Periodic Reporting for period 2 - IPaDEGAN (Integrable Partial Differential Equations: Geometry, Asymptotics, and Numerics.)
Berichtszeitraum: 2020-01-01 bis 2023-12-31
Partial Differential Equations (PDEs) are among the main tools for an efficient modelling of physical phenomena. Integrable PDEs, exhibiting infinite-dimensional analogues of regular (integrable) behaviour displayed by finite dimensional systems, began to be studied in the middle of the XX century in fluid dynamics, field theory and plasma physics.
It was observed that, in specific models and regimes, a balance between non-linearity and dispersion leads to the formation of stable patterns. The modern theory of integrable systems grew up around the study of the Korteweg de Vries (KdV) equation, with the discovery of the recurrence behaviour of soliton solutions, the Lax pair formulation, and the infinite number of conservation laws. Over the last 40 years refined analytical geometrical as well as numerical tools have been developed to study integrable or nearly integrable systems.
The idea that an integrable behaviour persists in real (non-integrable) systems, together with the combination of the state-of-the-art numerical methods with front-line geometrical and analytical techniques in the theory of Hamiltonian PDEs is the leitmotiv of this research project. The predictive power of numerics and scientific computing is used both as a testing tool for theoretical models and as a generator of new conjectures.
Random matrices, introduced in the study of the statistical properties of quantum systems with a great number of degrees of freedom (heavy nuclei), have been shown to successfully enter various fields such as number theory, combinatorics, quantum field theory, and even wireless telecommunications.
Just like the class of dispersive PDEs mentioned above, these random “objects” display in suitable asymptotic scalings, a completely integrable behaviour, conjectured to be universal, in the same way as the local statistical properties of random matrices become independent of the probability distribution when the size of the matrix grows indefinitely.
One of the principal scientific aim of the Network is to create a fertile research environment for younger scientists (especially Ph.D. students and Post-docs) working in the fields of Mathematical Physics by means of a steady and intense flow of personnel exchanges between the European and the Overseas nodes of the Network. It is expected that the broad interdisciplinary basis and intertwining of methods of Geometry and Mathematical Physics lead to solutions of hard geometrical problems and to formulations of new mathematical models of physical phenomena.
The overall methodological approach relies on combining analytical, geometrical and numerical techniques, so far often confined within different areas of Mathematics and Physics. The power of efficient numerical tools is instrumental to dismiss wrong conjectures, to formulate new ones and even to support some steps for their analytical proof. General tools of the theory of integrable systems such as Riemann – Hilbert boundary value problems, bi-Hamiltonian geometry, Frobenius manifolds, as well as analytical tools such as general complex analysis, potential theory in the complex plane, Deift-Zhou steepest descent method for oscillatory integrals are being used in the pursue of our goals.
It was observed that, in specific models and regimes, a balance between non-linearity and dispersion leads to the formation of stable patterns. The modern theory of integrable systems grew up around the study of the Korteweg de Vries (KdV) equation, with the discovery of the recurrence behaviour of soliton solutions, the Lax pair formulation, and the infinite number of conservation laws. Over the last 40 years refined analytical geometrical as well as numerical tools have been developed to study integrable or nearly integrable systems.
The idea that an integrable behaviour persists in real (non-integrable) systems, together with the combination of the state-of-the-art numerical methods with front-line geometrical and analytical techniques in the theory of Hamiltonian PDEs is the leitmotiv of this research project. The predictive power of numerics and scientific computing is used both as a testing tool for theoretical models and as a generator of new conjectures.
Random matrices, introduced in the study of the statistical properties of quantum systems with a great number of degrees of freedom (heavy nuclei), have been shown to successfully enter various fields such as number theory, combinatorics, quantum field theory, and even wireless telecommunications.
Just like the class of dispersive PDEs mentioned above, these random “objects” display in suitable asymptotic scalings, a completely integrable behaviour, conjectured to be universal, in the same way as the local statistical properties of random matrices become independent of the probability distribution when the size of the matrix grows indefinitely.
One of the principal scientific aim of the Network is to create a fertile research environment for younger scientists (especially Ph.D. students and Post-docs) working in the fields of Mathematical Physics by means of a steady and intense flow of personnel exchanges between the European and the Overseas nodes of the Network. It is expected that the broad interdisciplinary basis and intertwining of methods of Geometry and Mathematical Physics lead to solutions of hard geometrical problems and to formulations of new mathematical models of physical phenomena.
The overall methodological approach relies on combining analytical, geometrical and numerical techniques, so far often confined within different areas of Mathematics and Physics. The power of efficient numerical tools is instrumental to dismiss wrong conjectures, to formulate new ones and even to support some steps for their analytical proof. General tools of the theory of integrable systems such as Riemann – Hilbert boundary value problems, bi-Hamiltonian geometry, Frobenius manifolds, as well as analytical tools such as general complex analysis, potential theory in the complex plane, Deift-Zhou steepest descent method for oscillatory integrals are being used in the pursue of our goals.
In its whole life span (that extended from 2018 to 2023 with a one year suspension and a one year extension) the project's scientific production amounted to 66 peer-reviewed papers published in renowned Journals in Mathematics and Physics.
Among its main scientific achievements we mention:
-) In WP2, we obtained results about the continuation of solutions to the Airy equationsfor short times beyond the collapse at the physical boundary, developed and studied new models in the theory of stratified fluid dynamics. Also, we provided a
rigorous definition af a “Soliton gas” for the Korteweg-de Vries equation, and discovered the “soliton shileding” phenomenon in the Non Linear Schrödinger (NLS) equation,
-) In WP3, a novel representation of the joint moments of the characteristic polynomial of a random matrix in the Circular Unitary Ensemble was found.
The theory of τ-functions, and their applications in geometry was discussed also in relation with the theory of Fredholm determinants.
-) In WP4 we highlight results about Davey-Stewartson equations, and in particular the creation of a new method for the numerical solution of the eikonal problem to discuss the generation of oscillations from smooth initial data, as well as
progress in the numerical study of integrable equations such the KdV, NLS and the Camassa-Holm equations. Also, public libraries of codes for solving the above mentioned equations as well as Painlevé-type equations were released.
The dissemination of our scientific results within the Academia was achieved by the organisation (or co-organisation) of a total of 2 Conferences and 8 workshops (three of these events were among the Project's deliverable items). Members of the Network (Prof. T. Grava and Dr. M. Cafasso) organised the semester" Integrability and Randomness in Mathematical Physics" within the Jean-Morlet Chair scientific programme – Marseille (FR), January-June 2019. The Project's activities were also presented in a number of other Conferences and Workshops.
For what the dissemination to the public at large is concerned, IPaDEGAN has been represented in five major outreach events.
• The European Researchers’ Night event in Milano: MEETmeTONIGHT – September 2018 and 2019, Milano (IT).
• “Fete de la Science” event in Angers – and Dijon October 2019 (FR).
• ‘Trieste Next–2019’ – September 2019, Trieste (IT).
Lectures aimed at high school or perspective University students were give, in particular during Open days manifestations.
Among its main scientific achievements we mention:
-) In WP2, we obtained results about the continuation of solutions to the Airy equationsfor short times beyond the collapse at the physical boundary, developed and studied new models in the theory of stratified fluid dynamics. Also, we provided a
rigorous definition af a “Soliton gas” for the Korteweg-de Vries equation, and discovered the “soliton shileding” phenomenon in the Non Linear Schrödinger (NLS) equation,
-) In WP3, a novel representation of the joint moments of the characteristic polynomial of a random matrix in the Circular Unitary Ensemble was found.
The theory of τ-functions, and their applications in geometry was discussed also in relation with the theory of Fredholm determinants.
-) In WP4 we highlight results about Davey-Stewartson equations, and in particular the creation of a new method for the numerical solution of the eikonal problem to discuss the generation of oscillations from smooth initial data, as well as
progress in the numerical study of integrable equations such the KdV, NLS and the Camassa-Holm equations. Also, public libraries of codes for solving the above mentioned equations as well as Painlevé-type equations were released.
The dissemination of our scientific results within the Academia was achieved by the organisation (or co-organisation) of a total of 2 Conferences and 8 workshops (three of these events were among the Project's deliverable items). Members of the Network (Prof. T. Grava and Dr. M. Cafasso) organised the semester" Integrability and Randomness in Mathematical Physics" within the Jean-Morlet Chair scientific programme – Marseille (FR), January-June 2019. The Project's activities were also presented in a number of other Conferences and Workshops.
For what the dissemination to the public at large is concerned, IPaDEGAN has been represented in five major outreach events.
• The European Researchers’ Night event in Milano: MEETmeTONIGHT – September 2018 and 2019, Milano (IT).
• “Fete de la Science” event in Angers – and Dijon October 2019 (FR).
• ‘Trieste Next–2019’ – September 2019, Trieste (IT).
Lectures aimed at high school or perspective University students were give, in particular during Open days manifestations.
During all of the action lifespan, the Network’s activities have kept on developing novel methodologies combining techniques coming from apparently unrelated areas of Mathematics such as Geometry, Analysis and Numerical Analysis, applied to problems originating from Physics and Mathematics. For instance, asymptotic methods were applied in models of integrable and near-to integrable equations in one space dimension, and problems in the theory of random matrices have been solved by using techniques coming from Conformal Field Theory. Also, new numerical techniques were used to study semi-classical limits of integrable models in two space dimensions.
An immediate impact was given in terms of dissemination to Workshops (around 75 talks) of the project’s major results, as well as of the publications of papers in renowned Mathematical and Physical Journals.
Form a broader perspective, later impact will certainly come out of the realizations of the goals concerning the application of our results to a variety of unsolved problems in Mathematics and Mathematical Physics. Potential social and socio-economic by the development of our research about stratified fluid flows, to be potentially applied in oceanography and marine sciences. Not less important os the contribution given in terms of training and career enhancement of high level young scientists. Most of the younger participants in the project is developing a remarkable Academic career, but a few of them are applying skill refined during their secondments and their Ph.D. studies outside the Academia.
An immediate impact was given in terms of dissemination to Workshops (around 75 talks) of the project’s major results, as well as of the publications of papers in renowned Mathematical and Physical Journals.
Form a broader perspective, later impact will certainly come out of the realizations of the goals concerning the application of our results to a variety of unsolved problems in Mathematics and Mathematical Physics. Potential social and socio-economic by the development of our research about stratified fluid flows, to be potentially applied in oceanography and marine sciences. Not less important os the contribution given in terms of training and career enhancement of high level young scientists. Most of the younger participants in the project is developing a remarkable Academic career, but a few of them are applying skill refined during their secondments and their Ph.D. studies outside the Academia.