Variational principles and integrable PDEs (Ref: MA/MV-SF1/2026)

About the project

About the Project The focus of the project is to use variational principles, in particular those of Lagrangian multiform theory, to study particular solutions to integrable PDEs. Most nonlinear differential equations cannot be solved exactly. Integrable systems are the exceptions to this. They possess some hidden structure that allows one to obtain explicit formulas for at least some of their solutions. An important example of this are wave equations admitting soliton solutions. Integrable systems are relatively rare, but they provide valuable models in fundamental physics, signal processing, wave dynamics, and other fields. Integrability is the result of some structure underlying the differential equation. A wide variety of such structures exists, formulated through, for example, symmetry groups, differential geometry, spectral theory, or Hamiltonian systems. The aim of this project is to construct soliton solutions and other special solutions of integrable systems using variational principles. Variational principles for integrable systems are studied in Lagrangian multiform theory. This theory is a recent development in integrable systems, initiated in [1]. Its central object can be thought of as the Legendre transformation of a hierarchy of Hamiltonian PDEs [2]. So far, Lagrangian multiform theory has been used to study integrable equations and their symmetries, rather than specific solutions of those equations. The aim of the project is to construct solutions using Lagrangian multiform theory. There are some existing methods that will guide the way, such as the construction of soliton solutions using a constrained variational principle formulated in terms of the Hamiltonian structure [3,4], and the use of Lagrangian multiforms to describe systems of seemingly unrelated PDEs [5]. Name of primary supervisor/CDT lead: Mats Vermeeren m.vermeeren@lboro.ac.uk https://www.lboro.ac.uk/departments/maths/staff/mats-vermeeren/ Entry requirements: Students should have, or expect to achieve, at least a 2:1 honours (or international equivalent) in Mathematics or a related subject. A relevant master’s degree and a background in mathematical physics are desirable. English language requirements: Applicants must meet the minimum English language requirements. Further details are available on the International website ( http://www.lboro.ac.uk/international/applicants/english/ ). Bench fees required: No Closing date of advert: 1st April 2027 Start date: July 2026, October 2026, January 2027, July 2027 Full-time/part-time availability: Full-time 3 years Fee band: 2025/26 Band RA (UK £5,006, International £22,360) How to apply: All applications should be made online . Under programme name, select Mathematical Sciences. Please quote the advertised reference number: MA/MV-SF1/2026 in your application. To avoid delays in processing your application, please ensure that you submit a CV and the minimum supporting documents . The following selection criteria will be used by academic schools to help them make a decision on your application. Please note that this criteria is used for both funded and self-funded projects. Please note, applications for this project are considered on an ongoing basis once submitted and the project may be withdrawn prior to the application deadline, if a suitable candidate is chosen for the project. Project search terms: mathematics, pure mathematics, mathematical physics integrable systems Email Address Sci: sci-pgr@lboro.ac.uk

Research areas