Complete analytic solutions to quasi-continuous-wave four-wave mixing in nonlinear optical fibres are presented in terms of Weierstrass elliptic $\wp$, $\zeta$, and $\sigma$ functions, providing the full complex envelopes for all four waves under arbitrary initial conditions. A sequence of coordinate transformations reveals a canonical form with universal parameter-free structure. Remarkably, these transformations depend explicitly on the propagation variable yet preserve the structural form of the differential equations, an invariance property not previously reported for four-wave mixing. In the canonical coordinates, solutions become single-valued meromorphic Kronecker theta functions, establishing connections with other integrable nonlinear optical systems. The Hamiltonian conservation is shown to arise from the Frobenius-Stickelberger determinant. Numerical validation confirms the solutions using open-source Python libraries.

The Korteweg-de Vries (KdV) equation serves as a foundational model in nonlinear wave physics, describing the balance between dispersive spreading and nonlinear steepening that gives rise to solitons. This article introduces sangkuriang, an open-source Python library for solving this equation using Fourier pseudo-spectral spatial discretization coupled with adaptive high-order time integration. The implementation leverages just-in-time (JIT) compilation for computational efficiency while maintaining accessibility for instructional purposes. Validation encompasses progressively complex scenarios including isolated soliton propagation, symmetric two-wave configurations, overtaking collisions between waves of differing amplitudes, and three-body interactions. Conservation of the classical invariants is monitored throughout, with deviations remaining small across all test cases. Measured soliton velocities conform closely to theoretical predictions based on the amplitude-velocity relationship characteristic of integrable systems. Complementary diagnostics drawn from information theory and recurrence analysis confirm that computed solutions preserve the regular phase-space structure expected for completely integrable dynamics. The solver outputs data in standard scientific formats compatible with common analysis tools and generates visualizations of spatiotemporal wave evolution. By combining numerical accuracy with practical accessibility on modest computational resources, sangkuriang offers a platform suitable for both classroom demonstrations of nonlinear wave phenomena and exploratory research into soliton dynamics.

The Rulkov model, which simulates the behavior of biological neurons, is modified by replacing one of its control parameters with a memristive, sigmoid-type function of finite memory. This modification causes the parameter to vary according to the system's history throughout the simulation. Previous works usually modify the Rulkov model by introducing additional parameters altering its behavior. Here, by contrast, we retain the original equations and allow the control parameters to vary in time, thereby preserving the model's fundamental properties. In this sense, the proposed model is locally equivalent in time to the original one. However, unlike the original model, which reproduces a single neuronal regime per simulation, the new memristive version exhibits both uniform and chaotic transitions among multiple neuronal activity regimes. Its dynamics are examined with respect to the rate at which the memristive function changes and the number of internal states it stores. Three distinct scenarios emerge around a bifurcation point. Before the bifurcation, the system undergoes uniform transitions toward a stable bursting regime. After the bifurcation, it shows uniform transitions toward a final spiking or silent regime. At the bifurcation point, highly complex transitions arise. As examples, we present trajectories in which the neuron chaotically switches between regimes without ever settling, and trajectories for which it requires around 140000 map iterations to reach a stationary regime.

This work presents a numerical investigation of interacting chiral oscillators (COs), characterized by an intrinsic rotational handedness.

We study the escape of particles in the lemon billiard, a two-parameter family of billiard systems defined by the intersection of two identical circles. Using numerical simulations, we explore how the survival probability depends on the position and size of the hole, as well as on the billiard shape parameter. We find that the survival probability exhibits a two-stage decay pattern: an initial exponential regime followed by a long-time power-law tail, a signature of the stickiness effect. Our results show that the short-time exponential decay rate follows a power-law dependence on the hole size, with different scaling exponents for holes placed in chaotic regions versus mixed phase space regions. For holes located in mixed phase space regions, the decay exponent of the long-time power-law tail remains approximately constant, while the amplitude follows a power-law scaling with hole size. We also examine the dependence of short-time exponential decay rate on the billiard shape parameter and observe scaling behavior for small values of this parameter, which breaks down as the parameter increases.

