Abstracts

Exploring Non-local Equations in Korteweg–De Vries (KdV) Hierarchy and Its Solutions

Ysla Adans (Institute of Theoretical Physics-IFT/UNESP)

Integrable models in (1+1) dimensions are distinguished by possessing an infinite number of conservation laws, each directly linked to a nonlinear equation, forming an integrable hierarchy.  The systematic construction of integrable hierarchies has been shown to be closely related to graded Lie algebraic structures. For instance, the affine algebra sl(2) generates the mKdV hierarchy, which contains both the sinh-Gordon equation and the mKdV equation. In this talk, I will apply such techniques to explore negative (non-local) flows within the KdV Hierarchy, focusing on three key aspects: 

1) Generalized structure: We develop a generalized algebraic framework for the KdV Hierarchy, facilitating the derivation of novel non-local flows;

2) Mapping between KdV and mKdV flows: Through Gauge-Miura transformation, we establish a mapping between non-local flows in the KdV and mKdV hierarchies;

3) Novel Solutions: Utilizing the Miura transformation and different vacuum orbits, we obtain new solutions within the KdV Hierarchy.

Vortices in one- and two-component superfluid systems

Ryan Doran (Newcastle University)

Superfluids, such as those formed by ultra-cold atomic Bose-Einstein Condensates (BECs), have incredible properties, such as the ability to flow without viscous effects, and the fact that vorticity is quantized.

Although the problem of superfluid flow past a potential barrier is a well-studied problem in BECs, fewer studies have considered the case of superfluid flow through a disordered potential. We consider the case of a superfluid in a channel with multiple point-like barriers, randomly placed to form a disordered potential. We identify the relationship between the relative position of two point-like barriers, and the critical velocity for vortex nucleation of this arrangement, before considering a system with many obstacles. We then study how the flow of a superfluid in a point-like disordered potential is arrested through the nucleation of vortices and the breakdown of superfluidity. We then consider the vortex decay rate as the width of the barriers and show that vortex pinning becomes an important effect. 

We then turn our attention to a two-component BEC in the immiscible limit. In such a system, if vortices are formed in a "majority" component, atoms in the "minority" component will fill the vortex cores, modifying the vortex profile. We show that a variational approach can be employed to approximate the vortex profile for a range of atom numbers in the in-filling component, and that these solutions are stable to small perturbations. We then consider the dynamics of these in-filled vortices.

Hamiltonian aspects of the kinetic equation for soliton gas

Jenya Ferapontov (Loughborough University)

We investigate Hamiltonian aspects of the integro-differential kinetic equation for dense soliton gas which results as a thermodynamic limit of the Whitham equations. Under a delta-functional  ansatz, the kinetic equation  reduces to a non-diagonalisable  system of hydrodynamic type whose matrix consists of several  2x2 Jordan blocks. We demonstrate that the resulting system possesses local Hamiltonian structures of differential-geometric type, for all standard two-soliton interaction kernels (KdV, sinh-Gordon, hard-rod, Lieb-Liniger, DNLS, and separable cases). In the hard-rod case, we show that the continuum limit of these structures   provides a local multi-Hamiltonian formulation of the full kinetic equation.

Based on Joint work with Pierandrea Vergallo, P. Vergallo, E.V. Ferapontov, Hamiltonian aspects of the kinetic equation for soliton gas, arXiv:2403.20162.

The Ultra Discrete KdV equation and Soliton Gases.

Claire Gilson (University of Glasgow)

The ultra discrete KdV equation is a system which is discrete in both space and time.  Unlike normal lattice equations where standard operations of + and x are used,  in ultra discrete systems these operations are essentially replaced by Max and +.   The resulting ultra discrete KdV equation retains many of the properties of the original integrable KdV equation including multiple soliton solutions.   In this talk we investigate an ultra discrete soliton gas and look at some of the basic properties to see how closely they mimic those of the usual soliton gas.

On the N -wave hierarchy with constant boundary conditions

Georgi Grahovski (University of Essex)

In this talk, we will present the direct scattering transform for the N -wave resonant interaction equations with non-vanishing boundary conditions. For special choices of the boundary values Q±, we outline the spectral properties of L, the direct scattering transform and construct its fundamental analytic solutions. Then, we generalise Wronskian relations for the case of constant boundary conditions.

Next, using the Wronskian relations we derive the dispersion laws for the N-wave hierarchy and describe the NLEE related to the given Lax operator. We will present examples of 3- and 4-wave resonant interaction systems related to sl(3,C) and sp(4,C) Lie algebras. Finally, we outline the list of conserved quantities for the N-wave systems considered here.

Based on a joint work with Vladimir S. Gerdjikov [1, 2].

[1] V. S. Gerdjikov, G. G. Grahovski, On the 3-wave equations with constant boundary conditions, PLISKA Stud. Math. Bulgar. 21 (2012), 217–236 [E-print: arXiv.1204.5346].

[2] V. S. Gerdjikov, G. G. Grahovski, On the N -waves hierarchy with constant boundary conditions. Spectral properties, Int. J. of Geom. Methods in Modern Physics (2024) (to appear) [E-print: arXiv:2403.12925]

Quantisation ideals, classical limit and non-Abelian Hamiltonian dynamics.

