scholarly journals Channels of energy for the linear radial wave equation

2015 ◽  
Vol 285 ◽  
pp. 877-936 ◽  
Author(s):  
Carlos Kenig ◽  
Andrew Lawrie ◽  
Baoping Liu ◽  
Wilhelm Schlag
Keyword(s):  
Wave Equation ◽  
Radial Wave ◽  
Radial Wave Equation

Symmetry Reductions and Group-Invariant Radial Solutions to the n-Dimensional Wave Equation

2018 ◽  
Vol 73 (2) ◽  
pp. 161-170 ◽  
Author(s):  
Wei Feng ◽  
Songlin Zhao
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Symmetry Group ◽  
Group Method ◽  
Radial Solutions ◽  
Radial Wave ◽  
One Dimensional ◽  
Group Invariant ◽  
Dimensional Wave ◽  
Radial Wave Equation

AbstractIn this paper, we derive explicit group-invariant radial solutions to a class of wave equation via symmetry group method. The optimal systems of one-dimensional subalgebras for the corresponding radial wave equation are presented in terms of the known point symmetries. The reductions of the radial wave equation into second-order ordinary differential equations (ODEs) with respect to each symmetry in the optimal systems are shown. Then we solve the corresponding reduced ODEs explicitly in order to write out the group-invariant radial solutions for the wave equation. Finally, several analytical behaviours and smoothness of the resulting solutions are discussed.

scholarly journals Scalar Waves in the Exterior of a Schwarzschild Black Hole

1974 ◽  
Vol 64 ◽  
pp. 95-95
Author(s):  
S. Persides
Keyword(s):  
Black Hole ◽  
Wave Equation ◽  
Scalar Field ◽  
Schwarzschild Black Hole ◽  
Radial Wave ◽  
Point Test ◽  
Fourier And Laplace Transforms ◽  
Image Position ◽  
Interior Solutions ◽  
Radial Wave Equation

Fourier and Laplace transforms are used to study rigorously the properties of a test scalar field PS in the exterior of a Schwarzschild black hole of the mass m. In the Fourier analysis we examine the properties of the solutions of the radial wave equation and the relations of the exterior and interior solutions of the following four cases: (i) ω ≠ 0, m≠0, (ii) ω=0, m≠0, (iii) ω≠0, m=0, (iv) ω=0, m=0.In the Laplace analysis we show rigorously the following theorem: If ψ(t, r, τ, ϕ) is the field of a point test particle falling into the black hole, and lim Ψ exists, then lim Ψ = 0. The proof of this theorem is based on the facts that (a)t+2m ln (r − 2m) is finite for the particle even on the horizon, and (b) the behavior of Ψ as t → + ∞ is related to its Laplace transform near the origin of the complex plane.

scholarly journals Energy partition for the linear radial wave equation

2013 ◽  
Vol 358 (3-4) ◽  
pp. 573-607 ◽  
Author(s):  
Raphaël Côte ◽  
Carlos E. Kenig ◽  
Wilhelm Schlag
Keyword(s):  
Wave Equation ◽  
Energy Partition ◽  
Radial Wave ◽  
Radial Wave Equation
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