Papers

We provide an exact and complete solution for the dynamics of a rigid particle of uniform density with $C_{2v}$ and $S_4$ symmetry, settling under gravity in a viscous fluid at a Reynolds number much smaller than unity. The $S_4$ symmetry renders the problem exactly integrable, with all orbits labelled by a single conserved quantity $0\le C\le 1$. We show that, for $0<C<1$, there are two different time scales, which lead to quasi-periodic evolution, with a significant time-dependent horizontal displacement. We obtain the tilt $\theta$ and spin $\psi$ Euler angles as periodic Jacobi elliptic functions of time, and the azimuthal angle $\phi$ through an incomplete elliptic integral of the third kind with a complex characteristic, which splits $\phi$ into a uniform drift and a strictly periodic modulation. The orientation period and the drift rate (corresponding to a constant angular velocity around the gravity direction) follow in a closed form. The vertical centre-of-mass displacement is obtained in terms of periodic in time incomplete elliptic integrals, and the periodic orbit-averaged settling velocity reduces to a single ratio of complete elliptic integrals. The horizontal component of the motion in the laboratory frame of reference is obtained exactly and algebraically in terms of all three Euler angles, so it is quasi-periodic. The horizontal component of the centre-of-mass position traces rosette-like, in general open curves confined by two concentric `envelope' circles. We determine very simple exact expressions for radii of both envelopes and demonstrate that they tend to infinity for a family of shapes with the rotation-translation coupling decreasing to zero. We also explain the origin of cusps at the rosette-like trajectory and provide a commensurability condition selecting strictly periodic rosettes.

We study a nearest-neighbor non-Hermitian spin chain obtained from one member of an exact non-braided scalene Yang–Baxter triple. Its local Hamiltonian density violates both the difference-form Reshetikhin condition and its general non-difference counterpart, obstructing its realization by a differentiable homogeneous regular solution of the ordinary Yang–Baxter equation. The transfer matrices constructed from the regular member do not commute among themselves at distinct spectral parameters. Nevertheless, the scalene Yang–Baxter relation implies cross-commutativity with another transfer matrix constructed from the third member of the scalene triple. We evaluate the latter for arbitrary chain length and show that, on even periodic chains, it is a finite generating function of a non-obvious staggered nilpotent symmetry. The resulting conserved hierarchy belongs entirely to the algebra generated by this single symmetry and hence does not constitute an extensive family of algebraically independent charges. Nevertheless, this example demonstrates that scalene Yang–Baxter triples can act as an algebraic symmetry-discovery mechanism beyond the ordinary self-commuting transfer-matrix framework.

Bernoulli-type waveforms for modified Emden nonlinear oscillators of arbitrary natural power $q$ are obtained through a generalized commutative factorization approach. These oscillators display a well-defined odd-even dynamical dichotomy, which is discussed in detail: the odd-$q$ cases entail isochronous oscillators whose period $T = 2\pi/\omega$ is independent of amplitude and initial conditions, while the even-$q$ cases display underdamped behavior. The Lagrangian formulation is presented in the Lurie's dissipative description. The isochronous regime and the period of the solutions in the odd case are also confirmed through a generalized polar-coordinate analysis in the spirit of Sabatini's work. The absence of periodic orbits for even $q$ is shown to be a consequence of the Bendixson-Dulac criterion applied to the radial velocity function. Explicit waveforms and their phase portraits are presented for $q = 1, 2, 3, 4$, along with the non exponential envelope formulas for the damped cases and singular-region bounds for the isochronous ones. A few possible applications are also mentioned.

A class of elliptic solutions to the matrix CKP equation is studied. Equations of motion for their poles and matrix coefficients at the poles are obtained. As in the scalar case, they are of the first order, in contrast to what takes place in the KP and BKP hierarchies, where the equations of motion are of the second order.

We study annealed current fluctuations in the two-dimensional symmetric simple exclusion process (SSEP) across a circular passive counting boundary. The initial average density is $\rho_1$ inside a disk of radius $R$ and $\rho_2$ outside, and the observable is the net decrease of the particle number in the disk over a finite time. Using convexity of the macroscopic fluctuation theory action and rotational averaging, we show that the minimizer of the full two-dimensional variational problem may be chosen radially symmetric. For the resulting radial problem, a suitable change of variables leads to a nonisospectral formulation on the half-line. Combining the associated scattering construction with a scalar factorization, we obtain a closed expression for the scaled cumulant generating function and hence the annealed large-deviation statistics of the current.

