ترغب بنشر مسار تعليمي؟ اضغط هنا

A note on small periodic solutions of discrete nonlinear Klein-Gordon equations

154   0   0.0 ( 0 )
 نشر من قبل Masaya Maeda
 تاريخ النشر 2016
  مجال البحث
والبحث باللغة English
 تأليف Masaya Maeda




اسأل ChatGPT حول البحث

In this note, we consider discrete nonlinear Klein-Gordon equations with potential. By the pioneering work of Sigal, it is known that for the continuous nonlinear Klein-Gordon equation, no small time periodic solution exists generically. However, for the discrete nonlinear Klein-Gordon equations, we show that there exist small time periodic solutions.

قيم البحث

اقرأ أيضاً

Breathers are nontrivial time-periodic and spatially localized solutions of nonlinear dispersive partial differential equations (PDEs). Families of breathers have been found for certain integrable PDEs but are believed to be rare in non-integrable on es such as nonlinear Klein-Gordon equations. In this paper we consider semilinear Klein-Gordon equations and prove that single bump small amplitude breathers do not exist for generic analytic odd nonlinearities. Breathers with small amplitude can exist only when its temporal frequency is close to be resonant with the Klein-Gordon dispersion relation. For these frequencies, we identify the leading order term in the exponentially small (with respect to the small amplitude) obstruction to the existence of such small breathers in terms of the so-called Stokes constant. We also construct generalized breathers, which are periodic in time and spatially localized solutions up to exponentially small tails. We rely on the spatial dynamics approach where breathers can be seen as homoclinic orbits. The birth of such small homoclinics is analyzed via a singular perturbation setting where a Bogdanov-Takens bifurcation is coupled to infinitely many rapidly oscillatory directions. The leading order term of the exponentially small splitting between the stable/unstable invariant manifolds is obtained through a careful analysis of the analytic continuation of their parameterizations. This requires the study of another limit equation in the complexified evolution variable, the so-called inner equation.
90 - Shiwu Yang 2015
It is known that the Maxwell-Klein-Gordon equations in $mathbb{R}^{3+1}$ admit global solutions with finite energy data. In this paper, we present a new approach to study the asymptotic behavior of these global solutions. We show the quantitative ene rgy flux decay of the solutions with data merely bounded in some weighted energy space. We also establish an integrated local energy decay and a hierarchy of $r$-weighted energy decay. The results in particular hold in the presence of large total charge. This is the first result to give a complete and precise description of the global behavior of large nonlinear charged scalar fields.
We describe the long time behavior of small non-smooth solutions to the nonlinear Klein-Gordon equations on the sphere S^2. More precisely, we prove that the low harmonic energies (also called super-actions) are almost preserved for times of order $e psilon$^--r , where r >> 1 is an arbitrarily large number and $epsilon$ << 1 is the norm of the initial datum in the energy space H^1 x L^2. Roughly speaking, it means that, in order to exchange energy, modes have to oscillate at the same frequency. The proof relies on new multilinear estimates on Hamiltonian vector fields to put the system in Birkhoff normal form. They are derived from new probabilistic bounds on products of Laplace eigenfunctions that we obtain using Levys concentration inequality.
112 - Shiwu Yang 2015
It has been shown in the authors companion paper that solutions of Maxwell-Klein-Gordon equations in $mathbb{R}^{3+1}$ possess some form of global strong decay properties with data bounded in some weighted energy space. In this paper, we prove pointw ise decay estimates for the solutions for the case when the initial data are merely small on the scalar field but can be arbitrarily large on the Maxwell field. This extends the previous result of Lindblad-Sterbenz cite{LindbladMKG}, in which smallness was assumed both for the scalar field and the Maxwell field.
250 - Raphael C^ote , Xu Yuan 2021
We consider the nonlinear damped Klein-Gordon equation [ partial_{tt}u+2alphapartial_{t}u-Delta u+u-|u|^{p-1}u=0 quad text{on} [0,infty)times mathbb{R}^N ] with $alpha>0$, $2 le Nle 5$ and energy subcritical exponents $p>2$. We study the behavior o f solutions for which it is supposed that only one nonlinear object appears asymptotically for large times, at least for a sequence of times. We first prove that the nonlinear object is necessarily a bound state. Next, we show that when the nonlinear object is a non-degenerate state or a degenerate excited state satisfying a simplicity condition, the convergence holds for all positive times, with an exponential or algebraic rate respectively. Last, we provide an example where the solution converges exactly at the rate $t^{-1}$ to the excited state.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا