Solutions to a class of nonlinear Schrödinger equations involving a nonlocal term.
Lead Research Organisation:
Swansea University
Department Name: College of Science
Abstract
Background
The equations studied in project arise in Physics and Chemistry in relation to the so-called Density Functional Theory. The main idea of the so-called Density Functional Theory (DFT), consists in describing complex many-body effects with a single particle framework. This mathematically translates into working with a single instead of many strongly coupled equations, yielding an accurate description in instances where particles are weakly interacting (e.g. drug-protein and protein-protein interactions). This allows to explain the geometries and dissociation energies of molecules and is key to tackle problems in science and technology, such as understanding and design of catalytic processes in enzymes, drug design in medicine, etc.. As pointed out
in recent papers major challenges and inaccuracies in DFT arise when dealing with strongly interacting particle systems, e.g. multi-electron atoms. This is mainly due to the failure of the used computational methods in capturing the mathematical features of nonlinear equations set on unbounded domains. The proposed research aims at focussing on a class of these equations which arise as "mean field approximations" of systems of partial differential equations (Hartree-Fock equations) describing strongly interacting systems of particles, for instance the electrons of an atom. In this context DFT yields a single equation known as Nonlinear Schrödinger-Poisson-Slater equation.
Aims
The main aim of the proposed research is twofold. A first part of the project is to identify conditions in order to prove the existence of infinitely many solutions to a class of these equations which involve a function, say F. The presence of this function makes the equation under study more general, and in fact can be interpreted as a "charge-corrector" to equations available in the literature. In this first part a class of functions F will be identified in order to rigorously prove the existence of infinitely many solutions to the proposed equation. At a later stage the project aim is to implement the new knowledge developed earlier on the solutions found (characterised in terms of "energy"), in order to improve existing computational methods, informed by the new results of the theory. This requires significant improvement of the existing computational methods because the equations are set on unbounded domains.
The methodologies which will be followed in the first part of the project are related to the Nonlinear Analysis of Partial Differential Equations and Variational Methods. The second part will be computational.
The student progressive tasks
Target 1: To review the mathematical literature related to finding multiple solutions to systems of Schrödinger-Poisson type, including references [5, 6]. To consolidate knowledge of the abstract Lusternik-Schnirelman theory, from [1] chapters 9 and 10 and Brower's degree theory, familiarising with the rigorous proofs of the main results of the theory.
Target 2: Preparation of a first draft of the thesis on the existence of multiple solutions to a nonlinear Schrodinger-Poisson system involving a non-radial charge density conjectured in [3].
Target 3: To learn the Finite Element Method techniques applied to simple elliptic partial differential equations, and consolidate knowledge in the relevant functional spaces. Main references: [2,4].
This will involve more systematically the second supervisor Prof. M. Edwards.
Target 4: Implementation of the theory developed earlier to approximate computationally the solutions found within Target 1 and Target 2.
Main references:
[1] A. Ambrosetti and A. Malchiodi. Nonlinear Analysis and Semilinear Elliptic Problems.
[2] S. Brenner and L. Scott. The Mathematic
The equations studied in project arise in Physics and Chemistry in relation to the so-called Density Functional Theory. The main idea of the so-called Density Functional Theory (DFT), consists in describing complex many-body effects with a single particle framework. This mathematically translates into working with a single instead of many strongly coupled equations, yielding an accurate description in instances where particles are weakly interacting (e.g. drug-protein and protein-protein interactions). This allows to explain the geometries and dissociation energies of molecules and is key to tackle problems in science and technology, such as understanding and design of catalytic processes in enzymes, drug design in medicine, etc.. As pointed out
in recent papers major challenges and inaccuracies in DFT arise when dealing with strongly interacting particle systems, e.g. multi-electron atoms. This is mainly due to the failure of the used computational methods in capturing the mathematical features of nonlinear equations set on unbounded domains. The proposed research aims at focussing on a class of these equations which arise as "mean field approximations" of systems of partial differential equations (Hartree-Fock equations) describing strongly interacting systems of particles, for instance the electrons of an atom. In this context DFT yields a single equation known as Nonlinear Schrödinger-Poisson-Slater equation.
Aims
The main aim of the proposed research is twofold. A first part of the project is to identify conditions in order to prove the existence of infinitely many solutions to a class of these equations which involve a function, say F. The presence of this function makes the equation under study more general, and in fact can be interpreted as a "charge-corrector" to equations available in the literature. In this first part a class of functions F will be identified in order to rigorously prove the existence of infinitely many solutions to the proposed equation. At a later stage the project aim is to implement the new knowledge developed earlier on the solutions found (characterised in terms of "energy"), in order to improve existing computational methods, informed by the new results of the theory. This requires significant improvement of the existing computational methods because the equations are set on unbounded domains.
The methodologies which will be followed in the first part of the project are related to the Nonlinear Analysis of Partial Differential Equations and Variational Methods. The second part will be computational.
The student progressive tasks
Target 1: To review the mathematical literature related to finding multiple solutions to systems of Schrödinger-Poisson type, including references [5, 6]. To consolidate knowledge of the abstract Lusternik-Schnirelman theory, from [1] chapters 9 and 10 and Brower's degree theory, familiarising with the rigorous proofs of the main results of the theory.
Target 2: Preparation of a first draft of the thesis on the existence of multiple solutions to a nonlinear Schrodinger-Poisson system involving a non-radial charge density conjectured in [3].
Target 3: To learn the Finite Element Method techniques applied to simple elliptic partial differential equations, and consolidate knowledge in the relevant functional spaces. Main references: [2,4].
This will involve more systematically the second supervisor Prof. M. Edwards.
Target 4: Implementation of the theory developed earlier to approximate computationally the solutions found within Target 1 and Target 2.
Main references:
[1] A. Ambrosetti and A. Malchiodi. Nonlinear Analysis and Semilinear Elliptic Problems.
[2] S. Brenner and L. Scott. The Mathematic
