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

Filip Novkoski (1) , Eric Falcon (1) , Chi-Tuong Pham (2, 3)

1 MSC - Matière et Systèmes Complexes
2 COMET - COuplages Multiphysiques Et Transferts - LISN
3 LISN - Laboratoire Interdisciplinaire des Sciences du Numérique

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.

Mots clés: periodic sine-Gordon; direct scattering transform; Soliton; Kink; Breather; Algorithm; Scattering matrix; Integral system

Domaines: Systèmes Solubles et Intégrables [nlin.SI]; Formation de Structures et Solitons [nlin.PS]; Physique [physics]

Filip Novkoski, Eric Falcon, Chi-Tuong Pham. A numerical direct scattering method for the periodic sine-Gordon equation. The European Physical Journal Plus, 2023, 138, pp.1146. ⟨10.1140/epjp/s13360-023-04706-7⟩. ⟨hal-04338254⟩