In this paper, we investigate the theory of $R$-brackets, Baxter brackets and Nijenhuis brackets in the Banach setting, in particular in relation with Banach Poisson-Lie groups. The notion of Banach Lie--Poisson space with respect to an arbitrary duality pairing is crucial for the equations of motion to make sense. In the presence of a non-degenerate invariant pairing on a Banach Lie algebra, these equations of motion assume a Lax form. We prove a version of the Adler-Kostant-Symes theorem adapted to $R$-matrices on infinite-dimensional Banach algebras. Applications to the resolution of Lax equations associated to some Banach Manin triples are given. The semi-infinite Toda lattice is also presented as an example of this approach.

In this paper we consider the Toda lattice $(\boldsymbol{p}(t); \boldsymbol{q}(t))$at thermal equilibrium, meaning that its variables$(p_i)$ and $(e^{q_i-q_{i+1}})$ are independent Gaussian and Gamma random variables, respectively. We justify the notion from the physics literature that this model can be thought of as a dense collection of solitons (or "soliton gas'') by, (i) precisely defining the locations of these solitons; (ii) showing that local charges and currents for the Toda lattice are well-approximated by simple functions of the soliton data; and (iii) proving an asymptotic scattering relation that governs the dynamics of the soliton locations. Our arguments are based on analyzing properties about eigenvector entries of the Toda lattice's (random) Lax matrix, particularly, their rates of exponential decay and their evolutions under inverse scattering.

Generalised Hydrodynamics (GHD) describes the large-scale inhomogeneous dynamics of integrable (or close to integrable) systems in one dimension of space, based on a central equation for the fluid density or quasi-particle density: the GHD equation. We consider a new, general form of the GHD equation: we allow for spatially extended interaction kernels, generalising previous constructions. We show that the GHD equation, in our general form and hence also in its conventional form, is Hamiltonian. This holds also including force terms representing inhomogeneous external potentials coupled to conserved densities. To this end, we introduce a new Poisson bracket on functionals of the fluid density, which is seen as our dynamical field variable. The total energy is the Hamiltonian whose flow under this Poisson bracket generates the GHD equation. The fluid density depends on two (real and spectral) variables so the GHD equation can be seen as a $2+1$-dimensional classical field theory. In its $1+1$-dimensional reduction corresponding to the case without external forces, we further show the system admits an infinite set of conserved quantities that are in involution for our Poisson bracket, hinting at integrability of this field theory.

We study the link between the degree growth of integrable birational mappings of order higher than two and their singularity structures. The higher order mappings we use in this study are all obtained by coupling mappings that are integrable through spectral methods, typically belonging to the QRT family, to a variety of linearisable ones. We show that by judiciously choosing these linearisable mappings, it is possible to obtain higher order mappings that exhibit the maximal degree growth compatible with integrability, i.e. for which the degree grows as a polynomial of order equal to the order of the mapping. In all the cases we analysed, we found that maximal degree growth was associated with the existence of an unconfining singularity pattern. Several cases with submaximal growth but which still possess unconfining singularity patterns are also presented. In many cases the exact degrees of the iterates of the mappings were obtained by applying a method due to Halburd, based on the preimages of specific values that appear in the singularity patterns of the mapping, but we also present some examples where such a calculation appears to be impossible.

We introduce a novel systematic construction for integrable (3+1)-dimensional dispersionless systems using nonisospectral Lax pairs that involve contact vector fields. In particular, we present new large classes of (3+1)-dimensional integrable dispersionless systems associated to the Lax pairs which are polynomial and rational in the spectral parameter.

We derive and analyze a three dimensional model of a figure skater. We model the skater as a three-dimensional body moving in space subject to a non-holonomic constraint enforcing movement along the skate's direction and holonomic constraints of continuous contact with ice and pitch constancy of the skate. For a static (non-articulated) skater, we show that the system is integrable if and only if the projection of the center of mass on skate's direction coincides with the contact point with ice and some mild (and realistic) assumptions on the directions of inertia's axes. The integrability is proved by showing the existence of two new constants of motion linear in momenta, providing a new and highly nontrivial example of an integrable non-holonomic mechanical system. We also consider the case when the projection of the center of mass on skate's direction does not coincide with the contact point and show that this non-integrable case exhibits apparent chaotic behavior, by studying the divergence of nearby trajectories We also demonstrate the intricate behavior during the transition from the integrable to chaotic case. Our model shows many features of real-life skating, especially figure skating, and we conjecture that real-life skaters may intuitively use the discovered mechanical properties of the system for the control of the performance on ice.

