Let mu be a positive measure supported on a planar domain Omega. We consider the behavior of the balayage measure v := Bal(mu, partial derivative Omega) near a point z(0) is an element of partial derivative Omega at which Omega has an outward- pointing cusp. Assuming that the order and coefficient of tangency of the cusp are d > 0 and a > 0, respectively, and that d mu (z) asymptotic to|z- z(0)|(2b-2)d(2)z as z -> z(0) for some b > 0 (here d(2)z is the Lebesgue measure on C), we obtain the leading order term of v near z(0). This leading term is universal in the sense that it only depends on d, a, and b. We also treat the case when the domain has multiple corners and cusps at the same point. Finally, we obtain an explicit expression for the balayage of the uniform measure on the tacnodal region between two osculating circles, and we give an application of this result to two-dimensional Coulomb gases.

We consider the Boussinesq equation on the line for a broad class of Schwartz initial data relevant for water waves. In a recent work, we identified ten main sectors describing the asymptotic behavior of the solution, and for each of these sectors we gave an exact expression for the leading asymptotic term in the case when no solitons are present. In this paper, we derive an asymptotic formula in Sector IV, characterized by xt ∈ (√13, 1), in the case when solitons are present. In particular, our results provide an exact expression for the soliton-radiation interaction to leading order and a verification of the soliton resolution conjecture for the Boussinesq equation in Sector IV.

We construct a non-chiral conformal field theory (CFT) on the torus that accommodates a second quantization of the elliptic Calogero-Sutherland (eCS) model. We show that the CFT operator that provides this second quantization defines, at the same time, a quantum version of a soliton equation called the non-chiral intermediate long-wave (ncILW) equation. We also show that this CFT operator is a second quantization of a generalized eCS model which can describe arbitrary numbers of four different kinds of particles; we propose that these particles can be identified with solitons of the quantum ncILW equation.

We present a numerical scheme for the solution of the initial-value problem for the “bad” Boussinesq equation. The accuracy of the scheme is tested by comparison with exact soliton solutions as well as with recently obtained asymptotic formulas for the solution.

We prove a semiclassical asymptotic formula for the two elements M and Q lying at the bottom of the recently constructed non-polynomial hyperbolic q-Askey scheme. We also prove that the corresponding exponent is a generating function of the canonical transformation between pairs of Darboux coordinates on the monodromy manifold of the Painlevé I and III3 equations, respectively. Such pairs of coordinates characterize the asymptotics of the tau function of the corresponding Painlevé equation. We conjecture that the other members of the non-polynomial hyperbolic q-Askey scheme yield generating functions associated to the other Painlevé equations in the semiclassical limit.

We study the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation under the assumption that the solution vanishes asx→∞ and approaches an oscillatory plane wave asx→−∞. We first develop an inverse scattering transform formalism for solutions satisfying such step-like boundary conditions. Using this formalism, we prove that there exists a global solution of the corresponding Cauchy problem and establish a representation for this solution in terms of the solution of a Riemann--Hilbert problem. By performing a steepest descent analysis of this Riemann--Hilbert problem, we identify three asymptotic sectors in the half-planet≥0 of thext-plane and derive asymptotic formulas for the solution in each of these sectors. Finally, by restricting the constructed solutions to the half-linex≥0, we find a class of solutions with asymptotically time-periodic boundary values previously sought for in the context of the NLS half-line problem.

In a recent paper, we presented scenarios of long-time asymptotics for the solution of the focusing nonlinear Schrödinger equation with initial data approaching plane waves of the form A1eiϕ1e−2iB1x and A2eiϕ2e−2iB2x at minus and plus infinity, respectively. In the shock case B1<B2 some scenarios include sectors of genus 3, that is, sectors ξ1<ξ<ξ2, ξ≔x/t, where the leading term of the asymptotics is expressed in terms of hyperelliptic functions attached to a Riemann surface of genus 3. The present paper deals with the asymptotic analysis in a transition zone between two genus 3 sectors. The leading term is expressed in terms of elliptic functions attached to a Riemann surface of genus 1. A central step in the derivation is the construction of a local parametrix in a neighborhood of two merging branch points.

We study Hartree-Fock theory at half-filling for the 3D anisotropic Hubbard model on a cubic lattice with hopping parameter t in the x- and y-directions and a possibly different hopping parameter t(z) in the z-direction; this model interpolates between the 2D and 3D Hubbard models corresponding to the limiting cases t(z )= 0 and t(z )= t, respectively. We first derive all-order asymptotic expansions for the density of states. Using these expansions and units such that t = 1, we analyze how the Neel temperature and the antiferromagnetic mean field depend on the coupling parameter, U, and on the hopping parameter t(z). We derive asymptotic formulas valid in the weak coupling regime, and we study in particular the transition from the three-dimensional to the two-dimensional model as t(z )-> 0. It is found that the asymptotic formulas are qualitatively different for t(z )= 0 (the two-dimensional case) and t(z )> 0 (the case of nonzero hopping in the z-direction). Our results show that certain universality features of the three-dimensional Hubbard model are lost in the limit t(z )-> 0 in which the three-dimensional model reduces to the two-dimensional model.

We consider the 4D fixed length lattice Higgs model with Wilson action for the gauge field and structure group Zn. We study Wilson line observables in the strong coupling regime and compute their asymptotic behavior with error estimates. Our analysis is based on a high-temperature representation of the lattice Higgs measure combined with Poisson approximation. We also give a short proof of the folklore result that Wilson line (and loop) expectations exhibit perimeter law decay whenever the Higgs field coupling constant is positive.

In a recent study, we showed that the large ( x , t ) \left(x,t) behavior of a class of physically relevant solutions of Boussinesq's equation for water waves is described by ten main asymptotic sectors. In the sector adjacent to the positive x x -axis, referred to as Sector I, we stated without proof an exact expression for the leading asymptotic term together with an error estimate. Here, we provide a proof of this asymptotic formula.