We study the dynamics of a two-component perturbation of the Camassa–Holm equation arising from a reformulation of the Euler–Bernoulli beam problem, recently extended to a general Clifford algebra setting. We focus on the original case associated with a Clifford algebra with two generators and Minkowski signature, for which the resulting equation admits nonsmooth soliton solutions (peakons) carrying internal degrees of freedom. We investigate analytically and numerically the dynamics of a two-peakon solution and establish the existence of a synchronized exchange of energy between spatially separated peaks, a phenomenon previously observed only numerically. We obtain a complete description of the long-time dynamics: the amplitudes approach a periodic orbit determined by the spectral invariants, and the resulting hidden periodicity governs the persistent exchange between the two peakons. The limiting orbit and its averaged dynamics are described explicitly in terms of elliptic functions and complete elliptic integrals. We also derive an exact identity relating the peak separation to the accumulated imbalance of the two amplitudes. This identity, combined with a frozen-parameter comparison argument, yields exponential decay of the interaction between the peakons and exponential convergence of the amplitude variables to the limiting periodic orbit. Moreover, once the asymptotic regime is reached, the separation increases from one period of the internal oscillation to the next, even though its instantaneous rate may continue to change sign. These results reveal a dynamical feature absent from the scalar Camassa–Holm equation: a persistent oscillatory transfer of energy between increasingly separated peakons, coupled with quantitatively controlled asymptotic decoupling.

We investigate Hamiltonian curl-force systems with indefinite kinetic energy. We first reconsider Berry's polynomial Hamiltonian curl-force model, whose numerically observed closed trajectories motivated an integrability conjecture. A Painlevé analysis yields a non-principal resonance spectrum, so that the corresponding Laurent series cannot accommodate the required number of arbitrary constants of the general solution. The model therefore fails the standard Painlevé test. We then introduce a four-parameter curl-force family and identify the parameter locus on which this system is integrable. We construct a second Hamiltonian, compatible Poisson tensors, separated complex characteristic variables, and a Lax representation. More generally, the separated form yields polynomial integrable curl-force Hamiltonians of arbitrary degree. We also show that the same construction admits a higher time-derivative potentialisation whose free limit is the degenerate Pais-Uhlenbeck oscillator. Finally, we analyse zero-curl invariant reductions and elliptic periodic solutions, and exhibit an isolated periodic orbit outside the integrable regime. This illustrates that closed trajectories alone do not imply Liouville or Painlevé integrability.

We apply a machine learning framework to search for Lax connections in two-dimensional non-linear sigma models using local current data. For the $SU(2)$ principal chiral model and the symmetric coset $S^2 = SU(2)/U(1)$, the method successfully recovers the full spectral-parameter families without using the known spectral curves as training targets. For the non-symmetric coset $T^{1,1}$, the optimization converges to reproducible low-loss maps that distill into a compact block-diagonal ansatz. However, analytic verification shows that this candidate is a “fake Lax” connection which satisfies on-shell flatness but fails to encode the two-dimensional equations of motion, whereas its point-particle reduction yields a genuine mechanical Lax pair. These results demonstrate that machine learning can effectively propose candidate ansätze and identify spectral structures, but low flatness loss alone does not certify genuine integrability, underscoring the necessity of analytic validation.

Although Arnoldi reduction of a generally non-Hermitian Hamiltonian yields an upper Hessenberg matrix rather than the tridiagonal form of Hermitian Lanczos theory, we show that a closed Toda sector survives in its diagonal and subdiagonal coefficients. For a fixed finite-dimensional Hamiltonian and a cyclic state vector deformed holomorphically, the Krylov Gram determinants are $\tau$ functions of the finite two-dimensional Toda lattice, whose Flaschka variables coincide exactly with these Arnoldi coefficients. The Toda dynamics therefore closes on this sector without determining the remaining upper Hessenberg entries. The subdiagonal part of the same sector also has a direct geometric meaning: the squared subdiagonal coefficients determine both the Fubini–Study metric and the Berry curvature of holomorphic Krylov subspaces, whereas the geometric quantities associated with subspaces lost at Arnoldi breakdown cease to be defined. Along a smooth real path in the cyclic region, the Arnoldi-frame connection further provides a Hermitian tridiagonal generator of exact isospectral transport. When added to the Arnoldi matrix, this generator cancels transitions between instantaneous eigenspaces and realizes counterdiabatic driving whenever the matrix is diagonalizable with a nondegenerate spectrum.

In this research work, we consider a nonlinear fourth-order (1+2)-dimensional Kudryashov-Sinelshchikov (KS) equation which represents the wave propagation of pressures in liquids that contain gas bubbles. A direct Integrability of the KS equation is analysed using Painleve Analysis with Singular Manifold Method (SMM). With the help of WTC algorithm, we show that the (1+2) KS equation is Painleve integrable. Then by truncating the Painleve expansion we obtain the Auto-Backlund Transformation (ABT). By taking suitable forms of Manifold, various exact solutions based on the obtained Auto-Backlund transformation are derived. The consistancy check for these solutions are also performed. Representative solutions are presented in the from of 2D and 3D plots to understand the geometric perspective of the solutions.