We consider the Cauchy problem for the defocusing Schr$\ddot{\text{o}}$dinger (NLS) equation with a nonzero background $$\begin{align} &iq_t+q_{xx}-2(|q|^2-1)q=0, \nonumber\\ &q(x,0)=q_0(x), \quad \lim_{x \to \pm \infty}q_0(x)=\pm 1. \end{align}$$ Recently, for the space-time region |x/(2t)|<1 which is a solitonic region without stationary phase points on the jump contour, Cuccagna and Jenkins presented the asymptotic stability of the $N$-soliton solutions for the NLS equation by using the $\bar{\partial}$ generalization of the Deift-Zhou nonlinear steepest descent method. Their large-time asymptotic expansion takes the form \begin{align} q(x,t)= T(\infty)^{-2} q^{sol,N}(x,t) + \mathcal{O}(t^{-1 }),\label{res1} \end{align} whose leading term is N-soliton and the second term $\mathcal{O}(t^{-1})$ is a residual error from a $\overline\partial$-equation. In this paper, we are interested in the large-time asymptotics in the space-time region |x/(2t)|>1 which is outside the soliton region, but there will be two stationary points appearing on the jump contour $\mathbb{R}$. We found a asymptotic expansion that is different from (\ref{res1}) $$\begin{align} q(x,t)= e^{-i\alpha(\infty)} \left(1 +t^{-1/2} h(x,t) \right)+\mathcal{O}\left(t^{-3/4}\right),\label{res2} \end{align}$$ whose leading term is a nonzero background, the second $t^{-1/2}$ order term is from continuous spectrum and the third term $\mathcal{O}(t^{-3/4})$ is a residual error from a $\overline\partial$-this http URL above two asymptotic results (\ref{res1}) and (\ref{res2}) imply that the region |x/(2t)|<1 considered by Cuccagna and Jenkins is a fast decaying soliton solution region, while the region |x/(2t)|>1 considered by us is a slow decaying nonzero background region.

The integrable bootstrap program allows one to express the tempered distributions associated with the multipoint functions of the integrable 1+1 dimensional Sinh-Gordon quantum field theory by means of explicit series. The convergence of the latter is an open problem that was only solved for the two-point case. In this work, by taking for granted the convergence of these series, we show that these expressions satisfy all of the Wightman axioms. This thus shows that, upon a yet to be proven convergence property, the integrable bootstrap based construction of correlation functions does lead to a quantum field theory.

We present a new and relatively elementary method for studying the solution of the initial-value problem for dispersive linear and integrable equations in the large-$t$ limit, based on a generalization of steepest descent techniques for Riemann-Hilbert problems to the setting of $\overline{\partial}$-problems. Expanding upon prior work of the first two authors, we develop the method in detail for the linear and defocusing nonlinear Schr\"odinger equations, and show how in the case of the latter it gives sharper asymptotics than previously known under essentially minimal regularity assumptions on initial data.

We construct a Lagrangian multiform for the class of cyclotomic (rational) Gaudin models by formulating its hierarchy within the Lie dialgebra framework of Semenov-Tian-Shansky and by using the framework of Lagrangian multiforms on coadjoint orbits. This provides the first example of a Lagrangian multiform for an integrable hierarchy whose classical $r$-matrix is non-skew-symmetric and spectral parameter-dependent. As an important by-product of the construction, we obtain a Lagrangian multiform for the periodic Toda chain by choosing an appropriate realisation of the cyclotomic Gaudin Lax matrix. This fills a gap in the landscape of Toda models as only the open and infinite chains had been previously cast into the Lagrangian multiform framework. A slightly different choice of realisation produces the so-called discrete self-trapping (DST) model. We demonstrate the versatility of the framework by coupling the periodic Toda chain with the DST model and by obtaining a Lagrangian multiform for the corresponding integrable hierarchy.

We investigate theoretically the fundamental phenomenon of the spontaneous, noise-induced modulational instability (MI) of a plane wave. The long-term statistical properties of the noise-induced MI have been previously observed in experiments and in simulations but have not been explained so far. In the framework of inverse scattering transform (IST), we propose a model of the asymptotic stage of the noise-induced MI based on $N$-soliton solutions ($N$-SS) of the integrable focusing one-dimensional nonlinear Schr\"odinger equation (1D-NLSE). These $N$-SS are bound states of strongly interacting solitons having a specific distribution of the IST eigenvalues together with random phases. We use a special approach to construct ensembles of multi-soliton solutions with statistically large number of solitons $N\sim100$. Our investigation demonstrates complete agreement in spectral (Fourier) and statistical properties between the long-term evolution of the condensate perturbed by noise and the constructed multi-soliton bound states. Our results can be generalised to a broad class of integrable turbulence problems in the cases when the wave field dynamics is strongly nonlinear and driven by solitons.

The Kadomtsev-Petviashvili reduction method is a crucial method to derive the solitonic solutions of (1+1) dimensional integrable system from high dimensional system. In this work, we explore to use the solutions of lower dimensional system to construct the solutions in the high dimensional one with the Darboux transformation. Especially, we utilize this method to disclose the relationship between the rogue wave and lump solutions. Under one-constraint method, the asymptotic analysis to the lump pattern of Kadomtsev-Petviashvili equation is given

The nonlinear Schr\"odinger equation (NLSE) stands out as the dispersive nonlinear partial differential equation that plays a prominent role in the modeling and understanding of the wave phenomena relevant to many fields of nonlinear physics. The question of random input problems in the one-dimensional and integrable NLSE enters within the framework of integrable turbulence, and the specific question of the formation of rogue waves (RWs) has been recently extensively studied in this context. The determination of exact analytic solutions of the focusing 1D-NLSE prototyping RW events of statistical relevance is now considered as the problem of central importance. Here we address this question from the perspective of the inverse scattering transform (IST) method that relies on the integrable nature of the wave equation. We develop a conceptually new approach to the RW classification in which appropriate, locally coherent structures are specifically isolated from a globally incoherent wave train to be subsequently analyzed by implementing a numerical IST procedure relying on a spatial periodization of the object under consideration. Using this approach we extend the existing classifications of the prototypes of RWs from standard breathers and their collisions to more general nonlinear modes characterized by their nonlinear spectra.

