arXiv subjects

F. Ali Mehmeti

Publications and source records attributed to F. Ali Mehmeti.

4 recordsLinked to original sources

arXiv2015

Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation

We consider a version of the stationary phase method in one dimension of A. Erdélyi, allowing the phase to have stationary points of non-integer order and the amplitude to have integrable singularities. After having completed the original proof and improved the error estimate in the case of regular amplitude, we consider a modification of the method by replacing the smooth cut-off function employed in the source by a characteristic function, leading to more precise remainder estimates. We exploit this refinement to study the time-asymptotic behaviour of the solution of the free Schrödinger equation on the line, where the Fourier transform of the initial data is compactly supported and has a singularity. We obtain asymptotic expansions with respect to time in certain space-time cones as well as uniform and optimal estimates in curved regions which are asymptotically larger than any space-time cone. These results show the influence of the frequency band and of the singularity on the propagation and on the decay of the wave packets.

math.AP↗

arXiv2014

Explicit error estimates for the stationary phase method I: The influence of amplitude singularities

We consider a version of the stationary phase method in one dimension of A. Erdélyi, allowing the phase to have stationary points of non-integer order and the amplitude to have integrable singularities. We provide a complete proof and we improve the remainder estimates in the case of regular amplitude. Then we are interested in the time-asymptotic behaviour of the solution of the free Schrödinger equation on the line, where the Fourier transform of the initial data is compactly supported and has a singularity. Applying the above mentioned method, we obtain asymptotic expansions with respect to time in certain space-time cones, where the coefficients of the remainders are uniformly bounded. These results show the influence of the singularity on the decay.

math.AP↗

arXiv2014

Explicit error estimates for the stationary phase method II: Interaction of amplitude singularities with stationary points

In this paper, we improve slightly Erdélyi's version of the stationary phase method by replacing the employed smooth cut-off function by a characteristic function, leading to more precise remainder estimates. We exploit this refinement to study the time-asymptotic behaviour of the solution of the free Schrödinger equation on the line, where the Fourier transform of the initial data is compactly supported and has a singularity. We obtain uniform estimates of the solution in space-time regions which are asymptotically larger than any space-time cones. Moreover we expand the solution with respect to time on the boundaries of the above regions, showing the optimality of the decay rates of the estimates.

math.AP↗

arXiv2006

Global existence and causality for a transmission problem with a repulsive nonlinearity

It is well-known that the solution of the classical linear wave equation with compactly supported initial condition and vanishing initial velocity is also compactly supported in a set depending on time : the support of the solution at time t is causally related to that of the initially given condition. Reed and Simon have shown that for a real-valued Klein-Gordon equation with (nonlinear) right-hand side $- λu^3$, causality still holds. We show the same property for a one-dimensional Klein-Gordon problem but with transmission and with a more general repulsive nonlinear right-hand side $F$. We also prove the global existence of a solution using the repulsiveness of $F$. In the particular case $F(u) = - λu^3$, the problem is a physical model for a quantum particle submitted to self-interaction and to a potential step.

math.AP↗