arXiv subjects

Kenneth D. T-R McLaughlin

Publications and source records attributed to Kenneth D. T-R McLaughlin.

5 recordsLinked to original sources

arXiv2026

Painlev\'e Universality classes for the maximal amplitude solution of the Focusing Nonlinear Schr\"{o}dinger Equation with randomness

We establish universality for extremal solutions of the focusing nonlinear Schr\"{o}dinger equation. Extremal solutions are $N$-soliton solutions that achieve the theoretical maximal amplitude and diverge as $N \to \infty$. We consider extremal solutions with the discrete eigenvalues randomly drawn from sub-exponential distributions, and identify two distinct universality classes, determined by the macroscopic structure of the spectrum: the Painlev\'e--III rogue-wave solution, where the eigenvalues take the form $\lambda_j = v_j + i \mu_j$, and the Painlev\'e--V rogue wave solution, where $\lambda_j = -\zeta \, j + v_j + i \mu_j$, with $0 < \zeta < 1$. (In both cases, $\mu_{j}$ and $v_{j}$ are subexponential random variables.) Universality can then be summarized as follows: independently of the specific distribution of the eigenvalues, the rescaled solutions converge locally to a deterministic profile governed by the Painlev\'e-III equation in the first regime, and the Painlev\'e-V equation in the second. These results demonstrate that the formation of Painlev\'e-type rogue waves is a universal phenomenon robust to randomness.

math.AP↗

arXiv2025

Asymptotics and zeros of a special family of Jacobi polynomials

In this paper we study a family of non-classical Jacobi polynomials with varying parameters of the form $\alpha_n=n+1/2$ and $\beta_n=-n-1/2$. We obtain global asymptotics for these polynomials, and use this to establish results on the location their zeros. The analysis is based on the Riemann Hilbert formulation of Jacobi polynomials derived from the non-hermitian orthogonality introduced by Kuijlaars, et al. This family of polynomials arise in the symbolic evaluation integrals in the work of Boros and Moll and corresponds to a limitting case, which is not considered in the works of Kuijlaars, et al. A remarkable feature in the analyisis is encountered when performing the local analysis of the RHP near the origin, where the local parametrix introduces a pole.

math.CA↗

arXiv2025

The formation of a soliton gas condensate for the focusing Nonlinear Schr\"odinger equation

In this work, we carry out a rigorous analysis of a multi-soliton solution of the focusing nonlinear Schr\"{o}dinger equation as the number, $N$, of solitons grows to infinity. We discover configurations of $N$-soliton solutions which exhibit the formation (as $N \to \infty$) of a soliton gas condensate. Specifically, we show that when the eigenvalues of the Zakharov - Shabat operator for the NLS equation accumulate on two bounded horizontal segments in the complex plane with norming constants bounded away from $0$, then, asymptotically, the solution is described by a rapidly oscillatory elliptic-wave with constant velocity, on compact subsets of $(x,t)$. We then consider more complex solutions with an extra soliton component, and provide rigorous justification of the predictions of the kinetic theory of solitons in this deterministic setting. This is to be distinguished from previous analyses of soliton gasses where the norming constants were tending to zero with $N$, and the asymptotic description only included elliptic waves in the long-time asymptotics.

math.AP↗

arXiv2015

Asymptotics for the partition function in two-cut random matrix models

We obtain large N asymptotics for the Hermitian random matrix partition function \[Z_N(V)=\int_{\mathbb R^N}\prod_{i<j}(x_i-x_j)^2 \prod_{j=1}^N e^{-N V(x_j)}dx_j,\] in the case where the external potential $V$ is a polynomials such that the random matrix eigenvalues accumulate on two disjoint intervals (the two-cut case). We compute leading and sub-leading terms in the asymptotic expansion for $\log Z_N(V)$, up to terms that are small as $N$ goes to infinity. Our approach is based on the explicit computation of the first terms in the asymptotic expansion for a quartic symmetric potential $V$. Afterwards, we use deformation theory of the partition function and of the associated equilibrium measure to generalize our results to general two-cut potentials $V$. The asymptotic expansion of $\log Z_N(V)$ as $N$ goes to infinity contains terms that depend analytically on the potential $V$ and that have already appeared in the literature. In addition our method allows to compute the $V$-independent terms of the asymptotic expansion of $\log Z_N(V)$ which, to the best of our knowledge, had not appeared before in the literature. We use rigorous orthogonal polynomial and Riemann-Hilbert techniques which had so far been successful to compute asymptotics for the partition function only in the one-cut case.

math-ph↗

arXiv2013

Strong asymptotics of the orthogonal polynomials with respect to a measure supported on the plane

We consider the orthogonal polynomials $\{P_{n}(z)\}$ with respect to the measure $|z-a|^{2N c} {\rm e}^{-N |z|^2} \,{\rm d} A(z)$ over the whole complex plane. We obtain the strong asymptotic of the orthogonal polynomials in the complex plane and the location of their zeros in a scaling limit where $n$ grows to infinity with $N$. The asymptotics are described in terms of three (probability) measures associated with the problem. The first measure is the limit of the counting measure of zeros of the polynomials, which is captured by the $g$-function much in the spirit of ordinary orthogonal polynomials on the real line. The second measure is the equilibrium measure that minimizes a certain logarithmic potential energy, supported on a region $K$ of the complex plane. The third measure is the harmonic measure of $K^c$ with a pole at $\infty$. This appears as the limit of the probability measure given (up to the normalization constant) by the squared modulus of the $n$-th orthogonal polynomial times the orthogonality measure, i.e. $|P_n(z)|^2 |z-a|^{2N c} {\rm e}^{-N |z|^2} \,{\rm d} A(z)$. The compact region $K$ that is the support of the second measure undergoes a topological transition under the variation of the parameter $t=n/N$; in a double scaling limit near the critical point given by $t_c=a(a+2\sqrt c)$ we observe the Hastings-McLeod solution to Painlevé\ II in the asymptotics of the orthogonal polynomials.

math-ph↗