the same for any solution that follows that rhythm.
  • Think of it like a speedometer on a car. If you know the car is driving on a specific track (symmetry) and you know the fuel law (conservation), you can calculate the exact speed (the constant) without needing to know the exact shape of the road ahead.

4. How It Works (The "Magic Trick")

The paper provides an algorithm (a step-by-step recipe) to find these constants.

1. Identify the Rhythm: Find a symmetry (a pattern the solution must follow).
2. Identify the Rule: Find a conservation law (a rule that never breaks).
3. Check for Harmony: See if the rhythm and the rule "get along" (mathematically, if the symmetry leaves the conservation law unchanged).
4. Calculate the Constant: If they get along, the math automatically spits out a formula for a "constant of motion."

Why is this cool?

• No Map Changes: You don't need to invent new coordinates. You work with the original equations.
• Handles the Weird Stuff: It works even for "higher symmetries," which are abstract patterns that don't have a physical "flow" or movement you can easily visualize. It's like finding the rhythm in a song even if you can't see the musician.
• Simplifies the Puzzle: Once you have these constants, the massive, complex ocean of equations shrinks down to a much smaller, manageable pond (often just a system of ordinary equations).

5. Real-World Examples

The authors tested their "Motion Tracker" on famous equations used in physics:

• Burgers Equation: Models shock waves (like traffic jams or sonic booms).
• KdV Equation: Models shallow water waves (like tsunamis).
• Kaup-Boussinesq System: Models fluid dynamics.

In each case, they took a complex wave equation, found a hidden symmetry, paired it with a conservation law, and instantly generated a set of rules (constants) that describe exactly how those specific waves behave.

The Bottom Line

This paper is like giving mathematicians a universal remote control for solving complex physics puzzles. Instead of struggling to rewire the entire system (changing coordinates), they can now just press a button (run the algorithm) to find the "constants" that lock the system into a solvable state. It makes finding exact solutions for complex, real-world problems much faster and easier, even when the patterns involved are abstract and hard to visualize.

Technical Summary

1. Problem Statement

The paper addresses the challenge of finding constants of motion (first integrals) for symmetry-invariant solutions of systems of Partial Differential Equations (PDEs) with two independent variables (typically $t$ and $x$).

• Context: While symmetry reduction (using Lie point or contact symmetries) is a standard method to reduce PDEs to Ordinary Differential Equations (ODEs), finding exact solutions often requires additional constraints.
• Limitation of Existing Methods: Traditional approaches often rely on transforming the system into canonical coordinates where the symmetry generator becomes a simple translation (e.g., $\partial_s$). This approach fails or becomes computationally prohibitive for:
  • Higher symmetries: These do not generate geometric flows and thus lack canonical coordinates.
  • Complex point/contact symmetries: Where the transformation to canonical coordinates is algebraically intractable.
• Goal: To develop a coordinate-free, algorithmic procedure that utilizes a known local symmetry and a symmetry-invariant conservation law to directly compute constants of motion for invariant solutions, without requiring explicit coordinate transformations.

2. Methodology

The authors propose a reduction mechanism based on the properties of Lie derivatives acting on differential forms representing conservation laws.

Core Concepts

• Extended Kovalevskaya Form: The method applies to systems that can be rewritten as evolution equations ($u_t = f(u, u_x, \dots)$).
• Conservation Laws: Represented as differential 1-forms $\omega = P_1 dx - P_2 dt$ satisfying $D_t(P_1) + D_x(P_2) = 0$ on solutions.
• Symmetries: Represented by evolutionary vector fields $E_\phi$ (where $\phi$ is the characteristic).
• Invariant Conservation Laws: A conservation law $\omega$ is invariant under a symmetry $X$ if the Lie derivative $L_X \omega$ is a trivial conservation law (i.e., an exact differential).

The Algorithm

The central theoretical result (Theorem 1) states that if $X$ is a symmetry and $\omega$ is an $X$-invariant conservation law, there exists a function $\vartheta$ such that:
$$L_X \omega = d\vartheta = D_x(\vartheta)dx + D_t(\vartheta)dt$$
On any solution invariant under $X$ (where $X$ vanishes), the function $\vartheta$ is constant.

