Variational principles and integrable PDEs

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].

References [1] Lobb, S., & Nijhoff, F. (2009). Lagrangian multiforms and multidimensional consistency.

Journal of Physics A: Mathematical and Theoretical, 42(45), 454013. [2] Vermeeren, M. (2021). Hamiltonian structures for integrable hierarchies of Lagrangian PDEs.

Open Communications in Nonlinear Mathematical Physics, 1. [3] Lax, P. D. (1968).

Integrals of nonlinear equations of evolution and solitary waves. Communications on pure and applied mathematics, 21(5), 467-490. [4] Maddocks, J. H., & Sachs, R.

L. (1993). On the stability of KdV multi-solitons.

Communications on pure and applied mathematics, 46(6), 867-901. [5] Ferapontov, E. V., & Vermeeren, M. (2025).

Lagrangian multiforms and dispersionless integrable systems. Letters in Mathematical Physics, 115, 125. 94% of Loughborough’s research impact is rated world-leading or internationally excellent.

REF 2021