Large data scattering for NLKG on waveguide ℝd × 𝕋

2020 ◽  
Vol 17 (02) ◽  
pp. 355-394
Author(s):  
Luigi Forcella ◽  
Lysianne Hari
Keyword(s):  
Product Space ◽  
Gordon Equation ◽  
Large Data ◽  
Decomposition Theorem ◽  
Concentration Compactness ◽  
Flat Torus ◽  
Subcritical Case ◽  
One Dimensional ◽  
Klein Gordon ◽  
The One

We consider the pure-power defocusing nonlinear Klein–Gordon equation, in the [Formula: see text]-subcritical case, posed on the product space [Formula: see text], where [Formula: see text] is the one-dimensional flat torus. In this framework, we prove that scattering holds for any initial data belonging to the energy space [Formula: see text] for [Formula: see text]. The strategy consists in proving a suitable profile decomposition theorem on the whole manifold to pursue a concentration-compactness and rigidity method along with the proofs of (global in time) Strichartz estimates.

scholarly journals Bound states of a Klein–Gordon particle in the presence of a smooth potential well

2020 ◽  
Vol 35 (23) ◽  
pp. 2050140
Author(s):  
Eduardo López ◽  
Clara Rojas
Keyword(s):  
Bound States ◽  
Gordon Equation ◽  
Bound State ◽  
Potential Well ◽  
One Dimensional ◽  
Smooth Potential ◽  
Klein Gordon ◽  
The One ◽  
Potential Parameters ◽  
Klein Gordon Equation

We solve the one-dimensional time-independent Klein–Gordon equation in the presence of a smooth potential well. The bound state solutions are given in terms of the Whittaker [Formula: see text] function, and the antiparticle bound state is discussed in terms of potential parameters.

scholarly journals Scattering of a Klein–Gordon particle by a smooth barrier

2020 ◽  
Vol 98 (10) ◽  
pp. 939-943
Author(s):  
Eduardo López ◽  
Clara Rojas
Keyword(s):  
Potential Barrier ◽  
Gordon Equation ◽  
Reflection And Transmission ◽  
One Dimensional ◽  
Transmission Coefficients ◽  
Smoothness Parameter ◽  
Klein Gordon ◽  
Transmission Resonances ◽  
The One ◽  
Klein Gordon Equation

We present a study of the one-dimensional Klein–Gordon equation by a smooth barrier. The scattering solutions are given in terms of the Whittaker Mκ,μ(x) function. The reflection and transmission coefficients are calculated in terms of the energy, the height, and the smoothness of the potential barrier. For any value of the smoothness parameter we observed transmission resonances.

scholarly journals Scattering for the one-dimensional Klein–Gordon equation with exponential nonlinearity

2020 ◽  
Vol 17 (02) ◽  
pp. 295-354
Author(s):  
Masahiro Ikeda ◽  
Takahisa Inui ◽  
Mamoru Okamoto
Keyword(s):  
Gordon Equation ◽  
Spatial Dimension ◽  
Asymptotic Behavior Of Solutions ◽  
Small Initial Data ◽  
One Dimensional ◽  
Exponential Nonlinearity ◽  
The Cauchy Problem ◽  
Klein Gordon ◽  
The One ◽  
Klein Gordon Equation

We consider the asymptotic behavior of solutions to the Cauchy problem for the defocusing nonlinear Klein–Gordon equation (NLKG) with exponential nonlinearity in the one spatial dimension with data in the energy space [Formula: see text]. We prove that any energy solution has a global bound of the [Formula: see text] space-time norm, and hence, scatters in [Formula: see text] as [Formula: see text]. The proof is based on the argument by Killip–Stovall–Visan (Trans. Amer. Math. Soc. 364(3) (2012) 1571–1631). However, since well-posedness in [Formula: see text] for NLKG with the exponential nonlinearity holds only for small initial data, we use the [Formula: see text]-norm for some [Formula: see text] instead of the [Formula: see text]-norm, where [Formula: see text] denotes the [Formula: see text]th order [Formula: see text]-based Sobolev space.

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