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Hamiltonian systems of infinite dimension
Integrable systems of infinite dimension like the non-linear Schrödinger equation play a prominent role in the nonlinear applied sciences and are connected to many other subfields of mathematics. We spoke to Professor Thomas Kappeler about his research into solving these equations and analyzing the stability of solutions, which could open up new avenues of investigation.
The concept of an integrable system arose in the 19th century in the context of finite dimensional classical mechanics. Over the last 50 years or so researchers have started applying it to study systems with an infinite number of dimensions such as partial differential equations. Based at the University of Zurich, Professor Thomas Kappeler, together with his colleagues, is investigating a number of such equations, including the Korteweg-de Vries equation, the non-linear Schrödinger equation, the sine-Gordon equation, and the Benjamin-Ono equation. “These evolution equations in one space variable not only model a wide variety of wave phenomena, but astonishingly turn out to be useful in many other areas of the applied sciences such as fiber optics, crystallography or molecular biology,” he outlines. The common characteristics of these equations are that they are nonlinear and have infinitely many conservation laws. “Most strikingly,” explains Professor Kappeler “they admit solutions with specific features, relevant in applications, which cannot be observed in solutions of linear equations.”
Method of solving integrable systems
A powerful method of solving finite dimensional integrable systems consists in constructing special coordinates so that when expressed in these coordinates, the equation takes a particularly simple form. This method is referred to as the method of normal forms. It allows researchers to solve the considered equation explicitly and to study the properties of its solutions, such as being periodic or quasi-periodic in time. It also allows researchers to answer questions concerning the confinement of the orbits of its solutions. Similarly, for infinite dimensional linear systems, given by partial differential equations with constant coefficients, the renowned Fourier transform is such a change of coordinates. Very early on, the question arose whether the method of normal forms can also be developed for nonlinear integrable systems of infinite dimension. It turns out that for many of these equations, this is indeed the case, but that the transformation depends in a significant way on the equation considered. “One of the goals of my research,” says Professor Kappeler, “is to further develop the method of normal forms for solving nonlinear integral systems of infinite dimension in a setup where the space variable is periodic and to apply it to study properties of their solutions, such as their long time behaviour. In the cases considered, it can be shown that the transformations found share many properties with the Fourier transform and are in fact perturbations of the latter. Surprisingly, they allow us also to obtain solutions of very low regularity by providing a means of extending nonlinear expressions
Integrable systems of infinite dimension not only model a wide variety of wave phenomena, but turn out to be useful in many other areas of the applied sciences such as fiber
optics, crystallography or molecular biology.
which come up in these equations.”
Stability
The second part of the research centres on investigating the stability of the solutions of infinite dimensional integrable systems. “The two parts of this research are actually closely related,” says Professor Kappeler, ”since the normal form method is of great use for studying questions of stability. There are two types of stability issues. Both are relevant for applications when solutions of such equations are used to make predictions.”
The first concerns the stability with respect to initial conditions. “If you change the initial conditions a little bit, do you get a solution which stays nearby? And if so, for how long? It turns out that the normal form method can help answer such questions,” explains
Professor Kappeler. “For the equations studied, it can be shown that the orbits of two solutions, starting near each other, stay close at all times.”
The second stability issue concerns the stability of the structure of the equation: “If you perturb the equation considered, does the perturbed system have similar properties as the unperturbed one?” outlines Professor Kappeler. “For example, we have detailed knowledge about the solutions to the twobody problem in celestial mechanics, say the motions of the sun and the earth. In this model, which is an integrable system of six dimensions, we know the earth moves around the sun in an elliptical orbit. If you add a small planet to this system, what can you then say about the perturbed system? Is it possible that the orbits of sun and earth are significantly altered? If so, the system would be termed unstable. This is because even a little perturbation can change the whole system,” explains Professor Kappeler. The issue of structural stability of finite dimensional integrable systems turned out to be daunting and it took generations of researchers to make significant progress. Astonishingly, the answer, given by the celebrated theorem of Kolmogorov-ArnoldMoser (KAM), is of a probabilistic nature: for a typical integrable systems, the orbits of many, but possibly not all solutions of a slightly perturbed system remain almost unchanged at all times. “Typically, instability is related to resonances of the unperturbed system, since resonances channel the energy in different ways,” says Professor Kappeler.
