Papers

We study the formation and interaction of dispersive shock waves (DSWs) in the two-dimensional Toda lattice subject to wedge-type initial conditions, and show that their interaction gives rise to a discrete analog of Mach reflection for dispersive shock waves in discrete systems. The initial jump across each leg of the wedge acts locally as a Riemann problem for the one-dimensional Toda lattice, producing two oblique DSWs whose leading-edge soliton amplitude is determined explicitly by the one-dimensional Whitham modulation theory. The two-dimensional nature of the problem manifests when these oblique DSWs meet along the symmetry axis. We show that, for compressive wedges (i.e., when the initial conditions are such that two oblique DSWs that are generated propagate toward each other), a critical slope $q_{\mathrm{cr}}$ separates two regimes: in the subcritical regime ($q<q_{\mathrm{cr}}$) the interaction is resonant and it produces an expanding DSW whose amplitude, length and velocity are explicitly computed by using exact soliton solutions of the two-dimensional Toda lattice; in the supercritical regime ($q>q_{\mathrm{cr}}$) the interaction is ordinary and produces a localized peak whose amplitude is determined analytically. We also show qualitatively that a similar dichotomy between two regimes exists for expansive wedges (i.e., when the initial conditions are such that the two oblique DSWs propagate away from each other). We confirm all analytical predictions by comparing them with the results of direct numerical simulations. Finally, we show that the continuum limit of the result is consistent with the analogous theory for the Kadomtsev-Petviashvili equation, providing an independent validation of the analytical framework.

We introduce a numerical direct scattering transform scheme for dark solitons of the nonlinear Schrodinger equation, enabling the identification and complete characterization of nonlinear coherent structures in defocusing media with a continuous-wave (CW) background. Our scheme is based on numerically solving the auxiliary Zakharov-Shabat scattering problem with CW boundary conditions and on analytically derived expressions that relate the elements of the transfer matrix to the scattering data for dark solitons and continuous-spectrum waves. To test our approach, we consider two analytically solvable cases of the scattering problem: i) rectangular, and ii) hyperbolic tangent hollows in the CW background, which can contain an arbitrary number of dark solitons, with known scattering data. We revisit the analytical derivations and obtain a complete set of soliton parameters represented by discrete eigenvalues and norming constants. By supplementing the direct scattering transform algorithm with high-precision arithmetic to accurately recover soliton norming constants, we provide a robust method to analyze data from numerical or natural experiments on complex wave fields in optical, hydrodynamical, and other physical systems.

We develop the Sato-theoretic dressing framework for the anti-self-dual Yang-Mills (ASDYM) hierarchy based on a normalized Riemann-Hilbert decomposition. A four-sector expansion of the generating function yields a bi-infinite matrix array of relative coordinates, extending the affine coordinates on the big cell of the Sato Grassmannian. We derive the Sato-Wilson equations, continuous coordinate flows, and their discrete analogues. Several classical integrable hierarchies are recovered under dimensional reduction constraints, with their nonlinear variables identified as specific relative coordinates.

Different instances of the Riemann-Hilbert correspondence relate initial value spaces of Painlevé equations to affine varieties known as monodromy surfaces, built from monodromy invariants for associated linear ODEs. We show that these affine varieties admit realisations as embedded affine del Pezzo surfaces, characterised by their degree and a prescribed divisor at infinity, first in this paper for cases associated with Painlevé equations $\rm{VI},\rm{IV},\rm{II},\rm{I}$. We prove that the monodromy groups of these monodromy surfaces form the finite parts of the affine Weyl symmetry groups of the corresponding Painlevé equations. We realise the monodromy group in each case: analytically as permutations of lines induced by continuation along loops in parameter space, Galois-theoretically in terms of the function field of the incidence variety of lines, and combinatorially via their intersection graph. This in particular yields an interpretation of the parameter spaces of Painlevé equations, modulo symmetries, as moduli spaces of categories of embedded affine varieties. Further, it shows that, despite the affine Weyl group symmetries becoming trivial when conjugated by the corresponding Riemann-Hilbert map, the finite Weyl group part survives as monodromy of the monodromy surface.

