# Asymptotics for 1D Klein-Gordon Equations with Variable Coefficient Quadratic Nonlinearities

title={Asymptotics for 1D Klein-Gordon Equations with Variable Coefficient Quadratic Nonlinearities},
author={Hans Lindblad and Jonas L{\"u}hrmann and Avy Soffer},
journal={arXiv: Analysis of PDEs},
year={2020}
}
• Published 1 June 2020
• Mathematics, Physics
• arXiv: Analysis of PDEs
We initiate the study of the asymptotic behavior of small solutions to one-dimensional Klein-Gordon equations with variable coefficient quadratic nonlinearities. The main discovery in this work is a striking resonant interaction between specific spatial frequencies of the variable coefficient and the temporal oscillations of the solutions. In the resonant case a novel type of modified scattering behavior occurs that exhibits a logarithmic slow-down of the decay rate along certain rays. In the… Expand
4 Citations
On modified scattering for 1D quadratic Klein-Gordon equations with non-generic potentials
• Mathematics, Physics
• 2020
We consider the asymptotic behavior of small global-in-time solutions to a 1D KleinGordon equation with a spatially localized, variable coefficient quadratic nonlinearity and a nongeneric linearExpand
Sine-Gordon on a wormhole
• Physics, Mathematics
• Nonlinearity
• 2021
In an attempt to understand the soliton resolution conjecture, we consider the sine-Gordon equation on a spherically symmetric wormhole spacetime. We show that within each topological sector (indexedExpand
A Sufficient Condition for Asymptotic Stability of Kinks in General (1+1)-Scalar Field Models
• Mathematics, Physics
• 2020
We study stability properties of kinks for the (1+1)-dimensional nonlinear scalar field theory models \begin{aligned} \partial _t^2\phi -\partial _x^2\phi + W'(\phi ) = 0, \quad (t,x)\in \mathbbExpand Quadratic Klein-Gordon equations with a potential in one dimension • Mathematics, Physics • 2020 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 someExpand #### References SHOWING 1-10 OF 97 REFERENCES Normal forms and quadratic nonlinear Klein-Gordon equations, Comm • Pure Appl. Math • 1985 On the Asymptotic Behavior of Solutions to the Vlasov–Poisson System • Mathematics, Physics • 2020 We prove small data modified scattering for the Vlasov-Poisson system in dimension d=3 using a method inspired from dispersive analysis. In particular, we identify a simple asymptotic dynamicExpand Global existence for nonlinear wave equation In this work we propose a new approach for investigating the local and global existence of solutions of nonlinear wave equations. This approach gives new results. On the Landau damping • Physics, Mathematics • 2009 Going beyond the linearized study has been a longstanding problem in the theory of the Landau damping. In this paper we establish Landau damping for the nonlinear Vlasov equation, for any interactionExpand The Global Nonlinear Stability of the Minkowski Space. • Physics, Mathematics • 1994 The aim of this work is to provide a proof of the nonlinear gravitational stability of the Minkowski space-time. More precisely, the book offers a constructive proof of global, smooth solutions toExpand Decay and asymptotics for the 1D Klein-Gordon equation with variable coefficient cubic nonlinearities. • Mathematics, Physics • 2019 We obtain sharp decay estimates and asymptotics for small solutions to the one-dimensional Klein-Gordon equation with constant coefficient cubic and spatially localized, variable coefficient cubicExpand A Sufficient Condition for Asymptotic Stability of Kinks in General (1+1)-Scalar Field Models • Mathematics, Physics • 2020 We study stability properties of kinks for the (1+1)-dimensional nonlinear scalar field theory models\begin{aligned} \partial _t^2\phi -\partial _x^2\phi + W'(\phi ) = 0, \quad (t,x)\in \mathbbExpand
Decay and Asymptotics for the One-Dimensional Klein-Gordon Equation with Variable Coefficient Cubic Nonlinearities
• Computer Science, Mathematics
• SIAM J. Math. Anal.
• 2020
We obtain sharp decay estimates and asymptotics for small solutions to the one-dimensional Klein--Gordon equation with constant coefficient cubic and spatially localized, variable coefficient cubic...
Landau damping for analytic and Gevrey data
• Mathematics, Physics
• 2020
In this paper, we give an elementary proof of the nonlinear Landau damping for the Vlasov-Poisson system near Penrose stable equilibria on the torus $\mathbb{T}^d \times \mathbb{R}^d$ that was firstExpand
Long-time asymptotics and stability for the sine-Gordon equation
• Mathematics
• 2020
In this paper, we study the long-time dynamics and stability properties of the sine-Gordon equation $$f_{tt}-f_{xx}+\sin f=0.$$ Firstly, we use the nonlinear steepest descent for Riemann-HilbertExpand