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1. Large-parameter asymptotics of generalized Hastings–McLeod functions

   https://doi.org/10.1088/1361-6544/ae0230  

   Schmidt, Kurt; Buckingham, Robert (September 2025, Nonlinearity)

   Abstract The generalized Hastings–McLeod solutions to the inhomogeneous Painlevé-II equation arise in multi-critical unitary random matrix ensembles, the chiral two-matrix model for rectangular matrices, non-intersecting squared Bessel paths, and non-intersecting Brownian bridges on the circle. We establish the leading-order asymptotic behavior of the generalized Hastings–McLeod functions as the inhomogeneous parameter approaches infinity using the Deift–Zhou nonlinear steepest-descent method for Riemann–Hilbert problems. This analysis is done in both the pole-free region and pole region. The asymptotic formulae show excellent agreement with numerically computed solutions in both regions. 

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2. On the Asymptotics of Orthogonal Polynomials on Multiple Intervals with Non-Analytic Weights

   https://doi.org/10.3842/SIGMA.2026.006  

   Trogdon, Thomas (January 2026, Symmetry, Integrability and Geometry: Methods and Applications)

   We consider the asymptotics of orthogonal polynomials for measures that are differentiable, but not necessarily analytic, multiplicative perturbations of Jacobi-like measures supported on disjoint intervals. We analyze the Fokas-Its-Kitaev Riemann-Hilbert problem using the Deift-Zhou method of nonlinear steepest descent and its $$\overline{\partial}$$ extension due to Miller and McLaughlin. Our results extend that of Yattselev in the case of Chebyshev-like measures with error bounds that give similar rates while allowing less regular perturbations. For the general Jacobi-like case, we present, what appears to be the first result for asymptotics when the perturbation of the measure is only assumed to be differentiable with bounded second derivative. 

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3. Nonlinear steepest descent on a torus: a case study of the Landau–Lifshitz equation

   https://doi.org/10.1088/1361-6544/adbe22  

   Desiraju, Harini; Its, Alexander; Prokhorov, Andrei (March 2025, Nonlinearity)

   We obtain rigorous large time asymptotics for the Landau–Lifshitz (LL) equation in the soliton free case by extending the nonlinear steepest descent method to genus 1 surfaces. The methods presented in this paper pave the way to a rigorous analysis of other integrable equations on the torus and enable asymptotic analysis on different regimes of the LL equation. 

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4. Numerical inverse scattering transform for the defocusing nonlinear Schrödinger equation with box-type initial conditions on a nonzero background

   https://doi.org/10.1017/jnw.2025.10026  

   Gkogkou, Aikaterini; Prinari, Barbara; Trogdon, Thomas (January 2026, Journal of Nonlinear Waves)

   Abstract We present a method to solve numerically the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation with a box-type initial condition (IC) having a nontrivial background of amplitude$$q_o \gt 0$$as$$x\to \pm \infty$$by implementing numerically the corresponding inverse scattering transform (IST). The Riemann–Hilbert problem associated with the inverse transform is solved numerically by means of appropriate contour deformations in the complex plane following the numerical implementation of the Deift–Zhou nonlinear steepest descent method. In this work, the box parameters are chosen so that there is no discrete spectrum (i.e., no solitons). The numerical method is demonstrated to be accurate within the two asymptotic regimes corresponding to two different regions of the$$(x,t)$$-plane depending on whether$$|x/(2t)| \lt q_o$$or$$|x/(2t)| \gt q_o$$, as$$t \to \infty$$. 

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5. Schubert line defects in 3d GLSMs. Part II. Partial flag manifolds and parabolic quantum polynomials

   https://doi.org/10.1007/JHEP04(2026)075  

   Closset, Cyril; Gu, Wei; Khlaif, Osama; Sharpe, Eric; Zhang, Hao; Zou, Hao (April 2026, Journal of High Energy Physics)

   A<sc>bstract</sc> We construct Schubert line defects in the 3d$$ \mathcal{N}=2 $$ $N=2$supersymmetric gauged linear sigma model (GLSM) with target space a partial flag manifoldX= Fl(k;n), generalizing our construction for complete flag manifolds given in a companion paper (part I) [1]. In the context of the 3d GLSM/quantum K-theory correspondence, the Schubert line defects are constructed as 1d$$ \mathcal{N}=2 $$ $N=2$supersymmetric gauge theories coupled to the 3d field theory, and they flow to objects supported on Schubert varietiesX_w⊆Xin the quantum K-theory. The flavored Witten index of the 1d defect is expected to compute the Chern character of [𝒪_w] — more precisely, it gives us a polynomial representative of the Schubert class in the quantum K-theory ring. We give strong evidence for this claim by showing in examples that the Witten indices of Schubert defects indeed reproduce a recently-defined set of polynomials that represent the Schubert classes in the Whitney presentation, which we call the parabolic Whitney polynomials. Moreover, upon using the quantum ring relations, we can convert these polynomials into seemingly new polynomials in the Toda presentation, which we call the parabolic quantum Grothendieck polynomials. These new polynomials specialize to known polynomials in various limits, including to the quantum Grothendieck polynomials in the case of the complete flag. In the 2d limit, our construction also realizes the Schubert classes [X_w] in the quantum cohomology ring of the partial flag manifold, and the parabolic quantum Grothendieck polynomials then reduce to previously known parabolic quantum Schubert polynomials. 

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