Fu and Nijhoff's direct linearization scheme for the KP hierarchy and its reductions employs a system of evolution equations of an infinite matrix $U$ with quadratic nonlinearity. Part of the matrix elements of $U$ can be identified with affine coordinates $w_{ij}$ of the top cell of the Sato Grassmannian. The evolution equations of these matrix elements are identical to the evolution equations of $w_{ij}$ representing the KP hierarchy in geometric terms. This geometric interpretation can be extended to other matrix elements of $U$ by introducing negative flows to the evolution equations of $U$. The extended system turns out to be substantially equivalent to the two-component KP hierarchy. The Cauchy matrix approach to the KP hierarchy can be explained in this geometric perspective. Multi-component generalizations of Fu and Nijhoff's nonlinear system are related to the AKNS and ASDYM hierarchies. Multi-component Sato Grassmannians show up therein as the relevant geometric structure.

In this paper, we establish a connection between joint moments of characteristic polynomials and their derivatives in the Circular Jacobi Ensemble, a generalisation of the Circular Unitary Ensemble, and solutions of nonlinear Painlevé equations. For finite $N$, we show that these joint moments are characterised by a solution of the $\sigma$-Painlevé V equation for all real moment exponents in their admissible range. Under an appropriate large-$N$ scaling limit, we further prove that the limiting joint moments admit a representation in terms of a solution of the $\sigma$-Painlevé III$'$ equation for a certain range of moment exponents. As applications, we answer a question posed by Assiotis et al. in [Math. Physics. Anal. Geom. 25 (2022), no.2, Paper No. 15, 24pp, Remark 1.7] by showing that the characteristic function of a distinguished random variable is connected with the $\sigma$-Painlevé III$'$ equation for complex parameters. Furthermore, we investigate joint moments involving higher-order derivatives of characteristic polynomials in the Circular Jacobi Ensemble. As a consequence, we extend a result of Assiotis et al. in [Comm. Pure Appl. Math. 79 (2026), no. 7, 1771-1827, Theorem 1.11] concerning the joint moments of a sequence of random variables arising from the ergodic decomposition of Hua-Pickrell measures, from real parameters to complex parameters.

In this paper, we investigate the modified Toda (mToda) hierarchy, which can be regarded as the 2-component first modified Kadomtsev-Petviashvili (mKP) hierarchy. We first investigate the connection between the Toda and mToda tau functions. Based on this, we construct the transformations for the mToda tau functions and Lax operators. Furthermore, we present the mToda squared eigenfunction symmetries and derive the Adler-Shiota-van Moerbeke (ASvM) formula, which plays a crucial role by connecting the actions of the additional symmetries on the wave functions with the Sato–Bäcklund transformations of the tau functions. Finally, by establishing the equivalence between the actions of vertex operators on the mToda tau functions and the multi-step mToda transformations, we derive the mToda addition formulas, also known as the generalized Fay identities.

We develop a general procedure for constructing time-dependent integrable chiral field theories from autonomous unitary difference-form $S$-matrices satisfying the Yang-Baxter equation. Spectral parameters are transported along the free right- and left-moving characteristics, defining a map from physical spacetime to spectral space on which the two-body scattering data are evaluated. Requiring spatially homogeneous right-left scattering forces the characteristic map to be affine. The inverse Cayley transform then determines the local contact interaction, while evaluation along the affine spectral trajectory fixes its time dependence. Thus, the nonautonomous interaction is determined by the autonomous scattering data together with chiral kinematics and spatial homogeneity, rather than being introduced independently. The Yang–Baxter equation supplies the factorized many-body transport, and on a spatial circle periodicity leads to quantum Knizhnik–Zamolodchikov (qKZ) equations whose compatibility defines a flat discrete transport in spectral space. Pulling the corresponding spectral-space amplitudes back to physical coordinates gives the time-dependent many-body wavefunctions. Rational $SU(N)$, trigonometric $U_q(\widehat{\mathfrak{sl}}_2)$, and rational $O(N)$ scattering illustrate how the same mechanism generates distinct nonautonomous interactions. The resulting framework gives a geometric route from autonomous factorized scattering data to time-dependent integrable field theories.

Zhou and Yu [Appl. Math. Lett. 178 (2026) 109937] studied the space-time shifted nonlocal nonlinear Schrödinger (NLS) equation $iq_t(x,t)=q_{xx}(x,t)-2\sigma q^2(x,t) \bar{q}(x_0-x,t_0+\lambda t),\,\, \lambda=\pm1$, and claimed that for both choices of $\lambda$ the equation is integrable. Here we examine this claim from the perspective of reductions of the standard integrable NLS system. We find that the compatibility of the equation with the NLS system is true only when there is no time reversal in the argument. In other words, for this equation it requires $\lambda=1$ and $t_0=0$. The case with $\lambda=-1$ is a valid nonlocal equation, but cannot be derived from the standard NLS system by the corresponding complex conjugate shifted nonlocal reduction, and its integrability therefore does not follow from that integrable NLS system. We also discuss the equivalence between shifted (with real shifts) and unshifted nonlocal reductions and give the adjusted form of the Hirota bilinear equation.

We consider the open XXZ spin chain with unparallel boundary fields, in the special case in which the Vertex-IRF transform of the K-matrix labelling the boundary field at site 1 is diagonal and invariant under the value of the gauge parameters, the other K-matrix being arbitrary. We discuss here several properties related to this special gauge invariant boundary condition. Such a model can be solved in the Vertex-IRF framework with arbitrary gauge parameters: we can diagonalise the transfer matrix by means of generalised Sklyanin's Separation of Variable (SoV) approach in a family of SoV bases that depend on two arbitrary gauge parameters $\alpha$ and $\beta$, $\beta$ being the so-called dynamical parameter and $\alpha$ an arbitrary shift of the spectral parameter. We show here that such SoV bases depend actually on $\alpha$ and $\beta$ through the difference $\alpha-\beta$ only. Considering the ungauged limit in which $\alpha-\beta$ tends to infinity, we show how to compute all elementary building blocks for the correlation functions of this model. We finally show how to compute overlaps between eigenstates of this model and eigenstates of a spin chain sharing the same boundary condition at site $N$ and in which the gauge invariant boundary condition at site 1 has been changed to a more general one.

We study quantum dynamics through the lens of entanglement generation and characterize the underlying unitary evolution by the distinct signatures it imprints on the time-dependent entangling power. For a unitary operator, we characterize ergodicity by the equality between its long-time-averaged entangling power and the Haar-averaged linear entropy. We define mixing more stringently as the convergence of the time-dependent entangling power itself to the Haar value at long times. Within this framework, we establish the ergodic hierarchy of dynamical behavior, showing in particular that mixing implies ergodicity, whereas ergodicity does not necessarily imply mixing. As an application, we find that two-qubit unitary gates are neither ergodic nor mixing: their long-time-averaged entangling power can take only four discrete values, none of which coincides with the Haar average. We then investigate many-body dynamics using the kicked Ising chain and find that the long-time-averaged entangling power converges to the Haar value in both integrable and nonintegrable cases, indicating ergodicity. Remarkably, however, the nonintegrable chain exhibits mixing, whereas the integrable chain, despite being ergodic, is demonstrably nonmixing. We also introduce a Lyapunov-like exponent to characterize the rate at which the time-dependent entangling power approaches its saturation value. We find that this exponent increases systematically with the degree of integrability breaking in the many-body system. Our results establish entanglement generation as a useful framework for characterizing dynamical systems and reveal qualitatively different signatures of integrability beyond conventional diagnostics.

