Chaotic systems and their quantization
Researchers: Per Dahlqvist
Sponsors: NFR
Quantum systems whose classical counterparts are chaotic are studied. The emphasis is on bound systems. Two dimensional scattering systems, billiards, are mostly used as model systems. Tools used are e.g. periodic orbits, symbolic dynamics and zeta functions. In particular, asymptotics measures on periodic orbits are developed.
Publications: 14,16,109,110
Wave propagation models in hydroacoustics
Researchers: Bengt Enflo, Claes Hedberg (graduate student)
Sponsors: TFR
Within this project Bengt Enflo works as research leader and Claes Hedberg as research student. Bengt Enflo has studied propagation of nonlinear diffusive acoustic waves in simple geometries such as plane, cylindrical and spherical geometry and obtained new mathematical results on the behaviour of short pulses and periodic waves far from the sound source using Burgers and generalized Burgers equations. He continues studying the asymptotic behaviour of nonlinear limited acoustic beams, especially the decay of shock waves with application to crushers. Claes Hedberg has generalized the theory of propagation of a plane harmonic wave to a biharmonic wave, thus predicting the asymptotic behaviour of nonlinearly generated combination frequency waves. The influence of relative phase on solutions to problems with a boundary condition consisting of a sum of harmonic functions has been found. In a work together with S.N. Gurbatov, University of Nizhny Novgorod, Russia, he studies theoretically the propagation of modulated intense acoustic signals where initially phase modulated waves develops into apparently amplitude modulated waves and vice versa.
Publications: 15,22,78,79,87,115
Dynamical systems and multi-body modelling
Researchers: Martin Lesser, Hanno Essén, Arne Nordmark, Graduate Students: Mats Fredriksson, Anders Lennartsson
Sponsor: TFR, LUFT
The purpose of this study is to integrate modern methods of modelling multi-body systems with recent results in the theory of dynamical systems. Thus far most dynamical systems treated under the new heading of chaos theory has involved concentration on very low degree of freedom models derived in a somewhat ad-hoc fashion as representations of more complex mechanisms. Our aim is to combine our new techniques for dealing with complex mechanisms by computer algebra and Kanes Equations with the methods of dynamical systems theory to achive useful and readily interpretable representations, e.g. by means of center manifold ideas. One particular area of interest is problems of impact in subassemblies of complex mechanisms.
Publications: 17,18,34,35,55,68,96,97,100,120
Design of complex mechanical systems
Researchers: Martin Lesser, Sören Andersson (Machine Elements), Lennart Karlsson (Computer Aided Design, Luleå), Tore Risch, (Computer Science Linköping), Volvo Corporation. Graduate Student at KTH: Claes Tissel
Sponsor: NUTEK
Complex mechanical systems are treated by a combination of modern methods for simulation, computer aided design tools and object oriented data base technology. The aim of the project is to assemble all of these techniques into a usable design tool. Several particular problems are being used as test cases. These include blade mountings in jet engines and exhaust manifold in automobiles.
Publications: 104
Generation and focusing of converging shock waves of a desired shape. Shock wave propagation in liquid impact problems.
Researchers: Martin Lesser, Nicholas Apazidis, Josefina Dellby (graduate student)
Sponsors: TFR, LUFT.
This project deals with the calculation of various shapes and strengths of shock waves in liquids. Cylindrical shock waves are generated in a confined cylindrical reflector. It is shown that by an appropriate variation in the form of the reflector boundary one may obtain reflected shock waves having desirable shapes, for example a near-square shape. The shape of the converging reflected shock wave as well as its strength is then calculated by means of a numerical method based on Whitham's theory of geometrical shock dynamics. Numerical results are obtained for various reflector geometries and shock wave strengths. Technological and medical applications of the project are within the fields of shock wave propagation, shock induced collapce of cavities, desintegration of kidney stones by means of a shock wave attenuation in lithotripter devices.
The project also deals with shockwave propagation in liquid impact problems. Numerical calculations of propagation of shockwaves generated in a liquid drop impacting on a material surface are performed. Evolution of shockfront positions with the corresponding Mach number and pressure distributions in the liquid drop are obtained for various values of the drop impact Mach number.
Publications: 4,5,48,49
Non-Newtonian phenomena and kinetic theory of gases
Researcher: Lars Söderholm.
Sponsor: LUFT.
There are two basic limitations of the Navier-Stokes equations. One is that they are elliptic: some wave modes are lacking. The other one is that they are valid only if the mean free path is a factor 100 to 1000 smaller than the macroscopic scale. This is important for high altitude flight, especially when the mean free path is of the order of one tenth to one hundredth of the characteristic scale. General flow equations for gases can be obtained from the Boltzmann equation via the Chapman-Enskog method and the Grad method. In the Newtonian limit, the Chapman-Enskog method gives more accurate results than the corresponding Grad approximation. But in the non-Newtonian region, the Grad method is more reliable than the Chapman-Enskog method. Attempts are made at an intermediate method, in order to combine the advantages of both. A new 13 moments system of equations has been obtained, which is hyberbolic like the Grad system (so that a full set of wave modes is included), but which gives the correct values for viscocity and heat conduction, in contrast with the Grad system. A new type of regularization of the Burnett equations is also studied.
