101 citations · 322 across the 12 of their papers we have counts for
15 papers
Infinite-order rogue waves that are small (but not small in )
Deniz Bilman, Peter D. Miller
General rogue waves of infinite order constitute a family of solutions of the focusing nonlinear Schrödinger equation that have recently been identified in a variety of asymptotic…
Efficient computation of soliton gas primitive potentials
Cade Ballew, Deniz Bilman, Thomas Trogdon
We consider the problem of computing a class of soliton gas primitive potentials for the Korteweg--de Vries equation that arise from the accumulation of solitons on an infinite int…
General rogue waves of infinite order: Exact properties, asymptotic behavior, and effective numerical computation
Deniz Bilman, Peter D. Miller
This paper is devoted to a comprehensive analysis of a family of solutions of the focusing nonlinear Schrödinger equation called general rogue waves of infinite order. These soluti…
Computation of Large-Genus Solutions of the Korteweg-de Vries Equation
Deniz Bilman, Patrik Nabelek, Thomas Trogdon
We consider the numerical computation of finite-genus solutions of the Korteweg-de Vries equation when the genus is large. Our method applies both to the initial-value problem when…
Broader Universality of Rogue Waves of Infinite Order
Deniz Bilman, Peter D. Miller
We show that the same special solution of the focusing nonlinear Schrödinger equation that has been shown to arise in a certain near-field/large-order limit from soliton and Peregr…
Extreme Superposition: High-Order Fundamental Rogue Waves in the Far-Field Regime
Deniz Bilman, Peter D. Miller
We study fundamental rogue-wave solutions of the focusing nonlinear Schrödinger equation in the limit that the order of the rogue wave is large and the independent variables $(x,t)…
Far-Field Asymptotics for Multiple-Pole Solitons in the Large-Order Limit
Deniz Bilman, Robert Buckingham, Deng-Shan Wang
The integrable focusing nonlinear Schrodinger equation admits soliton solutions whose associated spectral data consist of a single pair of conjugate poles of arbitrary order. We st…
On numerical inverse scattering for the Korteweg-de Vries equation with discontinuous step-like data
Deniz Bilman, Thomas Trogdon
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 de…
Large-order asymptotics for multiple-pole solitons of the focusing nonlinear Schrödinger equation
Deniz Bilman, Robert Buckingham
We analyze the large- behavior of soliton solutions of the integrable focusing nonlinear Schrödinger equation with associated spectral data consisting of a single pair of conjug…
Extreme Superposition: Rogue Waves of Infinite Order and the Painlevé-III Hierarchy
Deniz Bilman, Liming Ling, Peter D. Miller
We study the fundamental rogue wave solutions of the focusing nonlinear Schrödinger equation in the limit of large order. Using a recently-proposed Riemann-Hilbert representation o…
A robust inverse scattering transform for the focusing nonlinear Schrödinger equation
Deniz Bilman, Peter D. Miller
We propose a modification of the standard inverse scattering transform for the focusing nonlinear Schrödinger equation (also other equations by natural generalization) formulated w…
Benchmarking Numerical Methods for Lattice Equations with the Toda Lattice
Deniz Bilman, Thomas Trogdon
We compare performances of well-known numerical time-stepping methods that are widely used to compute solutions of the doubly-infinite Fermi-Pasta-Ulam-Tsingou (FPUT) lattice equat…
Discrete integrable systems, Darboux transformations, and Yang-Baxter maps
Deniz Bilman, Sotiris Konstantinou-Rizos
These lecture notes are devoted to the integrability of discrete systems and their relation to the theory of Yang-Baxter (YB) maps. Lax pairs play a significant role in the integra…
Numerical Inverse Scattering for the Toda Lattice
Deniz Bilman, Thomas Trogdon
We present a method to compute the inverse scattering transform (IST) for the famed Toda lattice by solving the associated Riemann--Hilbert (RH) problem numerically. Deformations f…
On the evolution of scattering data under perturbations of the Toda lattice
Deniz Bilman, Irina Nenciu
We present the results of an analytical and numerical study of the long-time behavior for certain Fermi-Pasta-Ulam (FPU) lattices viewed as perturbations of the completely integrab…