We construct representation-independent families of solutions of the non-braided scalene Yang–Baxter equation from quadratic noncommutative algebras. Beginning with two anticommuting generators, we obtain a continuous deformation in terms of quantum-plane algebras and extend the construction to an arbitrary number of generators. In the latter case the scalene Yang–Baxter relation fixes the multiparameter exchange matrix to an exact multiplicative form, thereby selecting a distinguished subclass of multiparameter quantum affine spaces. We construct finite-dimensional realizations using singular matrices and finite Heisenberg–Weyl operators, as well as infinite-dimensional realizations in terms of bilateral weighted shifts and multiplicative-shift operators. The explicit realizations are generically purely scalene: although the ordered triple satisfies the scalene Yang–Baxter equation, its individual constituent operators do not satisfy the ordinary non-braided Yang–Baxter equation. These results provide a representation-independent algebraic framework for constructing Yang–Baxter-irreducible scalene triples and a starting point for investigating their possible applications to quantum integrability.

We define a map from solutions of the Bethe Ansatz equations (BAEs) of four-dimensional $\mathcal{N}=4$ super-Yang–Mills with arbitrary semisimple gauge algebra $\mathfrak{g}$ to extrema and poles of the potential of the untwisted elliptic Calogero–Moser system of type $\mathfrak{g}$. We conjecture the map to be a bijection on the preimage of the Calogero–Moser extrema, and show that it intertwines the symmetries of the two systems, both the gauge ones (torus, Weyl and center invariance) and a $\mathrm{PSL}(2,\mathbb{Z})$ action, so that solutions on both sides organize into orbits, each BAE orbit mapping onto a single orbit of Calogero–Moser extrema or poles. That the system produced is the untwisted one has a consequence: the conjectured correspondence between BAE solutions and vacua of the $\mathcal{N}=1^\ast$ deformation of $\mathcal{N}=4$ on $\mathbb{R}^{3,1}$, which are extrema of the twisted system, cannot extend to non-simply-laced $\mathfrak{g}$. It also fails within the simply-laced cases, though not for $\mathfrak{su}(N)$: we exhibit an $\mathfrak{so}(8)$ solution that flows to a pole of the Calogero–Moser potential rather than to an extremum, and so has no $\mathcal{N}=1^\ast$ counterpart. We illustrate the map in detail for every rank-two $\mathfrak{g}$, classical and exceptional alike, and use these cases as evidence for the conjecture.

We study the finite-genus algebro-geometric solutions of the Wadati-Konno-Ichikawa (WKI) equation with the saturable nonlinearity and long-time asymptotic behaviors of their short-range perturbations. First, for both the focusing and defocusing reductions, we formulate the finite-genus Baker-Akhiezer functions as explicitly solvable the matrix Riemann-Hilbert (RH) problems on the complex spectral plane and obtain theta-function representations together with the reconstruction formulae for the WKI field and the reciprocal coordinate. We then consider the Cauchy problem of the defocusing WKI equation on a finite-genus algebro-geometric background. We construct the scattering data and RH problem, and perform a Deift-Zhou nonlinear steepest descent analysis. The space-time plane is divided into two transition regions, a Zakharov-Manakov (ZM) region, and a fast-decay region. The leading term is a phase-shifted finite-genus WKI solution. The transition corrections are governed by a Painlevé-XXXIV model, while the ZM radiation is described by parabolic-cylinder functions. The reciprocal-coordinate asymptotics are obtained simultaneously.

We show that the Coulomb branches of 5d $\mathcal{N}=1$ necklace quiver gauge theories on $\mathbb{R}^3 \times T^2$ are identified with the phase spaces of elliptic spin Ruijsenaars-Schneider models with dynamical inhomogeneities. This identification resolves the long-standing problem of determining a Poisson structure and a quantization of these models. We further show that the resulting quantum system is the integrable system governing supersymmetric indices of 4d $\mathcal{N}=1$ theories of class $\mathcal{S}_k$.

