A numerical direct scattering method for the periodic sine-Gordon equation

Abstract

We propose a procedure for computing the direct scattering transform of the periodic sine-Gordon equation. This procedure, previously used within the periodic Korteweg–de Vries equation framework, is implemented for the case of the sine-Gordon equation and is validated numerically. In particular, we show that this algorithm works well with signals involving topological solitons, such as kink or anti-kink solitons, but also for non-topological solitons, such as breathers. It also has the ability to distinguish between these different solutions of the sine-Gordon equation within the complex plane of the eigenvalue spectrum of the scattering problem. The complex trace of the scattering matrix is made numerically accessible, and the influence of breathers on the latter is highlighted. Finally, periodic solutions of the sine-Gordon equation and their spectral signatures are explored in both the large-amplitude (cnoidal-like waves) and low-amplitude (radiative modes) limits.

Appendices

Appendix A

As a comparison with the results on the different solitons and periodic solutions to the pSG equation discussed in the main text, we here include the half-trace obtained for a sine wave of amplitude $A=0.01$, with $k=2$ and $\omega =1$ on a domain of length $L=60$. The real and imaginary parts of the trace are shown in Fig. 10 for the negative real part of the energy plane, where we observe that there is no pinching. In Fig. 11 we show the same quantity for the positive real part of the energy plane.

Fig. 10

a Real part of the trace of ${\textbf{M}}(E)$ in the complex plane of energy, for negative $\Re (E)$, for a low-amplitude sine wave. b Imaginary part of the trace of ${\textbf{M}}(E)$ for the same wave. Logscale colorbar

Fig. 11

a Real part of the trace of ${\textbf{M}}(E)$ in the complex plane of energy, for positive $\Re (E)$, for a low-amplitude sine wave. b Imaginary part of the trace of ${\textbf{M}}(E)$ for the same wave. Logscale colorbar

Appendix B

Fig. 12

a The eigenvalues of the truncated kink with energy $E=1$ for different values of the domain size. b Eigenvalues of the truncated kink on a domain with fixed $L=20$ for different values of the discretization $\Delta x$

In this appendix, we discuss how the domain size L and the spatial discretization $\Delta x$ affect the eigenvalues of a single infinite-line truncated kink soliton. We start varying L while keeping $\Delta x$ fixed at 0.02. Its energy has been chosen to be $E=1$. We can see in Fig. 12a that as the domain size L is increased, the results become more accurate. This is due to the fact that the errors due to periodicity (i.e. the mismatch at the domain ends) decreases. In Fig. 12b, we have kept the domain size value at $L=20$ and changed the discretization $\Delta x$. As expected, by decreasing the value of $\Delta x$, the energies converge to a single value.