Dynamics of the nonlinear Klein–Gordon equation in the nonrelativistic limit

@article{Pasquali2017DynamicsOT,
  title={Dynamics of the nonlinear Klein–Gordon equation in the nonrelativistic limit},
  author={Stefano Pasquali},
  journal={Annali di Matematica Pura ed Applicata (1923 -)},
  year={2017},
  volume={198},
  pages={903-972}
}
  • S. Pasquali
  • Published 5 March 2017
  • Mathematics
  • Annali di Matematica Pura ed Applicata (1923 -)
We study the nonlinear Klein–Gordon (NLKG) equation on a manifold M in the nonrelativistic limit, namely as the speed of light c tends to infinity. In particular, we consider a higher-order normalized approximation of NLKG (which corresponds to the NLS at order $$r=1$$r=1) and prove that when M is a smooth compact manifold or $$\mathbb {R}^d$$Rd, the solution of the approximating equation approximates the solution of the NLKG locally uniformly in time. When $$M=\mathbb {R}^d$$M=Rd, $$d \ge 2$$d… 

On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime

This work concerns the time averaging techniques for the nonlinear Klein-Gordon (KG) equation in the nonrelativistic limit regime which have recently gained a lot of attention in numerical analysis.

Complex valued semi-linear heat equations in super-critical spaces $$E^s_\sigma $$

We consider the Cauchy problem for the complex valued semi-linear heat equation where m ≥ 2 is an integer and the initial data belong to super-critical spaces E sσ for which the norms are defined by

Almost global existence for the nonlinear Klein-Gordon equation in the nonrelativistic limit

We study the one-dimensional nonlinear Klein-Gordon equation with a convolution potential, and we prove that solutions with small Hs norm remain small for long times. The result is uniform with

Metastability phenomena in two-dimensional rectangular lattices with nearest-neighbour interaction

We study analytically the dynamics of two-dimensional rectangular lattices with periodic boundary conditions. We consider anisotropic initial data supported on one low-frequency Fourier mode. We show

On global behaviour of classical effective field theories

We continue the rigorous study of classical effective field theories (EFTs) that was recently initiated in the work of Reall and Warnick [RW22]. We study a system with one light and one heavy field with

References

SHOWING 1-10 OF 101 REFERENCES

Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations

Abstract. We prove a Nekhoroshev type result [26,27] for the nonlinear Schrödinger equation \begin{eqnarray} iu_t=-u_{xx}-mu-u \varphi (|u|^2) , \end{eqnarray} with vanishing or periodic boundary

On the Semirelativistic Hartree-Type Equation

The global Cauchy problem and scattering problem for the semi-relativistic equation in R with nonlocal nonlinearity and asymptotic behavior of solutions as the mass tends to zero and infinity is studied.

A uniformly accurate (UA) multiscale time integrator Fourier pseudospectral method for the Klein–Gordon–Schrödinger equations in the nonrelativistic limit regime

The proposed multiscale time integrator Fourier pseudospectral method for solving the Klein–Gordon–Schrödinger equations in the nonrelativistic limit regime converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate.

On Scattering of Solitons for the Klein–Gordon Equation Coupled to a Particle

We establish the long time soliton asymptotics for the translation invariant nonlinear system consisting of the Klein–Gordon equation coupled to a charged relativistic particle. The coupled system

Strong Instability of Standing Waves for the Nonlinear Klein-Gordon Equation and the Klein-Gordon-Zakharov System

For the case when the frequency is equal to the critical frequency, the strong instability of ground state standing waves is proved for all radially symmetric standing waves $e^{i\omega_c t}\varphi(x)$.

Asymptotic preserving schemes for the Klein–Gordon equation in the non-relativistic limit regime

An asymptotic expansion for the solution of the Klein–Gordon equation with respect to the small parameter depending on the inverse of the square of the speed of light is constructed.

Nonrelativistic limit of Klein-Gordon-Maxwell to Schrödinger-Poisson

<abstract abstract-type="TeX"><p>We prove that in the nonrelativistic limit <i>c</i> → ∞, where <i>c</i> is the speed of light, solutions of the Klein-Gordon-Maxwell system on [inline-graphic

Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations

Abstract. We study the nonrelativistic limit of the Cauchy problem for the nonlinear Klein-Gordon equation and prove that any finite energy solution converges to the corresponding solution of the

The nonlinear Schrödinger equation as a resonant normal form

Averaging theory is used to study the dynamics of dispersive equations taking the nonlinear Klein Gordon equation on the line as a model problem: For approximatively monochromatic initial data of
...