Double ramification (DR) hierarchies associated to rank-$1$ F-CohFTs are important integrable perturbations of the Riemann--Hopf hierarchy. In this paper, we perform bihamiltonian tests for these DR hierarchies, and conjecture that the ones that are bihamiltonian form a $2$-parameter family. Remarkably, our computations suggest that there is a $1$-parameter subfamily of the rank-$1$ F-CohFTs, where the corresponding DR hierarchy is conjecturally Miura equivalent to the Camassa--Holm hierarchy. We also prove a conjecture regarding bihamiltonian Hodge hierarchies. Finally, we systematically study Miura invariants, and for another $1$-parameter subfamily propose a conjectural relation to the Degasperis--Procesi hierarchy.

Recently, researchers have proposed the Asymmetric Bethe ansatz method - a theoretical tool that extends the scope of Bethe ansatz-solvable models by "breaking" partial mirror symmetry via the introduction of a fully reflecting boundary. Within this framework, the integrability conditions which were originally put forward by Gaudin have been further generalized. In this work, building on Gaudin's generalized kaleidoscope model, we present a detailed investigation of the relationship between DN symmetry and its integrability. We demonstrate that the mathematical essence of integrability in this class of models is characterized by a newly proposed Kaleidoscope Yang-Baxter Equation. Furthermore, we show that the solvability of a model via the coordinate Bethe ansatz depends not only on the consistency relations satisfied by scattering matrices, but also on the model's boundary conditions and the symmetry of the subspace where solutions are sought. Through finite element method based numerical studies, we further confirm that Bethe ansatz integrability arises in a specific symmetry sector. Finally, by analyzing the algebraic structure of the Kaleidoscope Yang-Baxter Equation, we derive a series of novel quantum algebraic identities within the framework of quantum torus algebra.

We consider the novel nonlinear model in (1 + 1)-dimensions for Dirac spinors recently introduced by Alexeeva, Barashenkov, and Saxena [1] (ABS model), which admits an exact explicit solitary-wave (soliton for short) solution. The charge, the momentum, and the energy of this solution are conserved. We investigate the dynamics of the soliton subjected to several potentials: a ramp, a harmonic, and a periodic potential. We develop a Collective Coordinates Theory by making an ansatz for a moving soliton where the position, rapidity, and momentum, are functions of time. We insert the ansatz into the Lagrangian density of the model, integrate over space and obtain a Lagrangian as a function of the collective coordinates. This Lagrangian differs only in the charge and mass with the Lagrangian of a collective coordinates theory for the Gross-Neveu equation. Thus the soliton dynamics in the ABS spinor model is qualitatively the same as in the Gross-Neveu equation, but quantitatively it differs. These results of the collective coordinates theory are confirmed by simulations, i.e., by numerical solutions for solitons of the ABS spinor model, subjected to the above potentials.

We investigate neuronal excitability in the Hodgkin-Huxley model under temporal interference (TI) stimulation in a previously unexplored sub-Hz resonant regime and uncover a striking nonlinear response that we term 'multi-tongue frequency fractals'. Unlike single-frequency driving, which yields a smooth resonant valley, dual-frequency excitation fragments this response into a hierarchy of sharply modulated tongues whose number and structure grow with observation time, revealing clear self-similar architecture. These features emerge from transitions between non-cascaded and cascaded high-harmonic and sub-harmonic generation as detuning varies, and are maximized near the intrinsic ionic timescale at omega ~ 0.2 rad/s. Parameter sweeps show that the fractal count is higher for higher potassium conductances, lower sodium conductances and lower leak conductances. These results demonstrate that TI stimulation can elicit rich, hierarchically organized frequency responses even in classical excitable membranes, revealing fractal organization in Hodgkin-Huxley dynamics.