Hodograph equations for the n-dimensional Euler equations with the constant pressure and external force linear in velocity are presented. They provide us with solutions of the Euler in implicit form and information on existence or absence of gradient catastrophes. It is shown that in even dimensions the constructed solutions are periodic in time for particular subclasses of external forces. Several particular examples in one, two and three dimensions are considered, including the case of Coriolis external force.

Searching for integrable models is a central theme in theoretical and mathematical physics, as such systems offer valuable insights into the underlying structure and symmetries of complex physical phenomena. In this work, we contribute to this pursuit by proposing a new class of one-dimensional many-body integrable systems, which we refer to as the $q$-deformed Calogero's Goldfish system. Our construction employs $q$-deformation of logarithmic and exponential functions inspired by Tsallis' formalism in non-extensive statistical mechanics. Notably, the model satisfies the double-zero condition on its solutions, underscoring its integrable nature and offering a novel perspective on deformation techniques within exactly solvable systems.

Transport processes in crowded periodic structures are often mediated by cooperative movements of particles forming clusters. Recent theoretical and experimental studies of driven Brownian motion of hard spheres showed that cluster-mediated transport in one-dimensional periodic potentials can proceed in form of solitary waves. We here give a comprehensive description of these solitons. Fundamental for our analysis is a static presoliton state, which is formed by a periodic arrangement of basic stable clusters. Their size follows from a geometric principle of minimum free space. Adding one particle to the presoliton state gives rise to solitons. We derive the minimal number of particles needed for soliton formation, number of solitons at larger particle numbers, soliton velocities and soliton-mediated particle currents. Incomplete relaxations of the basic clusters are responsible for an effective repulsive soliton-soliton interaction seen in measurements. A dynamical phase transition is predicted to occur in current-density relations at low temperatures. Our results provide a theoretical basis for describing experiments on cluster-mediated particle transport in periodic potentials.

We investigate non-stationary heat transfer in the Kipnis-Marchioro-Presutti (KMP) lattice gas model at long times in one dimension when starting from a localized heat distribution. At large scales this initial condition can be described as a delta-function, $u(x,t=0)=W \delta(x)$. We characterize the process by the heat, transferred to the right of a specified point $x=X$ by time $T$, $$J=\int_X^\infty u(x,t=T)\,dx\,,$$ and study the full probability distribution $\mathcal{P}(J,X,T)$. The particular case of $X=0$ has been recently solved [Bettelheim \textit{et al}. Phys. Rev. Lett. \textbf{128}, 130602 (2022)]. At fixed $J$, the distribution $\mathcal{P}$ as a function of $X$ and $T$ has the same long-time dynamical scaling properties as the position of a tracer in a single-file diffusion. Here we evaluate $\mathcal{P}(J,X,T)$ by exploiting the recently uncovered complete integrability of the equations of the macroscopic fluctuation theory (MFT) for the KMP model and using the Zakharov-Shabat inverse scattering method. We also discuss asymptotics of $\mathcal{P}(J,X,T)$ which we extract from the exact solution, and also obtain by applying two different perturbation methods directly to the MFT equations.

We describe the most general ${\rm GL}_{NM}$ classical elliptic finite-dimensional integrable system, which Lax matrix has $n$ simple poles on elliptic curve. For $M=1$ it reproduces the classical inhomogeneous spin chain, for $N=1$ it is the Gaudin type (multispin) extension of the spin Ruijsenaars-Schneider model, and for $n=1$ the model of $M$ interacting relativistic ${\rm GL}_N$ tops emerges in some particular case. In this way we present a classification for relativistic Gaudin models on ${\rm GL}$-bundles over elliptic curve. As a by-product we describe the inhomogeneous Ruijsenaars chain. We show that this model can be considered as a particular case of multispin Ruijsenaars-Schneider model when residues of the Lax matrix are of rank one. An explicit parametrization of the classical spin variables through the canonical variables is obtained for this model. Finally, the most general ${\rm GL}_{NM}$ model is also described through $R$-matrices satisfying associative Yang-Baxter equation. This description provides the trigonometric and rational analogues of ${\rm GL}_{NM}$ models.

The classical Toda flow is a well-known integrable Hamiltonian system that diagonalizes matrices. By keeping track of the distribution of entries and precise scattering asymptotics, one can exhibit matrix models for log-gases on the real line. These types of scattering asymptotics date back to fundamental work of Moser. More precisely, using the classical Toda flow acting on symmetric real tridiagonal matrices, we give a "symplectic" proof of the fact that the Dumitriu-Edelman tridiagonal model has a spectrum following the Gaussian $\beta$-ensemble.