We study integrable variable-mass sine-Gordon model (vmSG) with point defects. Using the Lax and Bäcklund-gauge formulations, we construct type-I and type-II defect matrices, derive the corresponding sewing conditions, and generate the bulk and defect contributions to an infinite hierarchy of conserved charges. The lowest members reduce to the standard sine-Gordon energy and momentum in the homogeneous limit, while for inhomogeneous backgrounds they define integrability-generated energy- and momentum-type quantities. Defect compatibility imposes matching conditions on the variable-mass functions across the defect. An analytical transmission factor $z$ characterizes soliton transmission, topological conversion, and absorption/emission processes. We then deform the type-I sewing conditions by parameters $\alpha$ and $\beta$ and derive the associated defect anomalies. Consistent one-soliton transmission is recovered for $\alpha \beta =1$, whereas for $\alpha \beta \neq 1$ parity-centered kink-kink and kink-antikink transmissions display vanishing lowest order integrated anomalies but no generic vanishing at higher order. Numerical simulations reproduce the integrable transmission/conversion regimes and show that non-integrable defects generate weak radiative tails and lowest order quasi-conserved charges, while quasi-conservation of the higher order charges remains unestablished. These results elucidate how exact defect integrability deforms into charge-dependent quasi-conservation and establish a framework for defect-controlled soliton transport in inhomogeneous media, with potential applications to nonuniform Josephson junctions, magnetic and nonlinear-optical systems, and effective molecular and DNA models.

We investigate vector Akhmediev breathers (ABs) in the degenerate and nondegenerate regimes for the coupled Sasa-Satsuma system describing two coupled ultrashort optical pulse envelopes. Based on the carrier-wavenumber relations of the two-component background, we examine three configurations: a single vanishing wavenumber, a pair of opposite wavenumbers, and two nonzero wavenumbers of unequal magnitudes. These configurations admit distinct maximal degrees of nondegeneracy. For each configuration, we derive the reduced spectral equations, construct the existence diagrams, and compare the resulting branches with the modulation-instability gain spectra. Owing to the distinct carrier-wavenumber symmetries, nondegenerate-type modes can occur in spectrally degenerate regions, whereas degenerate-type ones may also appear in spectrally nondegenerate regions. Moreover, all admissible branches in the degenerate regimes satisfy the state-transition condition and correspond to periodic-wave states, while such branches are also present in nondegenerate regimes. Direct numerical simulations of representative solutions further confirm the analytical predictions.

We report new quasilinear and fully-nonlinear 5th-order Schwarzian evolution equations. These are symmetry-integrable evolution equations in 1+1 dimensions, i.e. equations that admit Lie-Bäcklund symmetries, whereby it is required that the equations are kept invariant under the Möbius transformation for their dependent variable.

We construct novel higher-order lump solutions of the modified Kadomtsev-Petviashvili-I (mKP-I) equation by applying the generalized long-wave limit method with spectral perturbations. By tuning the phase parameters in the soliton solution, we obtain second- and third-order lump solutions, which, to the best of our knowledge, are reported here for the first time for the mKP-I equation. A detailed asymptotic analysis reveals that these degenerate lumps exhibit anomalous scattering analogous to that in the KP-I equation. However, in contrast to the KP-I case, no complete energy equipartition occurs after the collision, and the two lumps remain distinguishable in the subleading terms of their asymptotic amplitudes. This distinction highlights a difference in the interaction dynamics between the modified and standard KP-I equations.

In this article we consider the Drinfeld-Sokolov hierarchy of type $A$ and its similarity reduction. Then the similarity reduction is expressed as $2$ types of Hamiltonian systems, one is with a Poisson bracket and another a Poisson bracket of canonical coordinates. We also investigate a connection between the similarity reduction and the isomonodromy deformation system.

We investigate integrability of Euler-Lagrange equations associated with 3D second-order Lagrangians of the form \begin{equation*} \int f(u_{xy},u_{xt},u_{yt})\ \text{d}x\text{d}y\text{d}t. \end{equation*} It is demonstrated that there are exactly four different types of such Lagrangian densities $f$: the first one is given by the formula $f=\sqrt{u_{xy}u_{xt}u_{yt}}$, the second and the third are more complicated (although still representable in elementary functions), whereas the most generic fourth one is expressible in terms of the Lobachevsky function, revealing unexpected links to spherical/hyperbolic trigonometry and Schläfly-type formulas. Dispersionless Lax pairs and integrable dispersive deformations of the corresponding Euler-Lagrange equations are also constructed.Remarkably, dispersive deformation of the first Lagrangian density coincides with the Lagrangian of the classical Darboux system arising in the theory of triply-orthogonal coordinate systems in $\mathbb{R}^3$. Dispersive deformations of the three other cases provide Lagrangian formulation of semi-discrete and fully discrete versions of the Darboux system, with one, two and three discrete variables, respectively.

