arXiv subjects

Per Kristen Jakobsen

Publications and source records attributed to Per Kristen Jakobsen.

13 recordsLinked to original sources

arXiv2026

On removing orders from amplitude equations

In this paper, we introduce a modified version of the renormalization group (RG) method and test its numerical accuracy. It has been tested on numerous scalar ODEs and systems of ODEs. Our method is primarily motivated by the possibility of simplifying amplitude equations. The key feature of our method is the introduction of a new homogeneous function at each order of the perturbation hierarchy, which is then used to remove terms from the amplitude equations. We have shown that there is a limit to how many terms can be removed, as doing so beyond a certain point would reintroduce linear growth. There is thus a \textit{core} in the amplitude equation, which consists of the terms that cannot be removed while avoiding linear growth. Using our modified RG method, higher accuracy can also be achieved while maintaining the same level of complexity in the amplitude equation.

math-ph↗

arXiv2026

General method for solving nonlinear optical scattering problems using fix point iterations

In this paper we introduce a new fix point iteration scheme for solving nonlinear electromagnetic scattering problems. The method is based on a spectral formulation of Maxwell's equations called the Bidirectional Pulse Propagation Equations. The scheme can be applied to a wide array of slab-like geometries, and for arbitrary material responses. We derive the scheme and investigated how it performs with respect to convergence and accuracy by applying it to the case of light scattering from a simple slab whose nonlinear material response is a sum a very fast electronic vibrational response, and a much slower molecular vibrational response.

physics.class-ph↗

arXiv2025

A rigorous coupled-wave analysis of birefringent holographic gratings with periodically-modulated dielectric tensor along an in-plane direction and tensor variations in the thickness direction

Diffraction of light upon interaction with thick slabs of a dielectric material having a periodic modulation of its refractive index (or dielectric tensor) is typically studied with the aid of the method known as the rigorous coupled-wave analysis (RCWA). The method involves solving Maxwell's equations for a large number of coupled electromagnetic plane-waves inside the dielectric slab, then matching the boundary conditions at the interface between the incidence medium and the slab, as well as those at the interface between the slab and the transmittance medium. In this way, one obtains the E-field and H-field amplitudes for all the reflected and transmitted plane-waves (i.e., diffraction orders as well as evanescent waves) that emerge within the incidence and transmittance media. If the refractive index (or dielectric tensor) of the holographic slab happens to vary in the thickness direction, one treats the slab as a number of thin layers stacked upon each other, then computes and combines the scattering matrices of these layers to arrive at the complete solution for the entire stack. The goal of the present paper is to extend the standard RCWA method to the case where the hologram's dielectric tensor varies in the thickness direction (in addition to being periodically modulated along an in-plane axis), without slicing up the thick hologram into a number of thin layers. The reflected and transmitted plane-waves in this case exhibit a large degree of degeneracy, but our numerical results confirm the validity and the accuracy of our proposed algorithm for handling such degeneracies.

physics.optics↗

arXiv2024

The continuum limit of k-space cavity angular momentum is controlled by an infinite range difference operator

A wavepacket (electromagnetic or otherwise) within an isotropic and homogeneous space can be quantized on a regular lattice of discrete k-vectors. Each k-vector is associated with a temporal frequency omega; together, k and omega represent a propagating plane-wave. While the total energy and total linear momentum of the packet can be readily apportioned among its individual plane-wave constituents, the same cannot be said about the packet's total angular momentum. One can show, in the case of a reasonably smooth (i.e., continuous and differentiable) wave packet, that the overall angular momentum is expressible as an integral over the k-space continuum involving only the Fourier transform of the field and its k-space gradients. In this sense, the angular momentum is a property not of individual plane-waves, but of plane-wave pairs that are adjacent neighbors in the space inhabited by the k-vectors, and can be said to be localized in the k-space. Strange as it might seem, this hallmark property of angular momentum does not automatically emerge from an analysis of a discretized k-space. In fact, the discrete analysis shows the angular momentum to be distributed among k-vectors that pair not only with nearby k-vectors but also with those that are far away. The goal of the present paper is to resolve the discrepancy between the discrete calculations and those performed on the continuum, by establishing the conditions under which the highly non-local sum over plane-wave pairs in the discrete k-space would approach the localized distribution of the angular momentum across the continuum of the k-space.

physics.class-ph↗

arXiv2023

Effects of Symmetry in a Diffusive Energy Balance Model

In this paper, we solve a North-type Energy Balance Model (EBM) using an analytical method, the Boundary Integral Method. This approach is discussed in light of existing analytical techniques for this type of equation. We use the method to demonstrate that the placement of a zonally symmetric continent, with an altered ice-albedo feedback dynamic, introduces new equilibrium states. A finite difference algorithm is implemented to solve the time-dependent equation and assess the stability of the equilibrium states, along with a numerical perturbation scheme. Bifurcation diagrams are drawn and we show that the bifurcation curve is extremely sensitive to the placement of a continent. The continent is initially configured with meridional symmetry, and we investigate how the system dynamics respond to a gradual reduction of the system's symmetry properties. We find that meridional symmetry increases the number of fold bifurcations and equilibria. Additionally, we discuss how the emerging bifurcation structures may provide insights into the complex dynamics involved as one ascends the climate model hierarchy.

math.DS↗

arXiv2021

On the numerical accuracy of the method of multiple scales for nonlinear dispersive wave equations

