We consider the family of (poly)continua 𝒦 in the upper half-plane ℍ that contain a preassigned finite anchor set E∈ℍ . For a given harmonic external field we define a Dirichlet energy functional ℐ(𝒦) and show that within each “connectivity class” of the family, there exists a minimizing compact 𝒦^* consisting of critical trajectories of a quadratic differential. In many cases this quadratic differential coincides with the square of the real normalized quasimomentum differential dp associated with the finite gap solutions of the focusing Nonlinear Schrödinger equation (fNLS) defined by a hyperelliptic Riemann surface ℜ branched at the points E∪E̅ . The motivation for this work lies in the problem of soliton condensate of least average intensity such that a given anchor set E belongs to the poly-continuum 𝒦 . An fNLS soliton condensate is defined by a compact 𝒦⊂ℍ (its spectral support) whereas the average intensity of the condensate is proportional to ℐ(𝒦) . We prove that the spectral support 𝒦^* provides the fNLS soliton condensate of the least average intensity within a given “connectivity class”.

The defocusing nonlinear Schrödinger hydrodynamics supports exact dark solitons under finite density boundary conditions. However, the dark soliton gas, an interacting ensemble of dark solitons, has not yet been studied. In this work, we introduce an arbitrary-genus potential of dark soliton gases by considering the limit of the $\mathcal{N}$-dark soliton as $\mathcal{N}\to \infty$. The large-space asymptotics and long-time evolution of this dark soliton gas potential are analytically investigated through Deift-Zhou nonlinear steepest descent approach. The genus-$N$ dark soliton gas potential approaches the genus-$N$ finite-gap solution as $x \to -\infty$ and the background $1$ as $x \to +\infty$. In the long-time evolution, as the self-similar variable $ξ=x/t$ increases, the gas configuration exhibits a cascade of behaviours, passing from unmodulated and modulated genus-$N$ regions and progressively reducing the genus down to the planar region (unmodulated genus-$0$ region). Notably, the evolution of lower-genus soliton gases can be embedded within that of higher-genus gases, exhibiting identical dynamics within specific regimes. This phenomenon is encoded by the underlying spectra. We also include numerical validations, in perfect agreement with the theoretical predictions.

In this paper, we analyze the asymptotic behaviour of the poles of certain rational solutions of the fifth Painlevé equation. These solutions are constructed by relating the corresponding tau function to a Hankel determinant of a certain sequence of moments. This approach was also used by one of the authors and collaborators in the study of the rational solutions of the second Painlevé equation. More specifically, we study the roots of the corresponding polynomial tau function, whose location corresponds to the poles of the associated rational solution. We show that, upon suitable rescaling, the roots asymptotically fill a region bounded by analytic arcs when the degree of the polynomial tau function tends to infinity and the other parameters are kept fixed. Moreover, we provide an approximate location of these roots within the region in terms of suitable quantization conditions.

We construct new sets of log-canonical coordinates on SL(2, ℂ) character varieties of compact Riemann surfaces. These coordinates are obtained by combining shear type coordinates with length-twist type coordinates. On the real component corresponding to the moduli space of compact Riemann surfaces ℳ_g, the generating function corresponding to the symplectomorphism between these new coordinates and Fenchel-Nielsen coordinates is explicitly computed.

We consider the problem of reconstruction of an n× n matrix with coefficients depending rationally on x∈ℙ^1 from the data of: (a) its characteristic polynomial and (b) a line bundle of degree g+n-1, with g the geometric genus of the spectral curve, represented by a choice of g+n+1 points forming a (non-positive) divisor of the given degree. We thus provide a reconstruction formula that does not involve transcendental functions; this includes formulas for the spectral projectors and for the change of line bundle, thus integrating the isospectral flows. The formula is a single residue formula which depends rationally on the coordinates of the points involved, the coefficients of the spectral curve, and the position of the finite poles of L. We also discuss the canonical bi-differential associated with the Lax matrix and its relationship with other bi-differentials that appear in Topological Recursion and integrable systems.

