Quadratic Klein-Gordon equations with a potential in one dimension

Abstract

This paper proposes a fairly general new point of view on the question of asymptotic stability of (topological) solitons. Our approach is based on the use of the distorted Fourier transform at the nonlinear level; it does not rely only on Strichartz or virial estimates and is therefore able to treat low-power nonlinearities (hence also nonlocalised solitons) and capture the global (in space and time) behaviour of solutions. More specifically, we consider quadratic nonlinear Klein-Gordon equations with a regular and decaying potential in one space dimension. Additional assumptions are made so that the distorted Fourier transform of the solution vanishes at zero frequency. Assuming also that the associated Schrödinger operator has no negative eigenvalues, we obtain global-in-time bounds, including sharp pointwise decay and modified asymptotics, for small solutions. These results have some direct applications to the asymptotic stability of (topological) solitons, as well as several other potential applications to a variety of related problems. For instance, we obtain full asymptotic stability of kinks with respect to odd perturbations for the double sine-Gordon problem (in an appropriate range of the deformation parameter). For the $\phi ^4$ problem, we obtain asymptotic stability for small odd solutions, provided the nonlinearity is projected on the continuous spectrum. Our results also go beyond these examples since our framework allows for the presence of a fully coherent phenomenon (a space-time resonance) at the level of quadratic interactions, which creates a degeneracy in distorted Fourier space. We devise a suitable framework that incorporates this and use multilinear harmonic analysis in the distorted setting to control all nonlinear interactions.

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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].…”

Asymptotic stability of solitary waves for the 1D cubic-quintic Schrödinger equation with no internal mode

“…We can go a little further and show, with the same proof, that there does not exist resonances under the same hypotheses, in the sense below. See [10] for similar arguments on the Klein-Gordon equation.…”

“…The literature on asymptotic stability is abondant. For wave-type equations, we refer to [13,16,26,27,28,34], which contain some of the most advanced results in different directions. Restricting now to Schrödinger-type models, we quote a few surveys [10,11,22,45] and some of the most recent articles in various settings [6,17,18,25,37].…”

Dynamics of kink clusters for scalar fields in dimension 1+1

“…Let us stress again that the main object of our study are solutions approaching multisoliton configurations in the strong energy norm, in other words we address the question of interaction of solitons in the absence of radiation. Allowing for a radiation term seems to be currently out of reach, the question of the asymptotic stability of the kink being still unresolved, see for example [9,24,13,16,3,28] for recent results on this and related problems.…”