In this paper we study dispersive wave equation using the method of multiple scales (MMS) and perform several numerical tests to investigate its accuracy. The key feature of our MMS solution is the linearity of the amplitude equation and the complex nature of the time-frequency. The MMS is tested as an initial value problem using three choices of the dispersion model, one toy and two Lorentz models. Depending on the parameters of the problem, the amplitude equation can be both well- or ill-posed. Despite the ill-posedness, the MMS solution remains a valid approximation of the solution to the original nonlinear model.

math.NA↗

arXiv2019

Topics in Applied Mathematics and Nonlinear Waves

The selection of topics in this text has formed the core of a one semester course in applied mathematics at the Arctic University of Norway that has been running continuously since the 1970s. The class has, during its existence, drawn participants from both applied mathematics and physics, and also to some extent from pure mathematics, analysis in particular. The material in these lecture notes can be covered by one semester's worth of five lecture hours a week. The work requirements for the students consists of seven obligatory projects whose content are taken from exercises and computational projects included in the text.

math.HO↗

arXiv2019

Constructing a partially transparent computational boundary for UPPE using leaky modes

In this paper we introduce a method for creating a transparent computational boundary for the simulation of unidirectional propagation of optical beams and pulses using leaky modes. The key element of the method is the introduction of an artificial-index material outside a chosen computational domain and utilization of the quasi-normal modes associated with such artificial structure. The method is tested on the free space propagation of TE electromagnetic waves. By choosing the material to have appropriate optical properties one can greatly reduce the reflection at the computational boundary. In contrast to the well-known approach based on a perfectly matched layer, our method is especially well suited for spectral propagators.

physics.optics↗

arXiv2019

An Introduction to Partial Differential Equations

The field of partial differential equations (PDEs) is vast in size and diversity. The basic reason for this is that essentially all fundamental laws of physics are formulated in terms of PDEs. In addition, approximations to these fundamental laws, that form a patchwork of mathematical models covering the range from the smallest to the largest observable space-time scales, are also formulated in terms of PDEs. The diverse applications of PDEs in science and technology testify to the flexibility and expressiveness of the language of PDEs, but it also makes it a hard topic to teach right. Exactly because of the diversity of applications, there are just so many different points of view when it comes to PDEs. These lecture notes view the subject through the lens of applied mathematics. From this point of view, the physical context for basic equations like the heat equation, the wave equation and the Laplace equation are introduced early on, and the focus of the lecture notes are on methods, rather than precise mathematical definitions and proofs. With respect to methods, both analytical and numerical approaches are discussed. These lecture notes has been succesfully used as the text for a master class in partial differential equations for several years. The students attending this class are assumed to have previously attended a standard beginners class in ordinary differential equations and a standard beginners class in numerical methods. It is also assumed that they are familiar with programming at the level of a beginners class in informatics at the university level.

math.HO↗

arXiv2018

On the EOS formulation for light scattering. Stability, Singularity and Parallelization

In this paper we discuss some of the mathematical and numerical issues that have to be addressed when calculating wave scattering using the EOS approach. The discussion is framed in context of light scattering by objects whose optical response can be of a nonlinear and/or inhomogeneous nature. The discussions address two issues that, more likely than not, will be part of any investigation of wave scattering using the EOS approach.

math.NA↗

arXiv2018

A 3D Nonlinear Maxwell's Equations Solver Based On A Hybrid Numerical Method

In this paper we explore the possibility for solving the 3D Maxwell's equations in the presence of nonlinear and/or inhomogeneous material response. We propose using a hybrid approach which combines a bound- ary integral representation with a domain-based method. This hybrid approach has previously been successfully applied to 1D linear and non- linear transient wave scattering problems. The basic idea of the approach is to propagate the Maxwell's equations inside the scattering objects for- ward in time by using a domain-based method, while a boundary integral representation of the electromagnetic field is used to supply the domain- based method with the required surface values. Thus no grids outside the scattering objects are needed and this greatly reduces the computational cost and complexity.

math.NA↗

arXiv2018

A boundary integral approach to linear and nonlinear transient wave scattering

In this paper we introduce a method for solving linear and nonlinear scattering problems for wave equations using a new hybrid approach. This new approach consists of a reformulation of the governing equations into a form that can be solved by a combination of a domain-based method and a boundary-integral method.Our reformulation is aimed at a situation where we have a collection of compact scattering objects located in an otherwise homogeneous unbounded space. The domain-based method is used to propagate the equations governing the wave field inside the scattering objects forward in time. The boundary integral method is used to supply the domain-based method with the required boundary values for the wave field. In this way the best features of both methods come into play; the response inside the scattering objects, which can be caused by both material inhomogeneity and nonlinearities, is easily taken into account using the domain-based method, and the boundary conditions supplied by the boundary integral method makes it possible to confine the domain based method to the inside of each scattering object.

math.NA↗

arXiv2018

The thermodynamics of light

These notes introduce some of the basic mathematical and physical tools neccessary for theoretical investigations into the thermodynamics properties of light in cavities. The notes were created while preparing for a project in this area were we noted that many of the tools neccessary for these investigations are spread out over the physics and mathematics litterature with videly varying styles, notation, technical level and expected mathematical and physical sofistication from the readers. We found that bringing together the basic tools from mathematical and theoretical physics neccessary for studing the thermodynamical properties of light in cavities, in a set of notes with a uniform style, notation and level of sofistication was useful for communication between the members of the research team. By publishing the notes we hope that they will also be useful for other research teams entering this field of science.

physics.optics↗