We consider soliton gas solutions of the focusing nonlinear Schrödinger (NLS) equation, where the point spectrum of the Zakharov–Shabat linear operator condenses in a bounded domain D in the upper half-plane. We show that the corresponding inverse scattering problem can be formulated as a ∂ ¯ -problem on the complex plane. We prove that the τ -function of the N soliton solution converges in the limit N → ∞ to the τ -function (a Fredholm determinant) of the ∂ ¯ -problem. Furthermore, we prove that such a τ -function is non-vanishing for all values of x and t , thus showing the existence of a solution of the ∂ ¯ -problem. Then we show that, when the domain D is an ellipse and the soliton gas spectral data are analytic, the inverse problem reduces to the soliton spectra concentrating on the segment connecting the foci of the ellipse (soliton shielding). The NLS solution for fixed times is asymptotically step-like oscillatory, and it is described by a periodic elliptic function as x → − ∞ while it vanishes exponentially fast as x → + ∞ .

We consider soliton gas solutions of the focusing nonlinear Schrodinger (NLS) equation, where the point spectrum of the Zakharov-Shabat linear operator condenses in a bounded domain D in the upper half-plane. We show that the corresponding inverse scattering problem can be formulated as a.-problem on the complex plane. We prove that the t -function of the N soliton solution converges in the limit N.8 to the t -function (a Fredholm determinant) of the.-problem. Furthermore, we prove that such a t -function is non-vanishing for all values of x and t, thus showing the existence of a solution of the.-problem. Then we show that, when the domain D is an ellipse and the soliton gas spectral data are analytic, the inverse problem reduces to the soliton spectra concentrating on the segment connecting the foci of the ellipse (soliton shielding). The NLS solution for fixed times is asymptotically step-like oscillatory, and it is described by a periodic elliptic function as x.-8 while it vanishes exponentially fast as x.+8.

We consider the extension to higher genus Riemann surfaces of the classical Chebotarev problem, with a view towards the development of the theory of Padé approximants on algebraic curves. To this end we define an appropriate notion of capacity that mimics the standard one, following works of Chirka and of the author and collaborators. The nontrivial topology of the Riemann surface requires further specification of the “homotopy class” of the continua in the solution of the Chebotarev problem. We also discuss the relationship of this problem to the theory of Jenkins-Strebel quadratic differentials.

Using WKB analysis, the paper addresses a conjecture of Shapiro and Tater on the similarity between two sets of points in the complex plane; on one side is the set the values of t∈ℂ for which the spectrum of the quartic anharmonic oscillator in the complex plane d^2 y/d x^2 - ( x^4 + tx^2 + 2Jx ) y = Λ y, with certain boundary conditions, has repeated eigenvalues. On the other side is the set of zeroes of the Vorob’ev–Yablonskii polynomials, i.e. the poles of rational solutions of the second Painlevé equation. Along the way, we indicate a surprising and deep connection between the anharmonic oscillator problem and certain degenerate orthogonal (monic) polynomials.

We develop the theory of integrable operators K acting on a domain of the complex plane with smooth boundary in analogy with the theory of integrable operators acting on contours of the complex plane. We show how the resolvent operator is obtained from the solution of a partial derivative -problem in the complex plane. When such a partial derivative -problem depends on auxiliary parameters we define its Malgrange one form in analogy with the theory of isomonodromic problems. We show that the Malgrange one form is closed and coincides with the exterior logarithmic differential of the Hilbert-Carleman determinant of the operator K . With suitable choices of the setup we show that the Hilbert-Carleman determinant is a tau-function of the Kadomtsev-Petviashvili (KP) or nonlinear Schrodinger hierarchies.