In the last thirty years or so significant progress has been made in extending the KAM theorem to integrable systems of infinite dimension: many of these systems admit integrable subsystems of finite dimension. Elements in these subsystems are sometimes referred to as periodic multi-solitons and might be large in size. Typically, these subsystems are
not invariant even under small perturbations, but for many of them one can show a KAM type theorem. Often, solutions of these integrable subsystems have specific properties, e.g., being a travelling wave solution. Hence a very natural question is, whether the corresponding solutions of the perturbed equation continue to have the same properties. “KAM type theorems can be used to answer such questions,” says Professor Kappeler.
Future research
While significant progress has been made in research into integrable systems of infinite dimension, there is still vast scope for further investigation.
A result for a class of finite dimensional integrable systems, which is complementary to the KAM theorem, has been obtained by Nekhoroshev. His theorem says that all solutions of a small perturbation of an integrable system in this class stay close to the orbits of the corresponding solutions of the unperturbed system. Up till now, corresponding theorems for integrable systems of infinite dimension have not been available. However, partial results have recently been obtained. “We are currently in the process of proving that solutions of small perturbations of the KdV equation, starting close to a large class of periodic multi-solitons, stay close to the orbit of the latter for a long period of time,” outlines Professor Kappeler.
The wider aim in this research is to develop ever more sophisticated mathematical tools, in order to deal with stronger perturbations and integrable subsystems of possibly infinite dimension of the above mentioned integrable systems. “Maybe the most challenging goal is to develop the normal form method and perturbation theory for integrable partial differential equations in two space dimensions such as the Kadomtsev-Petviashvili (KP) equation, which models a certain type of waves on the ocean,” explains Professor Kappeler.
HAMILTONIAN SYSTEMS OF INFINITE DIMENSION Hamiltonian systems of infinite dimension
Project Objectives
The concept of integrable system arose in the 19 th century in the context of finite dimensional classical mechanics. In the aftermath of groundbreaking numerical experiments on nonlinear systems by E. Fermi, J. Pasta, and S. Ulam, researchers have started applying it to infinite dimensional systems. The aim in the project is to advance the concept of integrable systems to evolution equations, relevant to the applied sciences, including the Korteweg-de Vries equation, the nonlinear Schrödinger equation, and the Benjamin-Ono equation.
Project Funding
The project is funded by the Swiss National Science Foundation
Contact Details
Project Coordinator, Professor Thomas Kappeler Institut für Mathematik Universität Zürich Winterthurerstrasse 190 CH-8057 Zürich E: thomas.kappeler@math.uzh.ch W: https://www.math.uzh.ch/index. php?id=people&semId=36&key1=113
References
(1) Kappeler, Thomas; Pöschel, Jürgen KdV & KAM. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 45. Springer-Verlag, Berlin, 2003. (2) Inci, Hasan; Kappeler, Thomas; Topalov, Peter , On the regularity of the composition of diffeomorphisms. Memoirs of the American Mathematical Society, 226 (2013) (3) Grébert, Benoît; Kappeler, Thomas The defocusing NLS equation and its normal form. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich, 2014. (4) Berti, Massimiliano; Kappeler, Thomas; Montalto, Riccardo. Large KAM tori for perturbations of the defocusing NLS equation. Astérisque 2018, no. 403.
Professor Thomas Kappeler
Thomas Kappeler is a Professor of Mathematics at the University of Zurich. He holds visiting positions at Universities in the US and also serves as an editor for the European Mathematical Society. His main research interests lie in global analysis and dynamical systems of infinite dimension.
