Geometric Transformations on Null Curves in the Anti-de Sitter 3-Space

TL;DR

This work constructs a geometric $\mathcal{T}$-transform on future-directed null curves in AdS that yields the Bäcklund transformation for the KdV bending $\kappa$ and proves a permutability property. The transform is formulated through a Riccati equation with parameter $\xi$ and two regimes distinguished by $\chi=0$ or $\chi\neq0$, providing explicit formulas for the transformed curve and bending. It then furnishes a geometric realization of the KdV BT via the LIEN flow, showing how the evolution of null curves by a third-order flow corresponds to KdV dynamics under the transform. Finally, the paper offers a constructive method for the $\chi=0$ case with constant bending, including extended frames, an explicit transforming function, and a procedure to generate higher-order transforms, with connections to holomorphic null curves and constant mean curvature surfaces in related geometries.

Abstract

We provide a geometric transformation on null curves in the anti-de Sitter 3-space (AdS) which induces the Bäcklund transformation for the KdV equation. In addition, we show that this geometric transformation satisfies a suitable permutability theorem. We also illustrate how to implement it when the original null curve has constant bending.

Figures (9)

• Figure 1: Left: The closed null curve $\gamma$ with constant bending $\kappa_{7,3}$ (see Example \ref{['example']}). This curve represents a torus knot of type $(2,5)$. Right: The associated pair of star-shaped cousins $(\eta_+,\eta_-)$ in blue and red, respectively.
• Figure 2: The star-shaped curve $\eta_+$ of Figure \ref{['constant']}, in blue and dotted in yellow, and the $\mathcal{T}$-transform $\widetilde{\eta}_+$ of $\eta_+$ in plain blue. The green triangles, which have the same area, are the ones with vertices $O$ (in black), $\eta_+(s)$ (in yellow) and $\widetilde{\eta}_+(s)$ (in blue). For each figure the triangle is shown at different values of the parameter $s\in J$.
• Figure 3: The star-shaped curve $\eta_-$ of Figure \ref{['constant']}, in red and dotted in yellow, and the $\mathcal{T}$-transform $\widetilde{\eta}_-$ of $\eta_-$ in plain red. The green triangles, which have the same area, are the ones with vertices $O$ (in black), $\eta_-(s)$ (in yellow) and $\widetilde{\eta}_-(s)$ (in red). For each figure the triangle is shown at different values of the parameter $s\in J$.
• Figure 4: Left: The $\mathcal{T}$-transform (for $\chi=0$) $\widetilde{\gamma}=\mathcal{T}_{\xi,f}(\gamma)$ of the null curve $\gamma$ with constant bending $\kappa_{7,3}$ (see Figure \ref{['constant']}). The parameter $\xi\neq 0$ of the $\mathcal{T}$-transform is $\xi=1.01$ and the transforming function $f$ is the solution of \ref{['Riccati']} with initial condition $0.1$. Right: The associated pair of star-shaped cousins $(\widetilde{\eta}_+,\widetilde{\eta}_-)$ in plain blue and red, respectively. The curves dotted in yellow are the corresponding $\mathcal{T}$-transforms, namely, $\eta_+$ and $\eta_-$ (cf. Figures \ref{['transform+']} and \ref{['transform-']}).
• Figure 5: Left: The transforming function $f_{m,n}$ of $\kappa_{m,n}$ with spectral parameter $\lambda_p$ and initial condition ${c=0}$. Right: The Bäcklund transform $\widetilde{\kappa}_{m,n}$ of $\kappa_{m,n}$ with spectral parameter $\lambda_p$ and transforming function $f_{m,n}$. The function $\widetilde{\kappa}_{m,n}$ represents a $1$-soliton solution of the KdV equation \ref{['KdV2']}. In both cases, $m=4$, $n=1$ and $p=1.4$.

Theorems & Definitions (24)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Theorem 2.4: MP
• Remark 2.5
• Example 2.6
• Definition 3.1
• Remark 3.2
• Definition 3.3
• Remark 3.4