Papers

Onsager solved the two-dimensional Ising model in zero field, but no solution is known in a magnetic field. We prove that in any dimension $d\ge2$, at every fixed dimensionless field outside a countable set, the free energy obeys no algebraic differential equation in the coupling, even locally. The obstruction arises from droplet geometry: the maximal droplet volume grows as the $d/(d-1)$ power of its boundary size, while an integer band in the pressure coefficients protects the extremal contributions against cancellation. Our result excludes Onsager-type solutions and closed forms in elliptic, hypergeometric, or Painlevé functions of the coupling at these fields.

We establish topological recursion for the spin enumeration of hypermaps, or, in the dual language, spin dessins d'enfants. This problem has remarkable features distinguishing it from other applications of topological recursion to enumerative combinatorics. First, the spectral curve is elliptic rather than rational, as in most cases studied before. Second, it contains degenerate ramification points, and one has to apply a version of degenerate topological recursion. Finally, the meromorphic correlator differentials have "moving" poles on the antidiagonals between pairs of distinct arguments, and one has to adjust the loop equations to this kind of degeneracy as well. The computations involve concepts of different kinds, such as BKP integrability, Gaussian integrals, the x-y swap, and symplectic duality.

A recent graph-theoretic criterion for a Hamiltonian to exhibit a free-parafermion spectrum is that its frustration graph be an indifference graph. We show that an arbitrary finite indifference graph can be realized as a frustration graph of the quantized six-vertex model at odd roots of unity, by a suitable choice of the defining wiring diagram and parameters. This provides a concrete realization of the criterion through the quantum Frobenius property of the quantized six-vertex model, recently established from its three-dimensional integrable structure.

In this paper, we solve the direct monodromy problem for the Borel-summed $0$-parameter solution of Painlevé~I. Using the auxiliary time equation, we first construct formal WKB wave matrices whose amplitudes are regular at the double turning point. Rigorous Borel summation then yields exact WKB matrices in each Stokes region. Uniform estimates on moving action strips provide a common exact wave matrix in a full neighbourhood of the double point, proving that all four incident jumps are identities for consistent column normalisations. We formulate the resulting normalised Riemann–Hilbert problem and prove its existence and uniqueness on the Borel domains of the construction. Finally, after Painlevé scaling, the five Borel-summed solutions coincide with the five rotated tritronquée solutions of Painlevé~I.

We analyze the KdV–ghost system arising from the BRST formulation of BF-type topological field theory, its symmetries, Lax pair, spectral problem, zero-curvature formulation and Darboux formalism. The system is invariant under shifts of the ghost field, whose generator does not commute with the BRST symmetry of the KdV–ghost system. Their generators allow the introduction of a systematic procedure for generating new solutions from a given one, and the corresponding expressions in the Grassmann algebra are obtained explicitly. We introduce a Grassmann-valued Lax pair and prove an isospectral theorem relating the spectral problem of the spatial Lax operator for the KdV equation to that of its KdV–ghost extension. We also construct a triangular zero-curvature representation and show that the Grassmann-valued spatial Lax operator is related to the KdV spatial Lax operator by a nilpotent conjugation. The natural variable for a local Darboux formulation is the differentiated ghost $v=C_x$, for which the ghost equation coincides with the Fréchet linearization of the KdV flow. We then construct a nilpotent lift of the scalar KdV Bäcklund–Darboux transformation. Using Riccati variables and off-shell factorization identities for the KdV residue and its Fréchet derivative, we prove that the transformation closes on the local KdV–ghost system. Successive Darboux transformations yield a recursive ghost extension of the bosonic Darboux chain, with a new odd homogeneous contribution introduced at each step. For the solitonic examples, the resulting local ghost fields are naturally identified with tangent directions associated with the soliton phase parameters. The construction provides a Darboux-theoretic interpretation of the ghost sector while remaining distinct from the standard supersymmetric KdV hierarchies.

