Asymptotics of the Klein-Gordon equation using the method of stationary phase
(2024) In Master's Theses in Mathematical Sciences FMAM05 20241Mathematics (Faculty of Engineering)
- Abstract
- The Klein--Gordon equation is a relativistic wave equation which models a relativistic spin-free particle. In this thesis we investigate the asymptotic behaviour of the linear Klein-Gordon equation with a rapidly decaying potential in one space dimension. This is done by first studying the case with zero potential and finding asymptotic estimates using the method of stationary phase. This is a method which makes it possible to determine the asymptotic behaviour of integrals of rapidly oscillating functions. Then the distorted Fourier transform is defined using tools from scattering theory, which transforms the Klein-Gordon equation into the zero potential case. Finally we state asymptotic bounds of the solution, which for bounded regions... (More)
- The Klein--Gordon equation is a relativistic wave equation which models a relativistic spin-free particle. In this thesis we investigate the asymptotic behaviour of the linear Klein-Gordon equation with a rapidly decaying potential in one space dimension. This is done by first studying the case with zero potential and finding asymptotic estimates using the method of stationary phase. This is a method which makes it possible to determine the asymptotic behaviour of integrals of rapidly oscillating functions. Then the distorted Fourier transform is defined using tools from scattering theory, which transforms the Klein-Gordon equation into the zero potential case. Finally we state asymptotic bounds of the solution, which for bounded regions in space are almost identical to the results for the zero potential. (Less)
- Popular Abstract
- Mathematics can be used to model nearly everything. Differential equations, and especially partial differential equations are ubiquitous in mathematical modelling of everything from biology to physics. A partial differential equation (or PDE) is an equation where the solution is a function which depends on multiple variables, usually time and space. The equation describes how this functions should change, both under small changes in time and in space.
PDE:s are known to be very difficult to solve (i.e. find a function which fulfills the given equation), but even if one cannot find an explicit solution for a given equation it is possible to deduce some behaviour of it. For instance, if one has a PDE which models a system for which the... (More) - Mathematics can be used to model nearly everything. Differential equations, and especially partial differential equations are ubiquitous in mathematical modelling of everything from biology to physics. A partial differential equation (or PDE) is an equation where the solution is a function which depends on multiple variables, usually time and space. The equation describes how this functions should change, both under small changes in time and in space.
PDE:s are known to be very difficult to solve (i.e. find a function which fulfills the given equation), but even if one cannot find an explicit solution for a given equation it is possible to deduce some behaviour of it. For instance, if one has a PDE which models a system for which the behaviour asymptotically (for large times) is of interest, then it would be of great value to be able to deduce some information about it. That is precisely what this thesis explores.
The Klein--Gordon equation is a partial differential equation which models a particle such as the Higgs boson moving at relativistic speeds through a potential field. It is known that the solution of the equation tends towards zero as time goes towards infinity, but an interesting question to pose might be how quickly it goes towards zero, and roughly how the solution looks for large times.
The answer to these questions were known when we had no potential field, such as that the solution goes towards zero at a rate of $1/\sqrt{t}$. What this thesis shows is that we can retain almost the same results as the previously known results when we apply a small potential field, as long as we bound ourselves how far away we are allowed to move in space. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9166464
- author
- Thorin, Nils LU
- supervisor
- organization
- course
- FMAM05 20241
- year
- 2024
- type
- H2 - Master's Degree (Two Years)
- subject
- keywords
- Klein-Gordon equation, PDE, Scattering theory, Method of stationary phase
- publication/series
- Master's Theses in Mathematical Sciences
- report number
- LUTFMA-3544-2024
- ISSN
- 1404-6342
- other publication id
- 2024:E41
- language
- English
- id
- 9166464
- date added to LUP
- 2024-11-26 16:48:03
- date last changed
- 2024-11-26 16:48:03
@misc{9166464,
abstract = {{The Klein--Gordon equation is a relativistic wave equation which models a relativistic spin-free particle. In this thesis we investigate the asymptotic behaviour of the linear Klein-Gordon equation with a rapidly decaying potential in one space dimension. This is done by first studying the case with zero potential and finding asymptotic estimates using the method of stationary phase. This is a method which makes it possible to determine the asymptotic behaviour of integrals of rapidly oscillating functions. Then the distorted Fourier transform is defined using tools from scattering theory, which transforms the Klein-Gordon equation into the zero potential case. Finally we state asymptotic bounds of the solution, which for bounded regions in space are almost identical to the results for the zero potential.}},
author = {{Thorin, Nils}},
issn = {{1404-6342}},
language = {{eng}},
note = {{Student Paper}},
series = {{Master's Theses in Mathematical Sciences}},
title = {{Asymptotics of the Klein-Gordon equation using the method of stationary phase}},
year = {{2024}},
}