Complex dynamical systems frequently encounter a recurrent structural instability: the collapse of the spectral gap, driving the system toward a low-dimensional "Zero-Mode Attractor" (e.g., spectral pile-up or over-smoothing). Building upon recent global well-posedness estimates [Hou, arXiv:2601.00638], this work generalizes the Multi-Scale Negative Coupled Information System (MNCIS) framework. We postulate that global stability requires an active topological operator -- Adaptive Spectral Negative Coupling (ASNC) -- functioning as a state-dependent high-pass filter that penalizes entropy accumulation at spectral boundaries. We validate this unified framework via three implementations:(1) Hydrodynamics: In 3D Navier-Stokes turbulence ($N=256^3$), ASNC acts as a global-enstrophy adaptive sub-grid scale (SGS) model, stabilizing the inviscid limit and preserving the Kolmogorov $-5/3$ inertial range without artificial hyper-viscosity.(2) Artificial Intelligence: Addressing Over-smoothing in Graph Neural Networks (GNNs), we implement ASNC as a parameter-free topological constraint. Unlike baselines (e.g., DeepGCNs) relying on dense residual connections to bypass signal decay, our framework enables the training of ultra-deep 64-layer networks without residual connections, maintaining perfectly stationary feature variance ($\sigma^2 \equiv 1.0$) on the ogbn-arxiv benchmark. (3) Biological Physics: In reaction-diffusion morphogenesis, it stabilizes Turing patterns against diffusive washout in high-entropy regimes. Our results suggest that the MNCIS framework provides a base-independent topological condition for distinguishing viable complex systems from those collapsing into thermal equilibrium, bridging physical stability and information persistence.

In many real-world applications of data assimilation (DA), the strategic placement of observers is crucial for effective and efficient forecasting. Motivated by practical constraints in sensor deployment, we show that global recovery of the flow field can be achieved using observations available only in a subregion of the domain, possibly far from the boundary. We focus on the two-dimensional incompressible Navier-Stokes equations posed in a bounded physical domain with Dirichlet boundary conditions. Building on the continuous data assimilation framework of Azouani, Olson, and Titi (2014), we rigorously prove that the assimilated solution converges globally to the true solution under suitable conditions on the nudging parameter, spatial resolution, and the geometry of the observation region, specifically, when the maximum distance from any point in the domain to the observational subregion is bounded by a constant multiple of \( \nu^{1/2} \) (in terms of scaling). Our computational results, conducted via finite element methods over complex geometries, support the theoretical findings and reveal even greater robustness in practice. Specifically, synchronization with the true solution is achieved even when the observational subregion lies farther from the rest of the domain than the theoretical threshold permits. Across all three tested scenarios, the local nudging algorithm performs comparably to full-domain assimilation, reaching global accuracy up to machine precision. Interestingly, observational data near the boundary are found to be largely uninformative. This demonstrates that full observability is not necessary: carefully chosen interior observations, even far from the boundary, can suffice.

The El Niño-Southern Oscillation (ENSO) is a dominant mode of interannual climate variability, yet the mechanisms limiting its long-lead predictability remain unclear. Here we develop a physics-guided Deep Echo State Network (DESN) that operates on physically interpretable climate modes selected from the extended recharge oscillator (XRO) framework. DESN achieves skillful Niño3.4 predictions up to 16-20 months ahead with minimal computational cost. Mechanistic experiments show that extended predictability arises from nonlinear coupling between warm water volume and inter-basin climate modes. Error-growth analysis further indicates a finite ENSO predictability horizon of approximately 30 months. These results demonstrate that physics-guided reservoir computing provides an efficient and interpretable framework for diagnosing and predicting ENSO at long lead times.