Publications: 56
Non-linear waves and integrability
Researcher: Lars Söderholm.
Sponsor: LUFT.
An important class of non-linear wave equations turns out to be completely integrable via some sort of non-linear superposition like inverse scattering. Most of these equations are the first non-linear corrections to the linear approximations. This usually applies to the Korteweg-de Vries equation and the non-linear Schrödinger equation (valid for long and short water waves, respectively). Remarkably enough, the fully non-linear Euler equations are exactly integrable. The same applies to the Einsteins field equations of general relativity when sufficient symmetry reduces the problem to two independent variables. In a sense, the integrable equations can be considered to be intrinsically linear. They are not linear in the usual variables, but via (often non-local) transformations, they can be reduced to linear equations. This appears to call for a geometric and "coordinate-independent" description (most likely in an infinity number of dimensions) of such equations. In general relativity coordinate-independent methods have been employed for a long time, especially in the so-called equivalence problem. Some important insight into the class of integrable equations should be possible from a study of the equations of general relativity. A result is that the relativistic Euler equations have been reduced to a linear equation, which is studied at present. - Some of these questions are pursued in cooperation with Michael Bradley, University of Umeå.
Asymptotic methods in dynamical stability theory and quantum mechanics
Researchers: K.-E. Thylwe, in collaboration with Ö. Dammert and S. Linnéus (theor. phys., UU).
Sponsor: 'Rörlig resurs, KTH'.
Higher-order asymptotic methods relevant for parametrically excited, linear and non-linear, coupled oscillator models are developed. Certain uniform matching techniques are needed to describe the behaviour near time-localised resonances of the coupled oscillators. The methods are expected to be powerful tools in stability theory for analysing linear and non-linear variational equations of more general dynamical systems.
Publications: 19,57,60,61,113
Semiclassical quantization of time-dependent Hamiltonian systems
Researchers: K.-E. Thylwe, in collaboration with F. Bensch (University of Campinas, Brazil)
Sponsor: 'Rörlig resurs', KTH.
Various time-dependent Hamiltonian systems may serve as mathematical models for analysing fundamental details of non-linear dynamics and semiclassical quantization. Generically such systems are known to show chaotic dynamics even for one (spatial) degree of freedom. A question of considerable interest to us is to what extent and in what way classical nonlinear phenomena reflect themselves in quantum mechanics. We believe that time-dependent Hamiltonian systems are easier to approach in these matters than are the autonomous systems with more spatial degrees of freedom.
The following items of research are in progress: -(1) The question of how the rich variety of subharmonic resonance phenomena in classical mechanics is reflected in quantum mechanics. In particular the quantum mechanical aspect of subharmonic cascades. -(2) The semiclassical quantization of quasi-periodically parameter-excited integrable oscillator. This model represents a simple step towards a quantization of an irregular classical motion. -(3) The semiclassical quantization of systems in the fully chaotic regions of phase space, approached by a generalisation of Gutzwiller's Periodic Orbit Theory to the extended phase space.
Publications: 58,59
Theoretical study of atom-molecule collisions
Researchers: K.-E. Thylwe, A. Amaha (UU) and J. N. L. Connor (Manchester, UK).
Sponsor: Swedish Natural Science Research Council (NFR).
Powerful novel methods in the framework of "sudden" approximations, implementing complex angular momentum (Regge pole) techniques and semiclassical considerations, are being developed to study in detail the mechanisms of diffraction and rotational rainbows in state-to-state differential cross sections. On the bases of the approximations used, semi-empirical inversion schemes can be constructed which relates properties of the diffraction and rotational rainbow phenomena to realistic structural parameters of the interaction.
Publications: 1,43
Black-hole scattering of scalar waves
Researchers: K.-E. Thylwe, in collaboration with N. Andersson (Cardiff University, Wales)
Sponsor: 'Rörlig resurs', KTH.
A description of scalar waves scattered off a Schwarzschild black hole in terms of complex angular momenta is discussed. In the new picture the scattering amplitude is split into a supposedly smooth background integral and a sum over the so-called Regge poles. It seems reasonable to hope that we can learn more about the intricate details of general relativity by studying how incoming waves are affected by the presence of a black hole. This is especially important considering the fact that we only have indirect evidence for the actual existence of astrophysical black holes. By studying how a black hole interacts with its environment one may understand whether there are ways of making direct observations of such, in principle invisible, objects. Most studies of black-hole scattering have used the standard partial-wave paradigm. In that description it is, however, difficult to interpret the results.
An alternative description of scattering, that is celebrated for its interpretative power, is based on the use of complex angular momenta (CAM). Considering the success of the CAM approach it is surprising that no attempts have been made to use it in the black-hole case. Chandrasekhar and Ferrari have discussed CAM in the context of stellar pulsations, but they do not make use of the full potential of the technique.