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

@article{Lindblad2020AsymptoticsF1, 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} }

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 linear… Expand

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 (indexed… Expand

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 \mathbb… Expand

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 some… Expand

#### 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 dynamic… Expand

Global existence for nonlinear wave equation

- Physics
- 2010

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 interaction… Expand

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 to… Expand

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 cubic… Expand

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 \mathbb… Expand

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 first… Expand

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-Hilbert… Expand