Multisoliton solutions for equivariant wave maps on a $2+1$ dimensional wormhole

TL;DR

The paper investigates the long-time behavior of equivariant wave maps from a $2+1$-dimensional wormhole to $S^2$, focusing on asymptotically static multi-kink/anti-kink chains. It combines a collective coordinates reduction to finite-dimensional ODEs with direct PDE simulations to predict and verify the dynamics of $N$-chain configurations, particularly for $N=2$ and $N=3$. The main contributions are the explicit zero-energy asymptotically static solutions for the reduced ODEs, the identification of unstable directions and codimension of stable manifolds, and the PDE validation that the threshold dynamics observed numerically align with the ODE predictions, including quantified expansion rates and energy radiative losses. This work advances the taxonomy of end states for wormhole-based wave maps, highlighting threshold phenomena and the role of radiation in the decay or preservation of multi-soliton chains.

Abstract

We study equivariant wave maps from the $2+1$ dimensional wormhole to the 2-sphere. This model has explicit harmonic map solutions which, in suitable coordinates, have the form of the sine-Gordon kinks/anti-kinks. We conjecture that there exist asymptotically static chains of $N\geq 2$ alternating kinks and anti-kinks whose subsequent rates of expansion increase in geometric progression as $t\rightarrow \infty$. Our argument employs the method of collective coordinates to derive effective finite-dimensional ODE models for the asymptotic dynamics of $N$-chains. For $N=2,3$ the predictions of these effective models are verified by direct PDE computations which demonstrate that the $N$-chains lie at the threshold of kink-anti-kink annihilation.

Figures (6)

• Figure 1: Snapshots of even solutions for marginally subcritical initial data \ref{['id1']} with $b=b_{*} - 6\times10^{-16}$ (in blue) and marginally supercritical data with $b=b_{*} + 6\times10^{-16}$ (in red).
• Figure 2: The Bondi energy of sub- and supercritical solutions from Fig. \ref{['fig:N2evolution']}. For intermediate times the energy is slightly above the energy $\mathcal{E}=8$ of the asymptotically static kink-anti-kink pair. For supercritical solutions the energy tends slowly to $\mathcal{E}=8$ for $s\rightarrow \infty$. For subcritical solutions the energy is rapidly radiated away during the collision of kink and anti-kink and tends to $0$ (see the inset).
• Figure 3: Position of the kink $c_1(t)$ for sub- and supercritical from Fig. \ref{['fig:N2evolution']}. The gray dashed curve is the fit of formula \ref{['fitN2']} to the numerical data for intermediate times. For later times the solutions deviate from criticality along the unstable directions.
• Figure 4: Snapshots of odd solutions for marginally subcritical initial data \ref{['id2']} with $b=b_{*} - 6\times10^{-16}$ (in blue) and marginally supercritical data with $b=b_{*} + 6\times10^{-16}$ (in red).
• Figure 5: The Bondi energy of sub- and supercritical solutions from Fig. \ref{['fig:N3evolution']}. For intermediate times the energy is slightly above the energy $\mathcal{E}=12$ of the asymptotically static kink-anti-kink-kink configuration. For supercritical solutions the energy tends to $\mathcal{E}=12$ for $s\rightarrow \infty$. For subcritical solutions the energy is rapidly radiated away during the collision of kinks with the anti-kink (see the inset) and tends slowly to $\mathcal{E}=4$ for $s\rightarrow \infty$.