Wave maps on a wormhole

We consider equivariant wave maps from a fixed wormhole spacetime into the 3-sphere. This toy model is designed for gaining insight into the dissipation-by-dispersion phenomena, in particular the soliton resolution conjecture. We first prove that for each topological degree of the map there exists a unique static solution (harmonic map) that is linearly stable. Then, using the hyperboloidal formulation of the initial value problem, we give numerical evidence that every solution starting from smooth initial data of any topological degree evolves asymptotically to the harmonic map of the same degree. The late-time asymptotics of this relaxation process is described in detail.

Figure 1

The profiles of the harmonic maps $U_n(r)$ for $\ell =1$ and $n=1,2$.

Figure 2

The potentials $V_n(r)$ for $\ell =1,2$ and $n=1$ (upper plot) and $n=2$ (lower plot).

Figure 3

The series of snapshots of $F(s,y)$ for $\ell =1$ and sample initial data (18) of degree 1 ($n=1$). The solution converges pointwise to the harmonic map $F_1(y)$ (red curve).

Figure 4

The evolution of the solution from Fig. 3 is observed at a sample interior point $y_0=\pi /4$. The quasinormal ringdown (for early times) and the polynomial tail (for late times) are clearly seen. Because of the slow falloff of initial data at infinity, the tail decays by one power more slowly than in Eq. (20).

Figure 5

Evolution of initial data (18) with $n=2$ and two values of $\ell =1,2$. For large perturbations of harmonic maps, the ringdown is preceded by nonlinear oscillations that are due to the nonlinear coupling between the quasinormal modes.

Figure 6

The ringdown for $\ell =2$ and $n=2$ initial data of the form $g(y)=F_2(y)+(\tanh y+{10}^{-5})\exp (-\frac{{\tan }^{2}y}{4}),h(y)=0$. Initially, the ringdown is governed by the first odd quasinormal mode, which is rapidly decaying. For later times, the fundamental, slowly decaying, even quasinormal mode takes over.

Figure 7

The log-log plot of the evolution of compactly supported large perturbations of harmonic maps of the form $g(y)=F_n(y)+10\exp (-\frac{{\tan }^{2}y}{4}),h(y)=0$ for $\ell =1$. For late times, ${\partial }_{s}F(s,\pi /4)$ decays as $s^{-6}$ independently of $n$, in agreement with Eq. (20).

Figure 8

The log-log plot of the coefficients $c_k^{+}(s)$ for $\ell =1$ and $n=0$ initial data $g(y)=\exp (-\frac{{\tan }^{2}y}{4})\tanh (y),h(y)=0$. The dashed lines depict the leading-order behavior (24) with $a_0=0$ and the fitted constants $A_1=1.458$, $C_4=0.365$.