This article aims to introduce the bosonic asymmetric quantum conditional mutual information (QCMI), $I_{B}(A; B|C)$, as a rigorously defined information-theoretic quantity for continuous-variable (bosonic) systems. After defining $I_{B}(A; B|C)$ through quantum instruments on infinite-dimensional Fock spaces, we prove that Bose-Einstein condensation (BEC) induces a logarithmic divergence $ I_{B}(A; B|C) \geq \alpha \log_2 (N_{0}^C+1)$, which further establishes $I_{B}(A; B|C)$ as a quantum-information order parameter for BEC with power-law scaling near the critical point. We provide the complete five-step proof of this result, connecting measurement back-action, off-diagonal long-range order, and Jensen's inequality. In the condensed phase, the effective causal propagation velocity is set by the Bogoliubov sound speed $c_s$ in the long-wavelength limit, and vanishes continuously at the superfluid–Mott-insulator transition. Applications to composite bosons, the BEC–BCS crossover, and a quantum-gas-microscope measurement protocol are presented. Finally, we demonstrate how the reduced bosonic QCMI serves as a finite-size scaling probe analogous to the Binder cumulant.

The focusing coupled Kundu-Eckhaus (CKE) equation extends the coupled nonlinear Schrödinger (CNLS) equation by adding quintic and derivative nonlinearities. We study the Cauchy problem for the focusing CKE equation with Schwartz initial data under the assumption of no discrete spectrum and no spectral singularities on the real axis. Based on the gauge equivalence between the focusing CKE and CNLS equations, we analyze the associated $3\times3$ matrix Riemann-Hilbert problem by the Deift-Zhou nonlinear steepest descent method. By establishing joint error estimates for the CNLS solution and the gauge phase, we obtain a uniform long-time asymptotic formula for the dispersive radiation of the CKE equation with an $O(t^{-1}\log t)$ remainder in bounded velocity cones.

Higher-order semi-rational solutions are constructed for the (2+1)-dimensional nonlocal multi-component Yajima-Oikawa system. This system is obtained from its local \(2M\)-component counterpart through a spatial-reflection reduction. Using Hirota's bilinear method and the Kadomtsev-Petviashvili hierarchy reduction, we derive a unified \(6\times6\) block Gram determinant. Its entries involve Schur polynomials and exponential functions. The complex-conjugation and spatial-reflection reductions are imposed through algebraic constraints on the spectral parameters and determinant blocks, with the former ensuring a real-valued denominator tau function. Separate spectral parameter sets are assigned to the rational, breather, and dark-soliton blocks. These solutions describe paired lumps or line rogue waves on breather, dark-soliton, or combined backgrounds. Pure rational, breather, and dark-soliton solutions follow by setting the appropriate block sizes to zero. Asymptotic analysis establishes the elastic interactions of the fundamental breather and dark-soliton pairs and determines their position and phase shifts.

In these lecture notes we review some constructions and results on the relation between topology of the moduli space of stable curves and integrable systems of evolutionary PDEs.

We establish an exact correspondence between the quasi-Wronskian and quasi-Grammian constructions for the anti-self-dual Yang–Mills (ASDYM) equations with complex \(2\times2\) matrix-valued fields. Starting from vacuum wave functions associated with \(2M\) pairwise distinct nonzero constant spectral parameters, we restrict attention to a nonempty local domain on which the corresponding interpolation matrix is invertible. We show that both constructions are governed by the same uniquely solvable matrix-polynomial interpolation problem. In particular, the quasi-Wronskian Darboux polynomial \(T(\lambda)\) and the quasi-Grammian rational dressing matrix \(W(\lambda)\) satisfy \[ T(\lambda)=q_\xi(\lambda)W(\lambda), \] where \(q_\xi(\lambda)\) is the monic scalar polynomial whose zeros are the prescribed poles of \(W(\lambda)\). This identity yields explicit relations between the corresponding Yang and potential matrices and shows that the two constructions generate identical ASDYM gauge potentials in the same gauge.

We establish that bond Hamiltonian densities, up to local difference terms whose sum vanishes under periodic boundary conditions, lying in the commutant of bond operators, $w_j\pm w_{j+1}$, is a necessary and sufficient condition for a one-dimensional nearest-neighbour Hamiltonian to conserve a uniform or staggered sum of the one-site operators $w_j$. Starting from these local commutants, we show that every such Hamiltonian and its conserved charge can be obtained from monodromy matrices constructed using regular and non-regular members of scalene Yang-Baxter triples. The results hold on periodic chains of length $L\geq3$, with even length required in the staggered case. The results do not require translation invariance, and hence the interaction on each bond can be chosen independently from the full corresponding commutant. Our representation-independent construction applies in arbitrary finite local dimension and exhausts the Hamiltonians with either prescribed additive symmetry, without requiring ordinary Yang-Baxter integrability. We illustrate the construction with spin-$\tfrac12$ and spin-1 systems and discuss its relation to the double-commutant theorem. Classifying the one-site generators $w$ and their associated bond commutants provides a route to classifying such Hamiltonians with additive conserved charges.

