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 on Strichartz or virial estimates and is therefore able to treat low power nonlinearities (hence also non-localized solitons) and capture the global (in space and time) behavior of solutions. More specifically, we consider quadratic nonlinear Klein-Gordon equations with a potential in one space dimension. The potential is assumed to be regular, decaying, and either generic or exceptional (with some additional parity assumptions). Assuming 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 implications for the asymptotic stability of solitons, or topological solitons, for a variety of 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 ${\varphi }^4$ problem, we obtain asymptotic stability of the kink (with respect to odd perturbations) when the coupling to the internal mode appearing in the linearization around it is neglected. Our results also go beyond these examples since our approach allows for the presence of a fully coherent phenomenon 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.

152 pages. v2: abstract revised. Minor changes in the presentation. Some typos corrected. Added an application to the asymptotic stability of kinks for the double sine-Gordon equation, inspired by the work of Kowalczyk, Martel, Muñoz and Van Den Bosch (arXiv:2008.01276). v3: Expanded parts of introduction and minor corrections

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Cited by in corpus (7)

• Asymptotics for 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities
• On the asymptotic stability of the sine-Gordon kink in the energy space
• On $1$d quadratic Klein-Gordon equations with a potential and symmetries
• Asymptotic stability near the soliton for quartic Klein-Gordon in 1D
• Inviscid damping of monotone shear flows for 2D inhomogeneous Euler equation with non-constant density in a finite channel
• Long-time dynamics of small solutions to 1$d$ cubic nonlinear Schrödinger equations with a trapping potential
• Asymptotic stability of small standing solitary waves of the one-dimensional cubic-quintic Schrödinger equation