On 1d Quadratic Klein–Gordon Equations with a Potential and Symmetries

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“…The main result of this work is closely related to the recent work of Kairzhan-Pusateri [53] on codimension one full asymptotic stability of the soliton for (1.6) in the quartic case and to the recent works concerning the full asymptotic stability of kinks under odd perturbations by Delort-Masmoudi [30] for the 𝜙 4 model up to times 𝜀 −4+𝑐 with 0 < 𝑐 ≪ 1, by Germain-Pusateri [39] on double sine-Gordon models, see also Germain-Pusateri-Zhang [42], and by the authors [76] on the sine-Gordon model. See also Chen-Liu-Lu [11], Chen-Pusateri [12,13], Chen [9], and Léger-Pusateri [66,67].…”

“…(i) Using the distorted Fourier transform adapted to the Schrödinger operator −𝜕 2 𝑥 + 𝑉, see for instance [12,13,35,39,40,42,66,89,90,95]. (ii) Applying the wave operator associated with the Schrödinger operator −𝜕 2 𝑥 + 𝑉 to conjugate to the flat case, see for instance [27,30].…”

“…It appears that the adapted functional framework introduced in [39] could handle contributions of such singular quadratic source terms down to 𝑞(𝑥)𝑒 𝑖2𝑠 𝜀 2 𝑠 1+𝜇 , 𝜇 > 0, and obtain sharp 𝐿 ∞ 𝑥 decay at the rate 𝑡 − 1 2 and asymptotics. For odd perturbations of the kink in double sine-Gordon theories (in an appropriate range of the deformation parameter), see for instance [39,42], the associated linearized operator does not exhibit a threshold resonance, and one can propagate improved local decay of the solutions of type ‖⟨𝑥⟩ −1 𝑢(𝑡)‖ 𝐿 2 𝑥 ≲ ⟨𝑡⟩ −1+𝛿 𝜀, 0 < 𝛿 ≪ 1. Correspondingly, a quadratic source term of the following schematic form arises (1.31)…”