This work provides explicit expressions, in all sectors of the Fock space: soliton, antisoliton and breather, for the fundamental building blocks -the form factors- of the generalised integral kernels which represent the exponential of the field operator in the 1+1 dimensional Sine-Gordon quantum field theory. The obtained form factors reproduce the various results obtained earlier in the literature using, in particular, the free field approach.

We develop a systematic adiabatic perturbation theory for the integrable $F=1$ spinor nonlinear Schrödinger equation under nonvanishing boundary conditions, formulated entirely within the framework of the associated Riemann–Hilbert problem. In this setting, localized nonlinear excitations are characterized by discrete spectral data consisting of a complex eigenvalue and an associated polarization vector. For a general class of small perturbations preserving the background, we derive the perturbation-induced evolution of the scattering data directly at the level of the Riemann–Hilbert problem. In the one-soliton sector, this yields a closed finite-dimensional dynamical system governing the slow evolution of the effective soliton parameters, including the spectral variables, the soliton center and phase, the residue amplitude, and the internal polarization state. The latter evolves according to a constrained dynamical equation with no scalar analogue. For localized perturbations, the modulation equations are expressed in explicit integral form in terms of the one-soliton eigenfunctions, providing a fully computable description of the dynamics. In the limit of vanishing boundary conditions, the resulting system reduces to the perturbation theory obtained by E. V. Doktorov, et al, Phys. Rev. A 77 (2008), no. 4, 043617.

We introduce a class of nonautonomous nonlinear oscillator equations of mixed Liénard type that arises from a parametric deformation of the commutative factorization procedure applied to second-order ordinary differential equations with periodic solutions. Their solutions, in particular the isochronous waveforms, are obtained in closed form through a Riccati reduction scheme for the power-law choice of the factorization function, $\phi(x)=kx^{q}$, $k\in \mathbb{R}$, and $q\in\mathbb{N}$, corresponding to the so-called modified Emden oscillators, and do not depend on the arbitrary deformation parameter $A_1$. The variational structure of the equation is characterised by a Lagrangian supplemented with a generalised Rayleigh dissipation function that contains a non-standard cubic term in $\dot{x}$. We also show that multiplying the equation of motion by the Jacobi multiplier $M(x)=x^{-2A_1}$ a position-dependent-mass (PDM) form is obtained with related friction and restoring force.

This paper derives the second-order positive Burgers' equation from the standard Burgers' hierarchy to explore its complete integrability and exact analytical solutions. We construct this higher-order nonlinear evolution equation by systematically applying the recursion operator to the classical Burgers' equation. Expanding on this structural framework, we then derive the explicit third-order equation and use Complete Bell Polynomials to generalize the $n$-th order hierarchy. Through the Cole-Hopf transformation, the second-order nonlinear equation rigorously maps to the linear third-order dispersion equation. We establish complete integrability by explicitly formulating the scalar Lax pair (zero-curvature representation), which allows us to directly derive the associated differential and algebraic Darboux transformations. To systematically classify explicit stationary, time-dependent traveling wave, and self-similar profiles, we apply several techniques to the linear domain: separation of variables, the Hirota perturbation method, traveling wave reduction via the Complete Discrimination System for Polynomial Method (CDSPM), and Lie similarity reduction. Crucially, our analysis demonstrates the exact finite truncation of the Hirota perturbation series. We conclude by outlining how this equation impacts the theoretical understanding of fluid dynamics, nonlinear transport phenomena, and higher-order wave propagation.