Step-by-Step Procedure:

1. Identify Inputs: A PDE system in evolution form, a symmetry characteristic $\phi$, and a conservation law characteristic $\psi$ (or the form $P_1, P_2$) that is invariant under the symmetry.
2. Compute Lie Derivative: Calculate the action of the symmetry on the conservation law density: $X(P_1)$.
3. Apply Homotopy Formula: Use the horizontal homotopy formula to integrate $X(P_1)$ with respect to the dependent variables to find a candidate function $\vartheta$ up to a function of $t$ and $x$.
   $$\vartheta = \int_0^1 \frac{G[\tau u]}{\tau} d\tau + h(t,x)$$
   where $G[u]$ is derived from the variational derivatives of $X(P_1)$.
4. Determine Integration Constants: Substitute the candidate $\vartheta$ back into the condition $L_X \omega = d\vartheta$. This yields a system of equations for the unknown function $h(t,x)$ (specifically $D_x h$ and $D_t h$).
5. Final Integration: Integrate the resulting total differential $dh = g_1 dx + g_2 dt$ to find the explicit form of $\vartheta$.
6. Result: The function $\vartheta$ restricted to the invariant surface (where the symmetry characteristic $\phi=0$) constitutes a constant of motion for the reduced system.

3. Key Contributions

• Coordinate-Free Reduction: The method eliminates the need for canonical coordinates, making it applicable to higher symmetries (which do not generate flows) and complex point symmetries.
• Algorithmic Implementation: The procedure is fully algorithmic and has been implemented in Maple (provided in Appendix A). It works for point, contact, and higher symmetries.
• Generalization: It generalizes previous reduction mechanisms (Refs. [21–24]) by treating invariant conservation laws in a coordinate-free manner using the Lie derivative on differential forms.
• Connection to Cosymmetries: The paper clarifies the relationship between invariant conservation laws (via cosymmetries) and the resulting constants of motion, showing that the constant of motion is essentially the "potential" of the Lie derivative of the conservation law.

4. Results and Examples

The authors demonstrate the algorithm on four distinct cases, successfully deriving constants of motion that allow for the integration of the reduced systems:

1. Burgers Equation:

   • Symmetry: A point symmetry generator $Y = -t^2\partial_t - tx\partial_x + (x+tu)\partial_u$.
   • Result: Derived a constant of motion $\vartheta = \frac{(x+tu)^2}{2} + t(1+t u_x)$. This was used to prove the non-existence of invariant solutions near $t=0$.

2. Korteweg-de Vries (KdV) Equation:

   • Symmetry: A higher symmetry (5th order).
   • Result: Derived three functionally independent constants of motion. These allowed the authors to express higher derivatives ($u_{xxxx}, u_{xxx}, u_x$) in terms of lower derivatives and constants, effectively reducing the PDE to a system of ODEs that can be integrated to find the local general solution.

3. Potential Kaup-Boussinesq System:

   • Symmetry: A higher symmetry for a coupled system.
   • Result: Derived two independent constants of motion, reducing the system to a set of algebraic/differential constraints.

4. Potential Boussinesq System:

   • Symmetry: A local symmetry.
   • Result: Demonstrated the method using the relation $\psi_i \phi_i = D_x(\vartheta)$, yielding constants of motion directly from the characteristic product.

5. Significance

• Computational Efficiency: The method provides a systematic way to find exact solutions for nonlinear PDEs without the heavy algebraic burden of solving for canonical coordinates, which is often impossible for higher symmetries.
• Global Analysis: The constants of motion act as first integrals for the reduced ODE systems, enabling the study of global qualitative properties (stability, existence of global solutions) of invariant solutions.
• Software Integration: The inclusion of Maple code (using the GeM package logic) makes the method immediately accessible for symbolic computation, facilitating the discovery of new exact solutions for complex physical models.
• Theoretical Unification: It unifies the treatment of point, contact, and higher symmetries under a single framework of differential forms and Lie derivatives, bridging the gap between geometric symmetry methods and conservation law theory.

In summary, this paper provides a powerful, algorithmic tool for the "double reduction" of PDEs (using both symmetry and conservation laws), significantly expanding the class of solvable nonlinear PDEs, particularly those admitting higher-order symmetries.