We describe a family of two-dimensional metric deformations that preserve integrability. The seeds are (classically) conformal integrable field theories and the deformations are directly related to different classes of auxiliary field sigma models. We introduce a master action connecting deformations by Lie-algebra-valued auxiliary fields, the so-called $\nu$-frame, to deformations by two Lorentz-charged scalars. These auxiliary scalar fields can be identified as Beltrami differentials of an auxiliary two-dimensional metric, providing a direct connection with the metric approach to $T\bar{T}$-like deformations. For Weyl-invariant seed matter theories, the auxiliary metric sector is described by a reduced gravitational potential. We show that the resulting framework preserves classical Lax integrability whenever the seed matter theory admits a Lax representation of its equations of motion on a generic curved two-dimensional spacetime. Finally, within the Lorentz-invariant family and the zweibein deformations of conformal matter theories considered here, we identify $T\bar{T}$ as the only deformation that can be interpreted as a field-dependent coordinate transformation.

The staggered order parameter of the gapped XYZ spin-$1/2$ chain follows from the eight-vertex solution, but in elliptic variables that fix the Hamiltonian couplings only implicitly. We derive a closed form for the coefficient governing the leading quadratic response of the spontaneous staggered magnetization to the XY anisotropy at fixed longitudinal anisotropy in the gapped antiferromagnetic regime. Starting from the Baxter–Kelland–Jimbo–Miwa–Nakayashiki polarization formula, we show that at fixed longitudinal anisotropy both the modular nome and the crossing-parameter variable of the formula move with the XY anisotropy, and that the motion of the crossing-parameter variable reverses the sign of the expansion at fixed elliptic variables. The coefficient is a rational term plus a Lambert series. It is strictly negative and bounded above by a rational function of the longitudinal anisotropy, to which it is asymptotic in the Ising limit. At the isotropic point it diverges as the inverse five-halves power of the distance to that point, and since the Lambert sum has an asymptotic expansion with exponentially small remainder, the subleading terms of the divergence also follow in closed form. When the XY anisotropy vanishes as the five-fourths power of the distance to the isotropic point, the ratio of the perturbed to the unperturbed order parameter tends to a Gaussian in the scaling variable, and the quadratic truncation is controlled when that variable is small. A fourth-order perturbative calculation in the spin chain, with no input from the eight-vertex solution, reproduces the leading strong-coupling behaviour, and infinite-DMRG calculations at four anisotropies in the gapped regime are consistent with the closed form.

We construct a family of self-dual $(1+1)$-dimensional $\mathbb{CP}^1$ sigma models whose potentials $V$ are determined from pre-potentials $U$ by Platonic geometry. Using the round metric on $\mathbb{S}^2\simeq\mathbb{CP}^1$ and the generalized self-duality formalism, we build symmetry-invariant pre-potentials $U$ from the unit vertices of each Platonic solid intersecting the unit circumsphere. The regular-lattice sites associated with vertex, face-center, and edge-midpoint directions are critical points of $U$ and degenerate vacua of the scalar potential $V$. Dual solids share the same regular lattice but generate different pre-potentials, charges, potentials, and metric-gradient flows. Numerical integration reveals distinct self-dual trajectories and energy-density profiles, including multi-peak structures for paths passing near saddle points, while solutions with the same endpoint difference $\Delta U$ have the same total energy. The positive maxima of $V$ are also organized into symmetry orbits. This construction provides a geometric prescription for controlling the vacuum structure and self-dual sectors of $\mathbb{CP}^1$ models with a potential.

We study magnets composed of molecules with an arbitrary number $m$ of magnetically active atoms in a chain configuration interacting with nearest-neighbor couplings. We derive the thermodynamics of a chain of spin-$j$ $SU(2)$ magnetic atoms using a mean-field approach and determine the equilibrium states and stability conditions. The system exhibits a rich phase structure with a cascade of ferromagnetic phase transitions, their number ranging from $m$, for maximal ferromagnets, to zero, for paramagnets, depending on the relative strengths of the couplings. In the presence of impurities (modified couplings) at the ends of the chain, boundary states emerge that coexist with the paramagnetic phase or the fully ferromagnetic phase. For long chains, a continuum approximation can be taken that clarifies the structure of the phases. In the continuum description, magnetized states at temperatures lower than the critical temperature at which they appear correspond to kink solitons.

