Perfect fluid equations with nonrelativistic conformal symmetry: Exact solutions

Abstract: The group-theoretic approach is used to construct exact solutions to perfect fluid equations invariant under the Schrodinger group, or the l-conformal Galilei group, or the Lifshitz group. In each respective case, the velocity vector field looks similar to the Bjorken flow. It is shown that one can reach an arbitrarily high density (and hence pressure) for a short period of time by adjusting the value of l and other free parameters available.

• The paper derives exact solutions for nonrelativistic perfect fluid equations exhibiting conformal symmetry using group-theoretic methods.
• It demonstrates how symmetry parameters like ℓ and z control fluid expansion, density evolution, and higher derivative effects.
• The analysis extends to viscous fluids, providing insights for modeling high-density, fast-evolving flows in various physical systems.

Group-Theoretic Exact Solutions to Nonrelativistic Perfect Fluid Equations with Conformal Symmetry

Introduction and Context

The paper "Perfect fluid equations with nonrelativistic conformal symmetry: Exact solutions" (2604.03621) explores the construction of exact solutions to nonrelativistic perfect fluid equations exhibiting invariance under three distinct symmetry groups: the Schrödinger group, the $\ell$-conformal Galilei group, and the Lifshitz group. The analytical approach centers on leveraging the group-theoretic method to systematically construct symmetry-invariant variables and fields, reducing complex PDEs to more tractable ODEs or algebraic systems. This strategy not only generalizes prior results in fluid mechanics with conformal symmetry but also highlights the interplay between symmetry parameters and fluid dynamical characteristics, such as expansion rate and density evolution.

Mathematical Framework: Symmetries and Invariant Equations

The study begins by defining the underlying symmetry algebras:

• Schrödinger group: Associated with scaling exponent $\ell = 1/2$.
• $\ell$-conformal Galilei group: Parameterized by (half-)integer $\ell$, allowing multiple acceleration generators; includes temporal translation, dilatation, special conformal transformation, spatial rotations and translations, Galilei boosts, and higher-order accelerations.
• Lifshitz group: Characterized by the dynamical critical exponent $z$, derived by omitting the special conformal generator.

The perfect fluid equations invariant under the $\ell$-conformal Galilei group are: $$\frac{\partial \rho}{\partial t} + \frac{\partial (\rho \upsilon_i)}{\partial x_i} = 0, \qquad \rho\, \mathcal{D}^{2\ell} \upsilon_i = -\frac{\partial p}{\partial x_i}, \qquad p = a\, \rho^{1+\frac{1}{\ell d}},$$ with $\mathcal{D}$ the material derivative. Notably, for $\ell>1/2$, the Euler equation involves higher-order derivatives, reflecting the expanded generator set.

Exact Solutions in One Spatial Dimension

The group-theoretic approach enables explicit characterization of solutions under different symmetry subgroups in $1+1$ dimensions:

• Scaling transformations: The generator $\ell = 1/2$0 leads to scale-invariant variables such as $\ell = 1/2$1 and invariant fields $\ell = 1/2$2, $\ell = 1/2$3. The analysis supports two distinct solution strategies: either fix the density via the continuity equation and solve for velocity, or vice versa.
• Velocity field structure: For $\ell = 1/2$4, the scaling-invariant velocity solution is $\ell = 1/2$5, generalizing Bjorken flow; higher $\ell = 1/2$6 values yield faster fluid motion.
• Density behavior: For integer $\ell = 1/2$7, $\ell = 1/2$8; for admissible half-integer $\ell = 1/2$9 ($\ell$0), the density solution exhibits more nuanced temporal and spatial dependence. Dramatic shifts in density are controlled by $\ell$1.

  Figure 1: Surface plot of $\ell$2 for multiple $\ell$3 values, showing density evolution in $\ell$4, $\ell$5.

Higher-Dimensional Generalization

The scaling subgroup analysis is extended to arbitrary spatial dimensions:

• Invariant velocity: $\ell$6 for $\ell$7, with scale-invariant variables $\ell$8.
• Density solutions: For integer $\ell$9, $\ell$0; for admissible half-integer $\ell$1,

$\ell$2

This manifests spatially localized density variations.

• Expansion rate: The parameter $\ell$3 emerges physically as a quantifier for fluid expansion.

  Figure 2: Streamline plot of the velocity field $\ell$4 for $\ell$5, $\ell$6 in two dimensions.

  

  Figure 3: Mass evolution of a unit disk in two spatial dimensions for $\ell$7 and $\ell$8.

Applying symmetry transformations--special conformal and acceleration generators--enables construction of new solution families, enabling velocity field manipulation and spatial patterning.

Figure 4: Example flow visualization after applying a specific acceleration transformation to the velocity field in 2D.

Lifshitz-Symmetric Fluid Equations

With the Lifshitz group (parameter $\ell$9), the analysis yields:

• Anisotropic scaling: Solutions invariant under $z$0 produce $z$1 invariants.
• Velocity field: $z$2; density exhibits explicit $z$3 dependence.
• Lower bound: $z$4 is necessary for physical density decay.

  Figure 5: Mass evolution of a unit disk for different $z$5 values, demonstrating dependence of density decay on Lifshitz exponent.

Inclusion of Viscosity

The methodology extends naturally to viscous fluids:

• Modified Euler equation: Incorporating the rate-of-strain tensor $z$6, with viscosity coefficients scaling as density, maintains symmetry invariance.
• Scale-invariant viscous solutions: For integer $z$7, explicit expressions for $z$8, $z$9, $\ell$0 are obtained; for half-integer $\ell$1, transcendent equations arise.

Implications and Prospective Directions

The analysis demonstrates that by tuning symmetry parameters ($\ell$2, $\ell$3), the density and pressure of these nonrelativistic fluids can be made arbitrarily large for short intervals, suggesting potential applications in quark-gluon plasma modeling, early-universe cosmology, and explosion dynamics. The mathematical connections to Bjorken flow, symmetry-adapted expansions, and exact multi-dimensional solutions open theoretical avenues for further study, including supersymmetric extensions and Hamiltonian/Lagrangian formulations.

Conclusion

The group-theoretic approach presented in this paper results in explicit, analytic families of exact solutions for perfect fluid equations under nonrelativistic conformal symmetry. Physical parameters $\ell$4 and $\ell$5 are shown to control fluid expansion and density scaling. Adjustments to these parameters, along with free coefficients of the solution, allow for precision in the modeling of fast-evolving, high-density fluids relevant to diverse areas of mathematical physics. The theoretical implications for symmetry-driven fluid dynamics are complemented by practical utility in modeling complex physical systems, meriting further exploration into viscous, quantum, and supersymmetric regimes.