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Full-Text Articles in Physical Sciences and Mathematics
Asymptotic Behaviour Of Hyperbolic Partial Differential Equations, Shi-Zhuo Looi
Asymptotic Behaviour Of Hyperbolic Partial Differential Equations, Shi-Zhuo Looi
Theses and Dissertations--Mathematics
We investigate the asymptotic behaviour of solutions to a range of linear and nonlinear hyperbolic equations on asymptotically flat spacetimes. We develop a comprehensive framework for the analysis of pointwise decay of linear and nonlinear wave equations on asymptotically flat manifolds of three space dimensions that are allowed to be time-varying or nonstationary, including quasilinear wave equations. The Minkowski space and time-varying perturbations thereof are included among these spacetimes. A result on scattering for a nonlinear wave equation with finite-energy solutions on nonstationary spacetimes is presented. This work was motivated in part by the investigation of more precise asymptotic behaviour …
A Scattering Result For The Fifth-Order Kp-Ii Equation, Camille Schuetz
A Scattering Result For The Fifth-Order Kp-Ii Equation, Camille Schuetz
Theses and Dissertations--Mathematics
We will prove scattering for the fifth-order Kadomtsev-Petviashvilli II (fifth-order KP-II) equation. The fifth-order KP-II equation is an example of a nonlinear dispersive equation which takes the form $u_t=Lu + NL(u)$ where $L$ is a linear differential operator and $NL$ is a nonlinear operator. One looks for solutions $u(t)$ in a space $C(\R,X)$ where $X$ is a Banach space. For a nonlinear dispersive differential equation, the associated linear problem is $v_t=Lv$. A solution $u(t)$ of the nonlinear equation is said to scatter if as $t \to \infty$, the solution $u(t)$ approaches a solution $v(t)$ to the linear problem in the …