Many machine learning algorithms rely on the Fourier transform, which is also the inner core of many analog computing devices and, specifically, of optical computing. The square of the modulus of the Fourier spectrum naturally maps onto pairwise interactions, as in the renowned Ising model, making it applicable to combinatorial optimization. Here we show how the nonlinear Fourier transform, related to integrable nonlinear partial differential equations, significantly widens the computational capability and also enables a wide variety of computing gates. We study specifically the case of the nonlinear Schrödinger equation; we introduce an input composed by “tokens”, i.e., batches of information encoded in a multi-step potential, and we show how the nonlinear spectral computing arising in the wave propagation not only generalizes combinatorial optimization to higher order problems, but also realizes boolean gates. In the weakly nonlinear regime, nonlinearity generates higher-order interactions as perturbative corrections to the pairwise terms. In the solitonic regime, the discrete nonlinear spectrum performs digital computation through soliton generation. By using four input tokens, we demonstrate the programmability of all 256 Boolean functions of three variables, including 3-SAT instances. Our results show that integrable nonlinear propagation provides an unprecedented single physical platform for higher-order and logical computation, opening the way to a rich variety of new algorithms for analog computing, machine learning, and classical and quantum information processing.

We study magnets composed of molecules with atoms interacting via $SU(N)$-invariant two-body interactions. Atoms carry states transforming in generically different irreducible representations of $SU(N)$, and the interaction strength between atoms depends on the specific atoms involved. We derive the thermodynamics of these magnets in the mean-field approach, and find that they can exhibit mixed ferro-antiferromagnetic behavior. We present a full analysis of the simplest nontrivial case of molecules consisting of two atoms in the spin-$j$ representation of $SU(2)$, which generalizes conventional magnetic systems, and derive its phase structure and transitions.

We complete the systematic derivation of the Quantum Spectral Curve (QSC) for strings on $\text{AdS}_3\times\text{S}^3\times \text{T}^4$ with pure Ramond–Ramond (RR) flux, building on the extended Y-system obtained in our preceding paper. We establish the precise relation between the two formalisms of the Thermodynamic Bethe Ansatz and the QSC, derive the previously conjectured QSC, and, along the way, uncover additional relations satisfied by its solutions. Our analysis also clarifies how massless excitations, although not explicit among the basic QSC building blocks, are encoded in their analytic structure. The derivation culminates in the so-called $\mathbf{P}{\mu}$-system, a compact formulation of the QSC in terms of a small set of functions and their analytic continuations.

The Davey-Stewartson (DS) system is a canonical model for a weakly nonlinear wave packet in two spatial dimensions that is resonantly coupled to the slowly varying mean flow it generates, and as such it governs modulated wave trains in settings as different as finite-depth water waves, quadratic optical media and magnetic films. Here we develop the Whitham modulation theory for the DS system and use it to study the stability of its periodic traveling wave solutions. Specifically: 1. We perform a detailed study of the parametric dependence of the periodic solutions of all four variants of the DS system, including their harmonic and soliton limits. 2. We present the formulation of Whitham modulation theory for the DS system and write down the resulting Whitham modulation equations for all four variants of the DS system. 3. We use the resulting modulation equations to study the stability of the periodic traveling wave solutions of all four variants of the DS system. 4. We validate the theoretical predictions by comparing them with a direct linearization of the DS system as well as with the direct numerical simulations of the DS system, showing excellent agreement. The results indicate that the periodic solutions of the defocusing DSII system are linearly stable, whereas those of all other variants of the DS system are unstable. The calculations also evidence many structural features (non-evolutionary modulation equations, constraints on the admissible initial data, and auxiliary mean-field constants absent from the traveling wave solutions yet indispensable to the modulation system) are expected to recur in any two-dimensional envelope equation with mean-field coupling.

