Elastic Flow Jellyfish Expanders

• Elastic Flow Jellyfish Expanders are smooth, closed, self-similarly expanding planar curves defined by a fourth-order elastic flow with arbitrary large dihedral symmetry.
• They are explicitly constructed by solving a first-order ODE system for fundamental arcs and then gluing these arcs via reflection to form m-fold symmetric loops.
• The resulting family exhibits complex bifurcation structures and dynamic properties, highlighting challenges in understanding stability in geometric evolution equations.

Elastic flow jellyfish expanders are smooth, closed, homothetically expanding immersed plane curves with arbitrary large dihedral symmetry, explicitly constructed as solutions to the fourth-order geometric evolution equation for the elastic flow. The defining property is that these curves, under the elastic flow, expand self-similarly—a phenomenon characterized by never-trivial, highly symmetric solutions called "jellyfish." Their construction and analysis arise at the intersection of geometric analysis, nonlinear PDEs, and the theory of plane curve evolutions (Andrews et al., 29 Jan 2026).

1. Elastic Flow and Homothetic Expanders

The elastic flow describes the $L^2(ds)$-steepest descent of the elastic energy for immersed plane curves $\gamma$: $E[\gamma]=\frac12\int_\gamma k^2\,ds,$ where $k$ is the curvature and $s$ is arc-length. The flow evolves $\gamma$ by

$$\partial_t \gamma = -\bigl(k_{ss}+\tfrac12\,k^3\bigr)\nu,$$

where $\nu$ is the unit normal, $k_{ss}$ the second arc-length derivative of curvature. This flow is scaling-subcritical: $E[\rho\gamma]=\rho^{-1}E[\gamma]$, which generically induces expanding behavior.

A homothetic expander is a solution evolving through dilations: $$\gamma(s,t) = \lambda(t)(\hat\gamma(s)-c) + c, \quad \lambda(t)>0,$$ centered at $c\in\mathbb{R}^2$, reducing the flow to a nonlinear profile equation: $$k_{ss} + \tfrac12 k^3 = -\sigma\langle\hat\gamma - c, \nu\rangle, \tag{H}$$ with expansion rate $\sigma>0$.

2. Analytical Construction: Fundamental Arcs and Dihedral Gluing

The explicit construction of jellyfish expanders follows a two-step method developed by Andrews and Wheeler:

Step 1: Fundamental Arc Construction. For fixed auxiliary parameters $(\alpha,\varepsilon)$, fundamental arcs are solutions of a first-order ODE system in the state $S(s)=(x(s),y(s),\theta(s),k(s),v(s))$: $$\begin{cases} x'=-\sin\theta, \ y'=\cos\theta, \ \theta'=k, \ k'=v, \ v'=-\tfrac12\,k^3-\alpha\Bigl(\cos\theta+\varepsilon(x\cos\theta+y\sin\theta)\Bigr) \end{cases}$$ subject to boundary conditions ensuring orthogonality and vanishing derivative of curvature at the endpoints. At $(\alpha,\varepsilon)=(0,0)$, this yields one half of Euler’s rectangular elastica.

Step 2: Dihedral Gluing. The fundamental arc, once constructed, is extended by reflection across a ray through the center, producing a curve segment of length $2L$. If the angle between the rays at the endpoints is $2\overline\theta(\varepsilon)$ and for specific $\varepsilon$,

$\overline\theta(\varepsilon) = \frac{\pi}{m},$

then dihedral concatenation of $m$ such $2L$-arcs closes to a $C^\infty$ simple loop invariant under $D_m$ (dihedral symmetry of order $m$).

3. Existence Theorem and Key Properties

The principal result asserts the existence of a countable family of geometrically distinct, closed, $m$-fold dihedrally symmetric homothetic expanders for all $m>m_0$: $$\gamma^j_m: S^1 \to \mathbb{R}^2,\quad k_{ss}+\tfrac12\,k^3 = -\sigma\langle\gamma^j_m-c, \nu\rangle$$ with expansion rate $\sigma>0$. For $m\ne m'$, $\gamma^j_m$ and $\gamma^j_{m'}$ are inequivalent under plane similarities (Andrews et al., 29 Jan 2026).

Geometric properties include:

• Exact $m$-fold dihedral symmetry.
• The number of arms equals $m$; total turn angle $2\pi m$.
• Curvature $k(s)$ decreases strictly from $1$ to $-1$ on each fundamental arc, so tentacles exhibit no further inflection.
• All solutions are genuinely expanding; as $\varepsilon\to0$, the expansion rate satisfies $$\sigma(\varepsilon) = -\alpha'(0)\varepsilon^2 + O(\varepsilon^3) > 0$$.

4. Explicit Formulas and Bifurcation Structure

Base elastica solutions satisfy $k_0''+\tfrac12 k_0^3=0$, with $k_0(0)=1$, $k_0'(0)=0$, equivalent to selecting $k_0(s) = \operatorname{cn}(s/\sqrt2;m)$ (Jacobi elliptic function). The first integral,

$(k_0')^2+\tfrac14k_0^4=\tfrac14,$

characterizes the energy landscape.

For small $\varepsilon$, the seam-angle satisfies

$$\overline\theta(\varepsilon) = y_0(L_0)\,\varepsilon + O(\varepsilon^2),\quad y_0(L_0) = \int_0^{L_0}k_0(s)^2\,ds.$$

To leading order, $\varepsilon \approx \pi/(m\,y_0(L_0))$ ensures closure under dihedral gluing.

This construction yields a genuinely non-trivial, one-sided analytic branch of solutions emanating from the rectangular elastica, with a complex bifurcation structure. Unlike integrable cases (shrinking circle or Bernoulli lemniscate), these expanders are non-integrable and are believed to be dynamically unstable under the elastic flow.

5. Comparative Context: Other Flows and Related Families

Analogous self-similar families arise in the curve-diffusion and ideal-curve flows. For those dynamics, “epicyclic” shrinkers or expanders can be constructed via a similar perturbation approach coupled with dihedral gluing. However, classical shrinking solitons (e.g., the Bernoulli lemniscate) and the round circle represent integrable cases lacking the rich bifurcation and geometric complexity of the jellyfish expanders.

No larger symmetry can arise than the prescribed $D_m$, guaranteed by monotonicity of the seam-angle map and technical lemmas precluding the appearance of additional interior seams.

6. Outlook and Open Directions

Open research directions include:

• Rigorous classification of all homothetic solutions to the elastic flow.
• Dynamic stability and instability analysis for jellyfish expanders, as numerical evidence suggests instability.
• Construction and understanding of non-symmetric “one-sided” self-similar expanders observed in numerics.

The explicit, infinite family of jellyfish expanders reveals deep connections between geometric evolution equations, symmetry, and bifurcation phenomena, and motivates further study in higher-order geometric flows and their singularity structures (Andrews et al., 29 Jan 2026).