Breathers and nonlinear waves on open vortex filaments in the nonrelativistic Abelian Higgs model

We demonstrate the existence of various nonlinear waves along open vortex filaments in the nonrelativistic Abelian Higgs model. These include nonlinear waves, which give the birth and death of helical filaments, as well breathers, which modify the amplitude of helical filaments periodically in time. Importantly, we demonstrate that some of the dynamics are quite particular to the equation of motion for vortex filaments under the nonrelativistic Abelian Higgs model, giving rise to vortex filament solutions not possible under the hydrodynamic vortex filament equation of motion (the local induction approximation) or generalizations used to study the dynamics of quantized vortex filaments in ${\mathrm{He}}^4$.

Figure 1

Plot of the Kelvin wave translational velocity ${\beta }_1$ as a function of the rotational velocity ${\omega }_1$, for various fixed values of wave number $k$.

Figure 2

Plot of the Kelvin wave translational velocity ${\beta }_1$ as a function of the wave number $k$, for various fixed values of the rotational velocity ${\omega }_1$.

Figure 3

Plot of the growth and decay of a helical vortex filament under the solution representation (33) for $k=1$, ${\beta }_1=-1$, ${\beta }_0=0$, and ${\omega }_0=0$. Time values taken are (left to right) $t=-5$, $-2$, 0, 2, 5.

Figure 4

Plot of the breather vortex filament under the solution representation (37) for $k=1$, $a=1/2$, ${\beta }_1=-1/8$, ${\beta }_0=0$, and ${\omega }_0=0$. Time values taken are (left to right) $t=0$, 3.2, 6.4, 10.0, 13.4.