It has been known that (the square of) Jacobi elliptic function can be expressed by an infinite sum of single solitons of $\sech^2$ shape. In this paper, we show that there exists a similar structure for higher genus cases. It turns out that this is just a consequence of the quasi-periodicity of the Riemann $\theta$-function. More precisely, we show that the second derivative of $\log \theta$ with the real Riemann $\theta$-function of genus $g$ near soliton limit can be well approximated by the sum of real and regular $g$-soliton solutions of Hirota-type in $g$-dimensional real space $\R^g$. This leads to a tessellation of $\R^g$, whose tile is an oblique prism divided into $2^g$ sections by hyper-planes of dominant exponents in the theta function. We apply the results to study quasi-periodic solutions to the KdV and KP equations. We construct the quasi-periodic solutions using the Schottky group, which uniformizes the corresponding Riemann surfaces. We also discuss solitons on quasi-periodic background by pinching some of the homological cycles

Harnessing integrability at arbitrary interaction strength, we build long-lived and spatially localized soliton-like excitations in the 1D Bose gas with contact repulsion (Lieb-Liniger model). We show how these density profiles, built from selective superpositions of multi-hole eigenstates, display long lifetimes tunable through parameters like dispersion width and number of holes. We further illustrate a scaling protocol that in principle enables unbounded increase in lifetimes in the thermodynamic limit. Our results showcase classical-like excitations originating from unitary quantum dynamics, away from the semi-classical regime, and in regions where interactions cannot be treated perturbatively.

We consider the interaction-round-a-face version of the nineteen-vertex model and derive a new quantum spin-$1$ chain. In the isotropic limit, the associated spin chain can be seen as three coupled Babujian-Taktahjan spin chains with different boundary twists. We compute the ground state short-distance correlation functions of the isotropic model in the thermodynamic limit by exploiting the shared integrable structure between the models.

We derive long-time asymptotics in different space-time regions for the Sawada-Kotera (SK) equation with admissible data involving continuous spectrum and finite discrete spectrum. Our result proves the soliton resolution conjecture, yields the asymptotics governed by the F-XVIII Painlevé transcendent, and establishes the asymptotic stability of \(N\)-soliton solutions. More importantly, this extends the long-time asymptotic theory to the coexistence of solitons and radiation, whereas previous results were restricted to the purely continuous spectrum. One key ingredient in our argument is a new meromorphic Riemann-Hilbert framework that incorporates the discrete spectrum through complete six-point pole orbits while preserving cofactor analyticity and the singular structure at the spectral origin. A second ingredient is the combination of an orbitwise, region-dependent pole reduction mechanism and the \(\bar{\partial}\)-nonlinear steepest descent method, which allows us to select the solitons contributing in each region and separate them from the continuous radiation. Our framework also resolves the coalescing stationary points at the singular spectral origin through the regular modified SK problem and the Miura map.

An extension with temporal modulation is presented of the canonical 2 + 1-dimensional solitonic Dym equation as originally introduced by Konopelchenko and Dubrovsky. Application is made of admitted Painleve' II symmetry reduction to derive exact solution to a class of associated 2 + 1-dimensional nonlinear moving boundary problems of generalized Stefan-type.

This work establishes finite-time blow-up for the completely integrable defocusing Ablowitz-Ladik equation on both infinite and finite lattices. We first establish local well-posedness for arbitrary initial data in the space $\ell^\infty$ of uniformly bounded sequences, with no decay assumptions and under general boundary conditions. We then show that the unit circle acts as a separatrix for the lattice dynamics: sites with an initial modulus inside, on, or outside the unit circle maintain their modulus within their respective region for as long as the solution exists. As a consequence, initial data in the closure of the $\ell^\infty$ unit ball generate global-in-time solutions. On the other hand, the unit circle separatrix opens up the possibility for finite-time blow-up, which is restricted to sites with initial modulus outside the unit circle. We establish this scenario by constructing a class of initial data whose corresponding solutions are singular, with explicitly given exact blow-up times. We also construct solutions emanating from initial data with an arbitrarily large single-site excitation outside the unit circle that exist globally in time, a behavior driven by the convergence of the corresponding modulus to the unit circle equilibrium. In the presence of such solutions, along with other important classes of analytical, global-in-time solutions that remain outside the closed $\ell^\infty$ unit ball for all times, we examine the fundamental question of the dominant dynamics for general initial data outside the closed unit ball. By utilizing the constructed class of singular solutions as subsolutions for the general problem, we prove that the dynamics for the completely integrable defocusing Ablowitz-Ladik lattice is dominated by finite-time blow-up. We conclude with numerical studies of illustrative examples of general initial data