We study the Dirac spectrum in a sine-Gordon soliton background, where the induced position-dependent mass reduces the spectral problem to a Heun-type differential equation. Bound and scattering sectors are treated within a unified framework, with spectral data encoded in Wronskians matching local Heun solutions and exhibiting explicit dependence on the soliton parameters and the bare fermion mass. This formulation enables a systematic analysis of spinor bound and scattering states, supported by analytic and numerical verification of wave function matching across the soliton domain. The present work is related to arXiv:2512.07658 and emphasizes a pedagogical treatment of scattering states within the Heun-equation formalism.

Long arrays of identical, self-propelling flapping flyers are inherently unstable and thus unlikely to exist without active control mechanisms. One approach to enable long in-line formations is to enforce a constant separation between the group members. The objective then becomes to determine the flapping strategies the flyers should adopt to achieve a certain separation. Using an aerodynamic model of vortex wake production and inter-flyer effects, we explore different flapping strategies for followers given the motion of the leader. The choice of tactic is dependent upon the aerodynamic, kinematic, and physical parameters of the system, and reflects an interplay between efficiency and stability. We find that whether a flyer flaps in or out of phase with its upstream neighbour, together with the target separation, strongly affect the flapping amplitude and, therefore, the energetic cost of group flight. In certain regimes, group flight is energetically favourable compared to isolated flight, while in others, flying in formation becomes less efficient. We also identify "goldilocks zones", ranges of separation in which one of the in- or out-of-phase motions can be simultaneously energetically efficient and dynamically stable. Outside these regions, energetically favourable flight is unstable and therefore unlikely to occur.

We propose a financial liquidity policy sharing method for firm-to-firm supply networks, introducing a scalable autonomous control function for viable complex adaptive supply networks. Cooperation and competition in supply chains is reconciled through overlapping collaborative sets, making firms interdependent and enabling distributed risk governance. How cooperative range - visibility - affects viability is studied using dynamic complex adaptive systems modelling. We find that viability needs cooperation; visibility and viability grow together in scale-free supply networks; and distributed control, where firms only have limited partner information, outperforms centralised control. This suggests that policy toward network viability should implement distributed supply chain financial governance, supporting interfirm collaboration, to enable autonomous control.

We study the thermalization properties of a fully nonlinear lattice model originally derived from the two-dimensional cubic defocusing nonlinear Schrödinger equation (NLS) using analytical and numerical methods. Our analysis reveals both ergodic and nonergodic regimes; importantly, we find broad parameter ranges where the dynamics is ergodic even though it lies outside the Gibbsian parameter regime (for both $D=0.25$ and $D=2$), and a higher-energy range where ergodicity breaks down. We observe that in a certain range of parameters, the system requires non-standard statistical descriptions, indicating a breakdown of conventional thermalization. We examine the influence of the nonlinear dispersion parameter $D$ on the system's behavior, showing that increasing $D$ enhances fluctuations and speeds up the crossover of $q(T)$ toward the $\sim 1/T$ scaling. By analyzing excursion times, probability density functions, and localization patterns, we characterize transitions between ergodic and nonergodic behavior. In long-time numerical simulations within the non-ergodic regime for $D>1$, stable localization over two sites is observed, while $D<1$ favors single-site localization in the high energy density regimes. Our results provide insights into the interplay between thermalization, localization, and non-standard statistical behavior in genuinely nonlinear systems.

The rarely used Hamilton-Jacobi equation has been utilized as an elegant way to find the trajectories of mechanical systems and to derive symplectic maps. Further, the exact solution in kick approximation of Hamilton's equations of motion in interaction representation is written as a generalized one-turn twist map. One can imagine that the nonlinear kick comes first, followed by the one-period rotation along the machine circumference, or a second alternative in which the one-period rotation occurs before the kick. There is a difference in the result of solving Hamilton's equations between the two cases, which is expressed in obtaining a standard forward twist map in the first case, or alternatively a backward map in the second one. This nontrivial and intuitively unclear peculiarity is usually ignored/overlooked in practically all specialized references on the topic. Finally, the statistical properties and the behavior of the density distribution of a particle beam in configuration space under the influence of an isolated sextupole have been studied.

