scholarly journals Construction of excited multi-solitons for the 5D energy-critical wave equation

2021 ◽  
Vol 18 (02) ◽  
pp. 397-434
Author(s):  
Xu Yuan
Keyword(s):  
Wave Equation ◽  
Excited State ◽  
Energy Method ◽  
Gordon Equation ◽  
Travelling Waves ◽  
Existence Proof ◽  
Klein Gordon ◽  
Linearized Operator ◽  
Klein Gordon Equation

For the 5D energy-critical wave equation, we construct excited [Formula: see text]-solitons with collinear speeds, i.e. solutions [Formula: see text] of the equation such that [Formula: see text] where for [Formula: see text], [Formula: see text] is the Lorentz transform of a non-degenerate and sufficiently decaying excited state, each with different but collinear speeds. The existence proof follows the ideas of Martel–Merle [Construction of multi-solitons for the energy-critical wave equation in dimension 5, Arch. Ration. Mech. Anal. 222(3) (2016) 1113–1160] and Côte–Martel [Multi-travelling waves for the nonlinear Klein–Gordon equation, Trans. Amer. Math. Soc. 370(10) (2018) 7461–7487] developed for the energy-critical wave and nonlinear Klein–Gordon equations. In particular, we rely on an energy method and on a general coercivity property for the linearized operator.

scholarly journals Symmetry analysis of the Klein–Gordon equation in Bianchi I spacetimes

2015 ◽  
Vol 12 (03) ◽  
pp. 1550033 ◽  
Author(s):  
A. Paliathanasis ◽  
M. Tsamparlis ◽  
M. T. Mustafa
Keyword(s):  
Wave Equation ◽  
Gordon Equation ◽  
Invariant Solution ◽  
Lie Symmetries ◽  
Conformal Algebra ◽  
Noether Symmetries ◽  
Bianchi I ◽  
Klein Gordon ◽  
Klein Gordon Equation

In this work we perform the symmetry classification of the Klein–Gordon equation in Bianchi I spacetime. We apply a geometric method which relates the Lie symmetries of the Klein–Gordon equation with the conformal algebra of the underlying geometry. Furthermore, we prove that the Lie symmetries which follow from the conformal algebra are also Noether symmetries for the Klein–Gordon equation. We use these results in order to determine all the potentials in which the Klein–Gordon admits Lie and Noether symmetries. Due to the large number of cases and for easy reference the results are presented in the form of tables. For some of the potentials we use the Lie admitted symmetries to determine the corresponding invariant solution of the Klein–Gordon equation. Finally, we show that the results also solve the problem of classification of Lie/Noether point symmetries of the wave equation in Bianchi I spacetime and can be used for the determination of invariant solutions of the wave equation.

scholarly journals Dirac Equation in Cosmological Inertial Frame

2021 ◽  
Author(s):  
Sangwha Yi
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Wave Equation ◽  
Dirac Equation ◽  
Inertial Frame ◽  
Gordon Equation ◽  
Probability Amplitude ◽  
Function Type ◽  
Klein Gordon ◽  
Klein Gordon Equation

Dirac equation is a one order-wave equation. Wave function uses as a probability amplitude in quantum mechanics. We make Dirac Equation from wave function, Type A in cosmological inertial frame.The Dirac equation satisfy Klein-Gordon equation in cosmological inertial frame.

scholarly journals A note on the spectrum of discrete Klein-Gordon s-wave equation with eigenparameter dependent boundary condition

Filomat ◽  
2019 ◽  
Vol 33 (2) ◽  
pp. 449-455 ◽  
Author(s):  
Nimet Coskun ◽  
Nihal Yokus
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Boundary Value Problem ◽  
Gordon Equation ◽  
S Wave ◽  
Klein Gordon ◽  
Complex Sequences ◽  
Quantitative Properties ◽  
Klein Gordon Equation ◽  
Jost Solution

This paper is concerned with the boundary value problem (BVP) for the discrete Klein-Gordon equation ?(an-1?yn-1)+(vn-?)2 yn = 0; n ? N and the boundary condition (?0+?1?)y1+(?0+?1)y0 = 0 where (an),(vn) are complex sequences, ?i, ?i ? C, i=0,1 and ? is a eigenparameter. The paper presents Jost solution, eigenvalues, spectral singularities and states some theorems concerning quantitative properties of the spectrum of this BVP under the condition ?n?N exp(?n?)(|1-an| + |vn|) < ? for ? > 0 and 1/2 ? ? ? 1.

scholarly journals Derivation of Solutions of the Klein-Gordon Equation from Solutions of the Wave Equation

1966 ◽  
Vol 15 (2) ◽  
pp. 125-129 ◽  
Author(s):  
LL. G. Chambers
Keyword(s):  
Wave Equation ◽  
Gravity Waves ◽  
Nuclear Physics ◽  
Constant Velocity ◽  
Gordon Equation ◽  
Laplacian Operator ◽  
Klein Gordon ◽  
Image Position ◽  
Klein Gordon Equation

The Klein–Gordon equationΩ being a constant of dimensions [time]-1 and c being a constant velocity, appears in nuclear physics (1) and, when the Laplacian operator is twodimensional, in the theory of long gravity waves on a rotating earth (2). If Ω is zero it reduces to the wave equation

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