We quantize the permanently bound state of a kink and a breather in the sine–Gordon model — the wobble — by extracting its moduli‑space symplectic form from the Hamiltonian inverse scattering transform (IST). The IST description yields a three‑soliton configuration with one real and one complex‑conjugate pair of eigenvalues; the restriction of the field‑theoretic symplectic form to the independent scattering data gives an additive expression. Imposing the bound‑state condition that the kink and breather share the same velocity and centre position reduces the symplectic form to the canonical product ₖ=dP da+d dI, with total momentum P=8\, (1+2) and internal action I=16. Deformation quantisation follows immediately. We explain why a direct geometric pullback fails for this permanently bound configuration and why the B\"acklund transformation does not provide an independent geometric alternative. Finally, we show how the spatial integral in the symplectic form can be rigorously evaluated by the loop‑group method of Beggs--Johnson, connecting the IST formula back to the field configuration. This work completes the wobble sector of the geometric subsystem quantisation programme and provides strong evidence that the programme extends to all soliton and bound‑state sectors of the sine–Gordon equation.
Timmermans et al. (Wed,) studied this question.
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