We investigate Hamiltonian fluid reductions of the one-dimensional Vlasov-Poisson equation. Our approach utilizes the hydrodynamic Poisson bracket framework, which allows us to systematically identify fundamental normal variables derived from the analysis of the Casimir invariants of the resulting Poisson bracket. This framework is then applied to analyze several well-established Hamiltonian closures of the onedimensional Vlasov equation, including the multi-delta distribution and the waterbag models. Our key finding is that all of these seemingly distinct closures consistently lead to the formulation of a unified form of parametric closures: When expressed in terms of the identified normal variables, the parameterization across all these closures is revealed to be polynomial and of the same degree. All these parametric closures are uniquely generated from one of the moments, called $\mu$2, a cubic polynomial in the normal variables. This result establishes a structural connection between these different physical models, offering a path toward a more unified and simplified description of the one-dimensional Vlasov-Poisson dynamics through its reduced hydrodynamic forms with an arbitrary number of fluid variables.

Estimating collision rates is of immense importance in particle-laden flows. An economical way of doing this is to directly identify incidences of caustics, or extreme clustering, by tracking particle velocity gradients in the neighborhoods of individual particles. The objective of this work is two-fold. (i) We find conditions under which caustics form, in point-vortex flow and in two-dimensional turbulence. While caustics are known to form in regions of strain, we show that the type of strain is key. Particles must remain in compressional strain throughout the process to form caustics, whereas survivor particles: which visit high strain but do not form caustics, briefly go through extensional strain during the early part of the process. This enables survivor particles to attain significantly straighter paths, and to move faster, whereas caustics particles follow paths of high curvature and move slower. As a result, caustics particles stay longer in high-strain regions than survivors. (ii) We ask about the effect of finite particle density, where the particle is denser than the background fluid. We show that finite-density particles need to sample stronger background strain than infinite-density ones to trigger caustics, but our other findings are universal across particle density.

It has recently been theoretically predicted and experimentally observed that a soliton resulting from nonlinearity can be pumped across an integer or fractional number of unit cells as a system parameter is slowly varied over a pump period. Nonlinear Thouless pumping is now understood as the flow of instantaneous Wannier functions, ruling out the possibility of pumping a soliton across a nonzero number of unit cells over one cycle when a corresponding Wannier function does not exhibit any flow, i.e., when the corresponding Bloch band that the soliton bifurcates from is topologically trivial. Here we surprisingly find an anomalous nonlinear Thouless pump where the displacement of a soliton over one cycle differs from the Chern number of the Bloch band from which the soliton comes. We develop a general theory showing that this anomalous behavior arises from a transition of a soliton between different Wannier functions by passing through an intersite-soliton (or dipole-soliton) state. Furthermore, we find a nonlinearity-induced integer quantized Thouless pump of a soliton, allowing a soliton to travel across one unit cell during a pump period, even when the corresponding band is topologically trivial. Our results open the door to studying nonlinearity-induced Thouless pumping of solitons.

Understanding the dynamics of excitation patterns in neural fields is an important topic in neuroscience. Neural field equations are mathematical models that describe the excitation dynamics of interacting neurons to perform the theoretical analysis. Although many analyses of neural field equations focus on the effect of neuronal interactions on the flat surface, the geometric constraint of the dynamics is also an attractive topic when modeling organs such as the brain. This paper reports pattern dynamics in a neural field equation defined on spheroids as model curved surfaces. We treat spot solutions as localized patterns and discuss how the geometric properties of the curved surface change their properties. To analyze spot patterns on spheroids with small flattening, we first construct exact stationary spot solutions on the spherical surface and reveal their stability. We then extend the analysis to show the existence and stability of stationary spot solutions in the spheroidal case. One of our theoretical results is the derivation of a stability criterion for stationary spot solutions localized at poles on oblate spheroids. The criterion determines whether a spot solution remains at a pole or moves away. Finally, we conduct numerical simulations to discuss the dynamics of spot solutions with the insight of our theoretical predictions. Our results show that the dynamics of spot solutions depend on the curved surface and the coordination of neural interactions.