We consider the moduli space of vector bundles of rank $n$ and degree $ng$ over a fixed Riemann surface of genus $g\geq 2$. We use the explicit parametrization in terms of the Tyurin data. In the moduli space there is a "non-abelian" Theta divisor, consisting of bundles with $h^1\geq 1$. On the complement of this divisor we construct a non-abelian Cauchy kernel explicitly in terms of the Tyurin data. With the additional datum of a non-special divisor, we can construct a reference flat holomorphic connection which is also dependent holomorphically on the moduli of the bundle. This allows us to identify the bundle of Higgs fields, i.e. the cotangent bundle of the moduli space, with the affine bundle of holomorphic connections and provide a monodromy map into the ${\rm GL}_n$ character variety. We show that the Goldman symplectic structure on the character variety pulls back along this map to the complex canonical symplectic structure on the cotangent bundle and hence also on the space of affine connections. The pull-back of the Liouville one-form to the affine bundle of connections is then shown to be a logarithmic form with poles along the non-abelian theta divisor and residue given by $h^1$.

A famous result of Stieltjes relates the zeroes of the classical orthogonal polynomials with the configurations of points on the line that minimize a suitable energy. The energy has logarithmic interactions and an external field whose exponential is related to the weight of the classical orthogonal polynomials. The optimal configuration satisfies an algebraic set of equations: we call this set of algebraic equations the Stieltjes--Fekete problem or equivalently the Stieltjes--Bethe equations. In this work we consider the Stieltjes-Fekete problem when the derivative of the external field is an arbitrary rational complex function. We show that its solutions are in one-to-one correspondence with the zeroes of certain non-hermitean orthogonal polynomials that satisfy an excess of orthogonality conditions and are thus termed "degenerate". This generalizes the original result of Stieltjes.

Inthis paper, we study the small-lambda spectral asymptotics of an integral operator k defined on two multi-intervals J and E, when the multi-intervals touch each other (but their interiors are disjoint). The operator k is closely related to the multi-interval finite Hilbert transform (FHT). This case can be viewed as a singular limit of self-adjoint Hilbert-Schmidt integral operators with so-called integrable kernels, where the limiting operator is still bounded, but has a continuous spectral component. The regular case when dist (J,E) > 0, and k is of the Hilbert-Schmidt class, was studied in an earlier paper by the authors. The main assumption in this paper is that U = J boolean OR E is a single interval (although part of our analysis is valid in a more general situation). We show that the eigenvalues of k, if they exist, do not accumulate at lambda=0. Combined with the results in an earlier paper by the authors, this implies that H-p, the subspace of discontinuity (the span of all eigenfunctions) of k, is finite dimensional and consists of functions that are smooth in the interiors of J and E. We also obtain an approximation to the kernel of the unitary transformation that diagonalizes k, and obtain a precise estimate of the exponential instability of inverting k. Our work is based on the method of Riemann-Hilbert problem and the nonlinear steepest descent method of Deift and Zhou.

The main goal of the paper is to connect matrix polynomial biorthogonality on a contour in the plane with a suitable notion of scalar, multi-point Padé approximation on an arbitrary Riemann surface endowed with a rational map to the Riemann sphere. To this end we introduce an appropriate notion of (scalar) multi-point Padé approximation on a Riemann surface and corresponding notion of biorthogonality of sections of the semi-canonical bundle (half-differentials). Several examples are offered in illustration of the new notions.

Critical measures in the complex plane are saddle points for the logarithmic energy with external field. Their local and global structure was described by Martinez-Finkelshtein and Rakhmanov. In this paper we start the development of a theory of critical measures on higher genus Riemann surfaces, where the logarithmic energy is replaced by the energy with respect to a bipolar Green's kernel. We study a max-min problem for the bipolar Green's energy with external fields Re V where dV is a meromorphic differential. Under reasonable assumptions the max-min problem has a solution and we show that the corresponding equilibrium measure is a critical measure in the external field. In a special genus one situation we are able to show that the critical measure is supported on maximal trajectories of a meromorphic quadratic differential. We are motivated by applications to random lozenge tilings of a hexagon with periodic weightings. Correlations in these models are expressible in terms of matrix valued orthogonal polynomials. The matrix orthogonality is interpreted as (partial) scalar orthogonality on a Riemann surface. The theory of critical measures will be useful for the asymptotic analysis of a corresponding Riemann-Hilbert problem as we outline in the paper.