We construct spectral-parameter solutions of the quantum Yang–Baxter equation from the associativity condition of the universal symmetric $2$-algebraic two-valued group, whose multiplication law depends on four coefficients $k_2,k_4,k_6,k_8$ subject to the single relation $4k_8-k_4^2+k_2k_6=0$ – which coincides with the three-dimensional WDVV equation for Dubrovin's normalised potential. Every three-dimensional Frobenius manifold with a cyclic vector field carries such a spectral two-valued group, whose finite branch points are the eigenvalues of the multiplication operator. For an arbitrary unital algebra we show that a two-parameter spectral $R$-operator built from multiplication and unit has Yang–Baxter defect proportional to an explicit tensor built from the associator; for tangent Frobenius algebras this yields a regular, unitary $9\times9$ $R$-matrix satisfying the quantum Yang–Baxter equation exactly at WDVV points, with spectral parameter the Abel coordinate of a bielliptic curve whose elliptic quotient is the spectral curve, together with an associated integrable chain. The $A_3$ Frobenius manifold is worked out explicitly.

In an earlier paper of P. Rossi and the first author, a construction of a wide class of quantum hierarchies was presented, with the input data given by an arbitrary cohomological field theory. A crucial role in the construction is played by the double ramification (DR) cycles on the moduli spaces of stable curves. As a special case, one gets a quantization of the hierarchy of the Intermediate Long Wave (ILW) equation. The ILW equation has two famous limits: the Korteweg–de Vries equation and the Benjamin–Ono (BO) equation, and for the latter one a quantum hierarchy was constructed by Nazarov and Sklyanin. In our paper, we propose a conjecture describing a relation between a limit of the quantum ILW hierarchy and the Nazarov–Sklyanin hierarchy and present various checks to support it.

Every scalar ordinary differential equation of order at least three with rational right-hand side that is locally linearizable by a point transformation admits a linear form with rational coefficients over the same coefficient field. We prove this by restricting the derived symmetry algebra to a coordinate line and recovering a scalar differential operator from rational symmetry-jet data. The construction uses differential elimination and linear algebra; it does not require the symmetry generators or a linearizing transformation to be solved for. A single integer parameter suffices to choose the line.

We study a real-valued max-plus particle system on a finite periodic lattice. Its three-dimensional fundamental diagram relates the mean flow to two densities: the conserved particle density and a second density defined by a local state function. We prove that the deviation from the diagram tends to zero for arbitrary initial data with all state values in [0,1]. The proof uses two nonnegative local quantities and their evolution relations. For rational initial data with all state values in $[0,1]$, the orbit is eventually periodic, and the fundamental-diagram relation holds exactly on the periodic orbit.

This paper studies the focusing Calogero–Moser derivative nonlinear Schrödinger (CMdNLS) equation under the nonzero boundary condition \(|u(t,x)|\to 1\) as \(|x|\to+\infty\). In sharp contrast to the classical paradigm for the cubic nonlinear Schrödinger equation, namely that focusing nonlinearities support only bright solitons on a zero background, we show that the focusing CMdNLS equation admits a rich family of dark multi-soliton solutions on a nonzero background, whose existence is driven by a spectral gap \([-1,0]\) in the point spectrum of the Lax operator, together with the Hardy space structure. The main results are threefold. First, we establish an explicit formula for general solutions in the Hardy–Zhidkov space \(\mathcal Z^2_+\), via a semigroup method based on new commutator estimates between the Lax–Beurling shift semigroup and Toeplitz operators. This approach circumvents the non-commutativity of the unbounded generator with the time derivative and requires no weighted hypothesis on solutions or on the domain of validity of commutator formulas. Second, the algebraic definition of dark multi-soliton potentials is shown to be equivalent to two spectral conditions on the Lax operator. Together with the conservation of a generating functional, this yields the invariance of the dark \(N\)-soliton set \(\mathcal U_N\) under the CMdNLS flow. Third, the action-angle variables on \(\mathcal U_N\) are constructed, which establishes the complete integrability of the focusing CMdNLS equation on \(\mathcal U_N\). Combining these results, we obtain an inverse spectral formula, from which it follows that the poles of dark multi-soliton solutions satisfy a complexified Calogero–Moser system. Long time asymptotics and uniform Sobolev estimates are also derived.