Challenges of dispersionless integrability: Hirota type equations

Dispersionless systems typically arise as long-wave approximations to equations governing various physical phenomena. Applications include shallow water theory, aerodynamics, Whitham averaging theory, Laplacian growth processes, general relativity, and differential geometry. In many particularly interesting cases the resulting dispersionless systems have an additional property of integrability (informally, this means that they are amenable to analytical, not just numerical, treatment). Recently, our group has proposed a novel approach to the classification of integrable models of this kind, known as the method of hydrodynamic reductions. It is based on the requirement that the original multi-dimensional system can be decoupled into a collection of consistent 1+1 dimensional systems of hydrodynamic type in an infinity of ways. It was demonstrated that this requirement provides an efficient classification criterion. Dispersionless integrability proved to be an exciting research area with deep links to generalised conformal geometry, theory of special functions, complex analysis, algebraic geometry, and twistor theory.

The key challenges of dispersionless integrability can be summarised as follows:

1. Prove that the moduli spaces of dispersionless integrable systems are finite-dimensional (that is, such systems depend on finitely many essential parameters). Prove that `generic' systems of this type can be parametrised by special functions such as generalised hypergeometric functions, elliptic functions, or modular forms.
2. Prove that in 3D, every dispersionless integrable system possesses an integrable dispersive regularisation (such regularisations are known to prevent breakdown of classical solutions by generating, near the point of gradient catastrophe, a zone of rapid modulated oscillations later transforming into solitons). For `generic' dispersionless integrable systems, such regularisations constitute a novel class of fully discrete integrable equations.
3. Prove that in 4D, every dispersionless integrable system is necessarily linearly degenerate (the property of linear degeneracy is closely related to the null condition of Klainerman that insures global existence of classical solutions, even without any dispersive regularisation).
4. Develop a general solution procedure for linearly degenerate dispersionless integrable systems (non-breaking character of a linearly degenerate evolution suggests a dispersionless analogue of the classical inverse scattering transform).
5. Generalise the method of hydrodynamic reductions to systems that are not translationally invariant (the main problem here is the lack of a general theory of integrability of translationally non-invariant systems of hydrodynamic type in 1+1 dimensions).
6. Relate dispersionless integrability to generalised conformal geometry (generalised Einstein-Weyl geometry in 3D, or generalised self-dual geometry in 4D).

In full generality, the problems formulated above are out of reach at present. This is primarily due to the complexity of the integrability conditions, as well as their subtle dependence on the type of system under study. In this project, we plan to address these challenges for the particularly interesting class of dispersionless Hirota type equations, which appear in applications in nonlinear acoustics (dispersionless Kadomtsev-Petviashvili equation), general relativity (Boyer-Finley equation), differential geometry (special Lagrangian submanifolds, affine hyperspheres), dispersionless limits of various integrable hierarchies of KP/Toda type, and so on. I strongly believe that successful solution of the above problems for Hirota type equations, and the relevant new analytic/geometric techniques, would significantly advance our understanding of multi-dimensional dispersionless integrability. In fact, the class of Hirota type equations is broad enough to contain all essential difficulties of general challenges.

There are several other areas (in addition to those mentioned in `Academic Beneficiaries'), where the proposed research is expected to make impact. Two of them are described below.

1. The Hele-Shaw dynamics describing the interface between viscous and ideal fluids (oil and water) is governed by a hierarchy of integrable dispersionless Hirota type equations. This dynamics is known to possess singular behaviour manifesting itself in fingering, bubbling, and the emergence of cusp-like singularities. A universal discrete/dispersive deformation of Hirota type equations (to be constructed in this project) will lead to a canonical integrable regularisation of the Hele-Shaw flow, and will provide a necessary background for the design of an optimal numerical scheme. Our discretisation procedure is expected to find important applications within a broader area of study known today as 'Laplacian growth'. In fact, the discrete counterpart itself can be viewed as an alternative model of the original problem, indeed, physical processes are generally smooth, although may exhibit large gradients in certain regions of space and time.

2. In applied mathematics, many properties of a complicated nonlinear system can be established by analysing its formal linearisation. In my work on geometric aspects of quasilinear PDEs, I came across a canonical procedure that approximates a given quasilinear system by a linearly degenerate system; in contrast to the formal linearisation, this approximation is second-order, and can capture more subtle properties of the original model. Taking into account non-breaking character of linearly degenerate dynamics, one can argue that this construction would provide a perfect regularisation of a fully nonlinear system. For Hirota type equations, this leads to a canonical approximation by a Monge-Ampere equation. I am convinced that, due to its invariance and simplicity, the approximation of this kind will become a useful tool in applied research. The only reason for not including this topic in the list of Objectives was the need to concentrate, at this stage, on the key theoretical challenges.

Educational impact
1. One of the outcomes of the project will be a well-trained RA with expertise in the areas of integrable systems, differential geometry, perturbative techniques, computer algebra, and potential industrial applications. We will do our best to ensure that RA becomes a valuable member of the academic community, or the scientific industrial community.
2. I plan to give lecture courses on the topic of dispersionless integrability (China, Ningbo University and Kazakhstan, Eurasian National University), with the aim to attract strong research students from abroad.
3. In the second year, I plan to organise a Research Workshop `Challenges of Dispersionless Integrability', with the aim of bringing together experts from both pure and applied communities. Along with invited talks, the programme will include lecture courses on the subject of dispersionless integrability and its applications, aimed at UK research students.

In order to broaden our vision of the impact of the proposed research, both PI and RA plan to present their results at two applied conferences:
1. SIAM Conference on Nonlinear Waves and Coherent Structures. This annual event fosters collaborations among applied mathematicians and engineers in the subjects as diverse as fluid mechanics, Bose-Einstein condensation, nonlinear optics, atmosphere and ocean dynamics, and chemical reactions. Some of these processes are governed by Hirota type equations.
2. IMA Conference on Mathematics of Surfaces. This annual conference is aimed at applications of differential geometry of surfaces in practical problems encountered in industry, surface design, and architecture. Our project is related to this area since many classes of surfaces with specific geometric constraints on their shape are governed by Hirota type equations.