A shell model can be considered as a chain of triads, where each triad can be interpreted as a nonlinear oscillator that can be mapped to a spinning top. Investigating the relation between phase dynamics and intermittency in a such a chain of nonlinear oscillators, it is found that synchronization is linked to increased energy transfer. In particular, the results provide evidence that the observed systematic increase of intermittency, as the shell spacing is decreased, is associated with strong phase alignment among consecutive triadic phases, facilitating the energy cascade. It is shown that while the overall level of synchronization can be quantified using a Kuramoto order parameter for the global phase coherence in the inertial range, a local, weighted Kuramoto parameter can be used for the detection of burst-like events propagating across shells in the inertial range. This novel analysis reveals how partially phase-locked states are associated with the passage of extreme events of energy flux. Applying this method to helical shell models, reveals that for a particular class of helical interactions, a reduction in phase coherence correlates with suppression of intermittency. When inverse cascade scenarios are considered using two different shell models including a non local helical shell model, and a local standard shell model with a modified conservation law, it was shown that a particular phase organization is needed in order to sustain the inverse energy cascade. It was also observed that the PDFs of the triadic phases were peaked in accordance with the basic considerations of the form of the flux, which suggests that a triadic phase of \pi/2 and -\pi/2 maximizes the forward and the inverse energy cascades respectively.

In this paper we revisit an integrable map of the plane which we obtained recently as a two-parameter family of deformed mutations in the cluster algebra of type D$_4$. The rational first integral for this map defines an invariant foliation of the plane by level curves, and we explain how this corresponds to a rational elliptic surface of rank 2. This leads us to construct another (independent) integrable map, commuting with the first, such that both maps lift to compositions of mutations in an enlarged cluster algebra, whose underlying quiver is equivalent to the one found by Okubo for the $q$-Painlevé VI equation. The degree growth of the two commuting maps is calculated in two different ways: firstly, from the tropical (max-plus) equations for the d-vectors of the cluster variables; and secondly, by constructing the minimal space of initial conditions for the two maps, via blowing up $\mathbb{P}^1 \times \mathbb{P}^1$.

Twin-wire laser directed energy deposition (TW-LDED) provides a promising route for alloying and fabrication of compositionally graded structures. However, inherent multiparameter coupling in twin-wire systems critically exacerbates both process instabilities and compositional inhomogeneity. This unresolved issue escalates into a fundamental technological bottleneck, as the underlying physical mechanisms remain poorly understood. This study developed a high-fidelity multi-physics and multiphase simulation framework coupled with experimental validation to reveal thermal-fluid behavior and heat-mass transfer mechanisms in TW-LDED using Inconel 718 and SS316L fine wires. Three distinct transition modes were identified: twin-wire melt droplet, twin-wire liquid bridge, and droplet-bridge mixed transitions, with the twin-wire liquid bridge regime delivering optimal stability and uniform mixing. Parametric analysis demonstrates that increasing wire feeding speed or decreasing wire initial height promotes stable liquid bridge formation, while small laser spots at low feeding speeds induce excessive volumetric energy density and bridge instability. Simulation and single-track experiments confirm that liquid bridge transitions reduce dimensional fluctuations by 85% while enhancing compositional homogeneity. Conversely, the melt droplet-bridge transition mode creates periodic flow switching and compositional discontinuities along the scan direction. Finally, a 60 mm functionally graded ring was successfully fabricated using optimized parameters, achieving uniform elemental distribution in the transition zone without significant segregation, validating the feasibility of TW-LDED for functionally graded components.