The wave propagation of pressures in liquids that contain gas bubbles are an important concern in fluid dynamics and mathematical physics. The Kudryashov Sinelshchikov equation offers a useful mathematical framework in the study of nonlinear wave motion in bubbly liquids with reference to the effects of viscosity and heat exchange between liquid and gaseous phases. This paper examines the dimensional reduced (1 + 2)-dimensional Kudryashov Sinelshchikov equation, a fourth-order nonlinear partial differential equation. Analyzing the Lie symmetry, an infinite dimension Lie algebra is obtained because of the presence of arbitrary functions. By applying the commutative relation between these vector fields and choosing the specific forms for the arbitrary functions, helps the governing PDE to reduce to fourth order ODEs. The reduced equations are then investigated using the Painleve analysis to give solutions in the form of Laurent series. In addition, multiplier approach is used to obtain the conserved vectors and to analyzed conservations properties of the equation. We obtain four cases and the Conservation laws were verified for all the cases.

We give an analytic construction of a tau function for the $q$-difference sixth Painlevé equation ($q$PVI) as a Fredholm determinant through the general Riemann-Hilbert problem associated with it. We show that the tau function is an analytic function on its domain of definition that vanishes at a particular point if and only if the corresponding Riemann-Hilbert problem is point-wise not solvable there. We express the corresponding $q$PVI transcendents in terms of the tau function as well as three copies of it with some of the parameters shifted. Then the vanishing of any of these four tau functions corresponds to the transcendents taking value in a specific corresponding exceptional line on the initial value space of $q$PVI. Finally, we derive an asymptotic expansion of the tau function for small times $t$.

This work introduces an algebraic framework yielding explicit, closed matrix differential-difference equations for eighteen models in the exactly solvable sector of the KPZ universality class across four scaling regimes. By organizing Fredholm determinant data into an overdetermined linear problem on a directed lattice graph, we derive a compatibility system termed the diamond equations. Elementary seed data extracted from the shift structure of the Fredholm kernel provides simple solutions to this system. We then construct a Darboux transformation to compress the infinite-dimensional Fredholm data into a finite-dimensional matrix observable. We show this dressing procedure preserves the diamond equations; consequently the resulting matrix observable obeys the same nonlinear structure as the initial seed data. Verifying a closed nonlinear equation for any specific model thus reduces to checking a handful of linear conditions on its kernel data. Under a scalar reduction, the framework produces variable-coefficient Hirota-Miwa equations for Fredholm determinants, recovering the one-point bilinear equations of the author's earlier work as specializations. To supply the necessary seed data, a product graph construction with admissible propagators builds multipoint data in the fully discrete regime, while Euclidean division in a polynomial quotient algebra handles vertex and polymer models. Finally, we demonstrate the diamond equations are a gauge-equivalent reparametrization of the non-abelian Hirota-Miwa system, a central system in classical integrability theory.

We investigate the quantum interference and energetic phenomena associated with classical caustics in the quantum bouncing ball problem, we refer to as quantum caustics. By considering an initial Gaussian wavepacket, we show that caustics associated with the underlying classical trajectory families are exhibited. We connect the associated phase singularity chains in the vicinity of the caustic with the semiclassical Pearcey function built on the cusp catastrophe lines. We also quantify the amount of superenergy exhibited in these solutions - regions of space where the local energy exceeds the largest constituent energy eigenvalue. We give a complimentary description of the caustic and superenergy behavior using the Madelung/Bohm trajectories, which gives additional insight about the energy of the trajectories and how they traverse the phase singularity chains.

We present a novel approach to the superintegrability of collective Hamiltonians invariant under a Hamiltonian action of a connected semisimple compact Lie group, $G$, on a symplectic manifold, $M$. By exploiting a Hamiltonian torus action that goes back to Guillemin and Sternberg [GS,1983], we demonstrate that the functional dimensions of $\mathfrak{H} := \mathcal{J}^*(C^\infty(\mathfrak{g}^*)^G)$, where $\mathcal{J}: M \to \mathfrak{g}^*$ is the momentum map of the $G$ action, and its centralizer $\mathfrak{F}$ in $C^\infty(M)$ satisfy the equality $\mathrm{ddim}(\mathfrak{H}) + \mathrm{ddim}(\mathfrak{F}) = \mathrm{dim}(M)$. Together with a non-triviality condition, this ensures that the Abelian Poisson algebra $\mathfrak{H}\subset C^\infty(M)$ represents a superintegrable system, and it also follows that the momentum map of the GS torus action yields action variables for the system. Our work provides a new insight into collective superintegrability complementing earlier results of Bolsinov and Jovanović.