In this paper, we travel through an unknown territory of dromions, to unearth and show the new properties/attributes of dromions which have never been brought to the fore since they were first discovered by Boiti etal [1]. These new attributes are brought out by investigating a generalized (2+1) dimensional Nonlinear Schrodinger (NLS) equation by exploiting Truncated Painleve approach. The signatures that can be attributed to dromions include existence of firewall and reflection at the boundary, uneven distribution of energy among different bound states, amplitude dependence on adjacent dromions, etc. We have corroborated the main analytical results with numerical simulations also. We categorically state that these properties are universal and can be extracted in any (2+1) dimensional nonlinear partial differential equation. We do believe that these properties which shed more light on the behaviour of dromions may have wider repercussions in nonlinear optics, Bose-Einstein condensates and plasma physics.

In this paper, we investigate the integrability of Lotka-Volterra (replicator) systems arising from interaction matrices generated from corresponding graph structures, continuing work started by Visomirski and Griffin [J. Phys. A., 58:015701, 2025] and Evripidou et al. [J. Phys. A., 55:325201, 2022] (among others). In particular, we define a new family of graphs, the spoked graphs, and show that all dynamics generated from this family are integrable. In reference to Evripidou et al. (2022), we define a new anti-cloning operator and show that its action on balanced tournament graphs (with odd vertex count) generates new graphs whose dynamics are integrable. Interestingly, we provide numerical evidence that this anti-cloning operation leads to chaotic behaviour when applied to other graph families (e.g., the directed cycles that generate the classically integrable Volterra lattice). This work completes a taxonomy of all integrable dynamics generated by directed graphs with up to six vertices started by Visomirski and Griffin (2025), and suggests several future directions of study on this topic.

In this work, we derive selection rules imposed by Liouville integrability conditions for monomial charge densities with arbitrary powers. For a certain monomial Hamiltonian system, the selection rules reduce to a negative Pell equation, and its solutions generate an infinite set of integrals of motion that are mutually in involution. Furthermore, we study the correspondence between combinatorial polynomial sequences and Liouville integrable Hamiltonian field theories in 1+1 dimensions. We show that the Motzkin system coincides with the dispersionless limit of the Levi system, while the binomial system is equivalent to the dispersionless derivative nonlinear Schrödinger equation. Additionally, we show that the binomial Hamiltonian model admits a reduction to the inviscid Burgers equation and its higher-order charges generate generalized Burgers-type conservation laws.

In recent years, integrable hierarchies have been used to great advantage for the enumeration of combinatorial maps. They have led to recurrence formulas with respect to the size and genus of the maps, e.g. for triangulations, bipartite quadrangulations and bipartite maps, and for constellations. These formulas are not only remarkably simple but also provide the fastest way of calculating these numbers of maps. With the exception of Louf's work on constellations, it has however remained a challenge to obtain recurrence formulas that control the degrees of the faces of the maps. Here we show how to achieve this for bipartite maps with bounded face degrees. By combining equations from the KP hierarchy and from the Virasoro constraints, a differentially algebraic system is obtained. It couples the generating functions of bipartite maps with bounded root face degrees while controlling the numbers of edges, black vertices, white vertices and number of faces of each degree (and in particular the genus). Finally, this system of ODEs is shown to give recurrence formulas that allows to calculate all the corresponding numbers of maps.

We consider fluctuations of the spin current in the stochastic XNOR process, a kinetically constrained hopping model in one spatial dimension. We formulate three conjectures with explicit amplitudes for the long-time current distributions: a Gaussian limit on the $t^{1/4}$ scale for homogeneous initial states with nonzero magnetization, a half-normal limit on the $t^{1/4}$ scale for domain-wall states with opposite magnetizations, and an M-Wright limit on the $t^{1/8}$ scale for homogeneous initial states at zero magnetization. Earlier work identified the tracer mechanism and anticipated the scaling exponents and limiting shapes in the domain-wall and zero-magnetization settings. Starting from the microscopic XNOR dynamics, we predict the missing amplitudes for the continuous-time process and extend the picture to homogeneous biased initial data. Simulations provide numerical support for all three conjectures without fitted parameters. A separate mathematical companion paper presents an extensively AI-generated candidate proof. Generative-AI tools were used in the research and writing workflow.