Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userStability and instability of the standing waves for the Klein-Gordon-Zakharov system in one space dimension
134
0
0.0
(
0
)
Authors
Silu Yin
Ask ChatGPT about the research
No Arabic abstract
The orbital instability of standing waves for the Klein-Gordon-Zakharov system has been established in two and three space dimensions under radially symmetric condition, see Ohta-Todorova (SIAM J. Math. Anal. 2007). In the one space dimensional case, for the non-degenerate situation, we first check that the Klein-Gordon-Zakharov system satisfies Grillakis-Shatah-Strauss assumptions on the stability and instability theorems for abstract Hamiltonian systems, see Grillakis-Shatah-Strauss (J. Funct. Anal. 1987). As to the degenerate case that the frequency $|omega|=1/sqrt{2}$, we follow Wu (ArXiv: 1705.04216, 2017) to describe the instability of the standing waves for the Klein-Gordon-Zakharov system, by using the modulation argument combining with the virial identity. For this purpose, we establish a modified virial identity to overcome several troublesome terms left in the traditional virial identity.
rate research
Read More
We investigate the orbital stability and instability of standing waves for two classes of Klein-Gordon equations in the semi-classical regime.
We consider a system of two coupled non-linear Klein-Gordon equations. We show the existence of standing waves solutions and the existence of a Lyapunov function for the ground state.
We prove definitive results on the global stability of the flat space among solutions of the Einstein-Klein-Gordon system. Our main theorems in this monograph include: (1) A proof of global regularity (in wave coordinates) of solutions of the Einstein-Klein-Gordon coupled system, in the case of small, smooth, and localized perturbations of the stationary Minkowski solution; (2) Precise asymptotics of the metric components and the Klein-Gordon field as the time goes to infinity, including the construction of modified (nonlinear) scattering profiles and quantitative bounds for convergence; (3) Classical estimates on the solutions at null and timelike infinity, such as bounds on the metric components, weak peeling estimates of the Riemann curvature tensor, ADM and Bondi energy identities and estimates, and asymptotic description of null and timelike geodesics.
We consider the nonlinear Klein-Gordon equation in $R^d$. We call multi-solitary waves a solution behaving at large time as a sum of boosted standing waves. Our main result is the existence of such multi-solitary waves, provided the composing boosted standing waves are stable. It is obtained by solving the equation backward in time around a sequence of approximate multi-solitary waves and showing convergence to a solution with the desired property. The main ingredients of the proof are finite speed of propagation, variational characterizations of the profiles, modulation theory and energy estimates.
We are going to study the standing waves for a generalized Choquard equation with potential: $$ -ipartial_t u-Delta u+V(x)u=(|x|^{-mu}ast|u|^p)|u|^{p-2}u, hbox{in} mathbb{R}timesmathbb{R}^3, $$ where $V(x)$ is a real function, $0<mu<3$, $2-mu/3<p<6-mu$ and $ast$ stands for convolution. Under suitable assumptions on the potential and appropriate frequency $omega$ , the stability and instability of the standing waves $u=e^{i omega t}varphi(x)$ are investigated .
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