We prove the holding of chaos in the sense of Li-Yorke for a family of four-dimensional discrete dynamical systems that are naturally associated to ODEs systems describing coupled oscillators subject to an external non-conservative force, also giving an example of a discrete map that is Li-Yorke chaotic but not topologically transitive. Analytical results are generalized to a modular definition of the problem and to a system of nonlinear oscillators described by polynomial potentials in one coordinate. We perform numerical simulations looking for a strange attractor of the system; furthermore, we perform a bifurcation analysis of the system presenting 1D and 2D bifurcation diagrams, together with spectra of Lyapunov exponents and basins of attraction.

Associative memory models retrieve stored information through content-based addressing, mimicking the neural processes of animal brains. The classical Hopfield network-based models store memories as vectors of discrete values and have good storage capacity but do not consider the role of neuronal synchronization in memory storage and retrieval as observed in brains. This is addressed in phase-oscillator-based models which store memories as time-dependent phase-synchronized states, but suffer from instability and low capacity. The present study addresses these challenges through a novel chaotic oscillator-based associative memory model, by defining a phase relationship in chaotic systems and encoding memory as synchronized states of these phases. The underlying chaos in the network is shown to significantly improve both storage and retrieval and offer insights into the dynamics of memory retrieval.

We numerically study the nonlinear interactions of high-frequency circularly polarized electromagnetic (EM) waves and low-frequency electron-acoustic (EA) density perturbations driven by the EM wave ponderomotive force in relativistic plasmas {(moderate, strong, and ultra-relativistic)} with two groups of electrons--the population of relativistic degenerate dense electrons (bulk plasma) and the sparse relativistic nondegenerate (classical) electrons, and immobile singly charged positive ions. By pattern selection, we show that many solitary patterns can be generated and drenched through modulational instability of EM waves at different spatial length scales and that the EM wave radiation spectra emanating from compact astrophysical objects may not settle into stable envelope solitons but into different incoherent states, including the emergence of temporal and spatiotemporal chaos due to collisions and fusions among the patterns with strong EA wave emission. The appearance of these states is confirmed by analyzing the Lyapunov exponent spectra, correlation function, and mutual information {as quantitative evidence}. As a result, the redistribution of wave energy from initially exciting many solitary patterns at large scales to a few new incoherent patterns with small wavelengths in the system occurs, leading to the onset of turbulence in astrophysical plasmas.

We prove equivalence of two integral representations for the wave functions of hyperbolic Calogero-Sutherland system. For this we study two families of Baxter operators related to hyperbolic Calogero-Sutherland and rational Ruijsenaars models; the first one as a limit from hyperbolic Ruijsenaars system, while the second one independently. Besides, computing asymptotics of integral representations and also the value at zero point, we identify them with renormalized Heckman-Opdam $\mathfrak{gl}_n$ hypergeometric function.

A consistent kinetic modeling and discretization strategy for compressible flows across all Prandtl numbers and specific heat ratios is developed using the quasi-equilibrium approach within two of the most widely used double-distribution frameworks. The methodology ensures accurate recovery of the Navier-Stokes-Fourier equations, including all macroscopic moments and dissipation rates, through detailed hydrodynamic limit analysis and careful construction of equilibrium and quasi-equilibrium attractors. Discretization is performed using high-order velocity lattices with a static reference frame in a discrete velocity Boltzmann context to isolate key modeling aspects such as the necessary requirements on expansion and quadrature orders. The proposed models demonstrate high accuracy, numerical stability and Galilean invariance across a wide range of Mach numbers and temperature ratios. Separate tests for strict conservation and measurements of all dissipation rates confirm these insights for all Prandtl numbers and specific heat ratios. Simulations of a thermal Couette flow and a sensitive two-dimensional shock-vortex interaction excellently reproduce viscous Navier-Stokes-Fourier-level physics. The proposed models establish an accurate, efficient and scalable framework for kinetic simulations of compressible flows with moderate supersonic speeds and discontinuities at arbitrary Prandtl numbers and specific heat ratios, offering a valuable tool for studying complex problems in fluid dynamics and paving the way for future extensions to the lattice Boltzmann context, by application of correction terms, as well as high-Mach and hypersonic regimes, employing target-designed reference frames.