Related papers: Airy kernel determinant solutions to the KdV equat…
We study double integral representations of Christoffel-Darboux kernels associated with two examples of Hermite-type matrix orthogonal polynomials. We show that the Fredholm determinants connected with these kernels are related through the…
In a 1977 paper of McCoy, Tracy and Wu there appeared for the first time the solution of a Painlev\'e equation in terms of Fredholm determinants of integral operators. This equation is $\psi''(t)+t^{-1}\psi'(t)=(1/2) \sinh 2\psi+2\alpha…
We obtain asymptotics for the Airy kernel Fredholm determinant on two intervals. We give explicit formulas for all the terms up to and including the oscillations of order $1$, which are expressed in terms of Jacobi $\theta$-functions.
The perturbed Korteweg--de Vries equation is considered. This equation is used for the description of one--dimensional viscous gas dynamics, nonlinear waves in a liquid with gas bubbles and nonlinear acoustic waves. The integrability of…
The purpose of this article is to develop a theory behind the occurrence of "path-integral" kernels in the study of extended determinantal point processes and non-intersecting line ensembles. Our first result shows how determinants…
The Korteweg-de Vries equation is a fundamental nonlinear equation that describes solitons with constant velocity. On the contrary, here we show that this equation also presents accelerated wavepacket solutions. This behavior is achieved by…
The parabolic Airy process is the Airy$_2$ process minus a parabola, initially defined by its finite-dimensional distributions, which are given by a Fredholm determinant formula with the extended Airy kernel. This process is also the…
In this article, Lie symmetry analysis is used to investigate invariance properties of some nonlinear fractional partial differential equations with conformable fractional time and space derivatives. The analysis is applied to Korteweg-de…
The distance between the solutions to the integrable Korteweg-de Vries (KdV) equation and a broad class of non-integrable generalized KdV (gKdV) equations is estimated in appropriate Sobolev spaces. This family of equations includes, as…
We prove the existence of a Lax pair for the Calogero Korteweg-de Vries (CKdV) equation. Moreover, we modify the T operator in the the Lax pair of the CKdV equation, in the search of a (2+1)-dimensional case and thereby propose a new…
We modify the classical Paley-Wiener spaces $PW_x$ of entire functions of finite exponential type at most $x>0$, which are square integrable on the real line, via the additional condition of vanishing at finitely many complex points $z_1,…
In this paper, we give an improvement of the Strichartz estimate for Airy equation in the non-diagonal case. As an application, we prove the small data scattering and existence of a special non-scattering solutions, which are minimal in…
We present a method of determining a Lax representation for similarity reductions of autonomous and non-autonomous partial difference equations. This method may be used to obtain Lax representations that are general enough to provide the…
This paper addresses the problem of finding an asymptotic solution for first and second order integro-differential equations containing an arbitrary kernel, by evaluating the corresponding inverse Laplace and Fourier transforms. The aim of…
We derive Cordes-Nirenberg type results for nonlocal elliptic integro-differential equations with deforming kernels comparable to sections of a convex solution of a Monge-Amp\`ere equation. Under a natural integrability assumption on the…
We introduce a collection of nonlinear integrable partial differential-difference equations that are satisfied by the one-point distribution functions of some classical integrable KPZ models. Moreover, these equations can be regarded as…
We consider the linearized Korteweg-de-Vries equa- tions, sometimes called Airy equation, on general metric graphs with edge lengths bounded away from zero. We show that pro- perties of the induced dynamics can be obtained by studying…
A Fredholm integral equation of the second kind with the generalized Neumann kernel associated with the Riemann-Hilbert problem on unbounded multiply connected regions will be derived and studied in this paper. The derived integral equation…
We study numerically the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+\epsilon^{2}u_{xxx}=0$ for $\epsilon\ll1$ and give a quantitative comparison of the numerical solution with various asymptotic formulae for…
This manuscript develops a novel understanding of non-polar solutions of the discrete Painlev\'e I equation (dP1). As the non-autonomous counterpart of an analytically completely integrable difference equation, this system is endowed with a…