We revisit the symplectic aspects of the spectral transform for matrix-valued rational functions with simple poles. We construct eigenvectors of such matrices in terms of the Szegő kernel on the spectral curve. Using variational formulas for the Szegő kernel we construct a new system of action-angle variables for the canonical symplectic form on the space of such functions. Comparison with previously known action-angle variables shows that the vector of Riemann constants is the gradient of some function on the moduli space of spectral curves; this function is found in the case of matrix dimension 2, when the spectral curve is hyperelliptic.

We obtain Fredholm type formulas for partial degenerations of Theta functions on (irreducible) nodal curves of arbitrary genus, with emphasis on nodal curves of genus one. An application is the study of "many-soliton" solutions on an elliptic (cnoidal) background standing wave for the Korteweg-de Vries (KdV) equation starting from a formula that is reminiscent of the classical Kay-Moses formula for $N$-solitons. In particular, we represent such a solution as a sum of the following two terms: a ``shifted" elliptic (cnoidal) background wave and a Kay-Moses type determinant containing Jacobi theta functions for the solitonic content, which can be viewed as a collection of solitary disturbances on the cnoidal background. The expressions for the traveling (group) speed of these solitary disturbances, as well as for the interaction kernel describing the scattering of pairs of such solitary disturbances, are obtained explicitly in terms of Jacobi theta functions. We also show that genus $N+1$ finite gap solutions with random initial phases converge in probability to the deterministic cnoidal wave solution as $N$ bands degenerate to a nodal curve of genus one. Finally, we derive the nonlinear dispersion relations and the equation of states for the KdV soliton gas on the residual elliptic background.

We first consider a deterministic gas of N solitons for the focusing nonlinear Schrödinger (FNLS) equation in the limit N→∞ with a point spectrum chosen to interpolate a given spectral soliton density over a bounded domain of the complex spectral plane. We show that when the domain is a disk and the soliton density is an analytic function, then the corresponding deterministic soliton gas surprisingly yields the one-soliton solution with the point spectrum the center of the disk. We call this effect soliton shielding. We show that this behavior is robust and survives also for a stochastic soliton gas: indeed, when the N-soliton spectrum is chosen as random variables either uniformly distributed on the circle, or chosen according to the statistics of the eigenvalues of the Ginibre random matrix the phenomenon of soliton shielding persists in the limit N→∞. When the domain is an ellipse, the soliton shielding reduces the spectral data to the soliton density concentrating between the foci of the ellipse. The physical solution is asymptotically steplike oscillatory, namely, the initial profile is a periodic elliptic function in the negative x direction while it vanishes exponentially fast in the opposite direction.

The goal of this paper is to express the extended Goldman symplectic structure on the SL(n) character variety of a punctured Riemann surface in terms of Fock-Goncharov coordinates. The associated symplectic form has integer coefficients expressed via the inverse of the Cartan matrix. The main technical tool is a canonical two-form associated to a flat graph connection. We discuss the relationship between the extension of the Goldman Poisson structure and the Poisson structure defined by Fock and Goncharov. We elucidate the role of the Rogers' dilogarithm as generating function of the symplectomorphism defined by a graph transformation.

We study the WKB expansion of $2\times 2$ system of linear differential equations with four fuchsian singularities. The main focus is on the generating function of the monodromy symplectomorphism which, according to a recent paper is closely related to the Jimbo-Miwa tau-function. We compute the first three terms of the WKB expansion of the generating function and establish the link to the Bergman tau-function.