Sasha Mikhailov (University of Leeds)

We revisit the problem of quantisation and look at it from an entirely new angle, focusing on the quantisation of dynamical systems themselves, rather than of their Poisson structures [1]. Our starting point is a dynamical system defined on a free associative algebra A generated by non-commutative dynamical variables. The dynamical system defines a derivation of the algebra ∂ : A |→ A. In our approach the problem of quantisation reduces to the problem of finding a two- sided quantisation ideal in A. By definition, a two-sided ideal I ⊂ A is said to be a quantisation ideal for (A,∂) if it satisfies the following two properties:

(1) The ideal I is ∂-stable: ∂(I) ⊂ I;

(2) The quantum algebra A/I admits a basis of normally ordered monomials in the dynamical variables.

In application to the Volterra hierarchy we found first examples of bi-quantum systems which are quantum counterparts of bi-Hamiltonian systems in the classical theory, as well as first examples of quantum systems that cannot be obtained as deformations of classical dynamical systems with commutative variables [2, 3]. The quantisation ideals approach sheds light on the long standing problem of operator’s ordering.

By a well-known procedure, usually referred to as “taking the classical limit”, a quantum system becomes classical system with commutative variables, equipped with a Poisson bracket, Poisson algebra, and a Hamiltonian derivation defined by the Poisson bracket. With Pol Vanhaecke we found a generalisation of it to the case of formal deformations of an arbitrary noncommutative algebra A [4]. We show that the deformation leads to a commutative Poisson algebra structure on Π(A) := Z(A) × (A/Z(A)) and a Π(A)-Poisson module A, where Z(A) denotes the centre of A. The Poisson module defines Hamiltonian derivations ∂H : A → A, and non-Abelian Hamiltonian equations with Hamiltonians H belonging to Π(A) instead of A as in the commutative case. We illustrate it with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the quantised Volterra chains, Kontsevich integrable map, the quantum plane and the quantised Grassmann algebra.

[1] A.V. Mikhailov. Quantisation ideals of nonabelian integrable systems. Russ. Math. Surv., 75(5):199, 2020.

[2] S. Carpentier, A.V. Mikhailov and J.P. Wang. Quantisation of the Volterra hierarchy. Lett. Math. Phys., 112:94, 2022.

[3] S. Carpentier, A.V. Mikhailov and J.P. Wang. Hamiltonians for the quantised Volterra hierarchy. arXiv:2312.12077, 2023. (Submitted to Nonlinearity)

[4] A.V. Mikhailov and P. Vanhaecke. Commutative Poisson algebras from deformations of non- commutative algebras. arXiv:2402.16191v2, 2024. (Submitted to CMP)

Hamiltonian and recursion operators for a discrete analogue of the Kaup-Kupershmidt equation

Edoardo Peroni (University of Kent)

In this talk we will discuss the algebraic properties of a discrete analogue of the Kaup-Kupershmidt equation. This equation can be seen as a deformation of the modified Narita-Itoh-Bogoyavlensky equation and has the Kaup-Kupershmidt equation in its continuous limit. Using its Lax representation we explicitly construct a recursion operator for this equation and prove that it is a Nijenhuis operator via the use of pre-Hamiltonian operators. Moreover, we present the bi-Hamiltonian structures for this equation. This talk is bases on a homonymous paper written in collaboration with Jing Ping Wang.

New quantum spin chains

Ana Retore (Durham University)

Integrable quantum spin chains have applications in several areas of physics. Examples are the Heisenberg spin chain and the Potts model in magnetism, Hubbard model in superconductivity and the recent applications in the several instances of the gauge/gravity duality.

Integrability was used to construct very effective techniques to solve these models even when at intermediate and strong coupling. As a consequence, having a way to determine if a model is integrable or not, and to classify integrable models within certain classes is currently a very relevant problem.

In this talk, I will give an introduction to quantum spin chains and show a new method to construct quantum integrable models. I will present some of the new systems including a model where the electrons only move in the chain when they are in pairs.

From Semiclassical to Quantum Solitons in Superfluids

Hayder Salman (University of East Anglia)

The simplest system that can exhibit superfluid properties is a weakly interacting ultracold atomic Bose gas. Such systems are well described within a mean-field description in terms of the Gross-Pitaevskii equation, also known as the Nonlinear Schrödinger (NLS) equation. Nowadays, ultracold Bose gases can be realized in tightly confined traps that can be engineered to reduce the effective dynamics to 1D.  In this limit, we expect the system to exhibit localised excitations in the form of the well-known soliton solutions of the integrable NLS equation.

In the first part of the talk,  I will discuss these 1D solutions in the context of what can be realized in current experiments, and their relevance to excitations that can arise in higher dimensional 2D and 3D systems.  However,  the underlying fundamental description from which the NLS equation is derived is a linear many-body Schrödinger equation describing a fully quantum mechanical system of indistinguishable particles.  A question that arises, is how do these semi-classical soliton solutions emerge from a quantum description.  In the second part,  I will aim to address these questions within the framework of the 1D NLS and Lieb-Liniger models. We will demonstrate that the relevant excitation spectra for the soliton modes can also be recovered in the fully quantum description and discuss the implications on how soliton density profiles may emerge in this description.