Related papers: Degasperis-Procesi peakons and the discrete cubic …
In this paper, we study the Cauchy problem for a generalized integrable Camassa-Holm equation with both quadratic and cubic nonlinearity. By overcoming the difficulties caused by the complicated mixed nonlinear structure, we firstly…
We develop a variational method of deriving stochastic partial differential equations whose solutions follow the flow of a stochastic vector field. As an example in one spatial dimension we numerically simulate singular solutions (peakons)…
We consider the hermitian random matrix model with external source and general polynomial potential, when the source has two distinct eigenvalues but is otherwise arbitrary. All such models studied so far have a common feature: an…
We consider a new partial differential equation, of a similar form to the Camassa-Holm shallow water wave equation, which was recently obtained by Degasperis and Procesi using the method of asymptotic integrability. We prove the exact…
Peakons (peaked solitons) are particular solutions admitted by certain nonlinear PDEs, most famously the Camassa-Holm shallow water wave equation. These solutions take the form of a train of peak-shaped waves, interacting in a particle-like…
We present a method to compute dispersive shock wave solutions of the Korteweg-de Vries equation that emerge from initial data with step-like boundary conditions at infinity. We derive two different Riemann-Hilbert problems associated with…
Rotating charged black strings are solutions of four-dimensional Einstein-Maxwell equations with a negative cosmological constant and a non-trivial topology. According to the AdS/CFT correspondence, these black strings are dual to rotating…
We consider one-dimensional inverse scattering in attenuating media where both the reflectivity and loss distributions are unknown. Mathematically, this corresponds to recovering the coefficients of a damped wave operator, or equivalently,…
The squared eigenfunctions of the spectral problem associated with the Camassa-Holm (CH) equation represent a complete basis of functions, which helps to describe the inverse scattering transform for the CH hierarchy as a generalized…
In this announcement we present a general and new approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems. Such problems arise, in particular, in evaluating the long-time behavior of nonlinear wave equations solvable…
Peakons are special weak solutions of a class of nonlinear partial differential equations modelling non-linear phenomena such as the breakdown of regularity and the onset of shocks. We show that the natural concept of weak solutions in the…
Discrete integrable systems are closely related to orthogonal polynomials and isospectral matrix transformations. In this paper, we use these relationships to propose a nonautonomous time-discretization of the Camassa-Holm (CH) peakon…
The inverse scattering approach for the defocusing Davey-Stewartson II equation is given by a system of D-bar equations. We present a numerical approach to semi-classical D-bar problems for real analytic rapidly decreasing potentials. We…
The isospectral deformations of the Frobenius-Stickelberger-Thiele (FST) polynomials introduced in [32](Spiridonov et al. Commun. Math. Phys. 272:139--165, 2007 ) are studied. For a specific choice of the deformation of the spectral…
Generalized indefinite strings provide a canonical model for self-adjoint operators with simple spectrum (other classical models are Jacobi matrices, Krein strings and 2x2 canonical systems). We prove a number of Szeg\H{o}-type theorems for…
Important illustration to the principle ``partition functions in string theory are $\tau$-functions of integrable equations'' is the fact that the (dual) partition functions of $4d$ $\mathcal{N}=2$ gauge theories solve Painlev\'e equations.…
We solve two inverse spectral problems for star graphs of Stieltjes strings with Dirichlet and Neumann boundary conditions, respectively, at a selected vertex called root. The root is either the central vertex or, in the more challenging…
We consider the direct and inverse scattering problem for a penetrable, isotropic obstacle with a second-order Robin boundary condition, which asymptotically models the delamination of the boundary of the scatterer. We develop a direct…
The paper deals with the construction of the asymptotic soliton-like and the asymptotic peakon-like solutions to the modified Camassa-Holm equation with variable coefficicents and a singular perturbation. This equation is a generalization…
The squared eigenfunctions of the spectral problem associated to the Camassa-Holm equation represent a complete basis of functions, which helps to describe the Inverse Scattering Transform for the Camassa-Holm hierarchy as a Generalised…