Abstract : In the last decade, there have been some new developments in the study of rogue-waves of nonlinear integrable evolutionary equations, such as their long-time asymptotics, their stability, their universal patterns, and their onset mechanisms. This minisymposium aims to bring together a group of world-leading researchers to discuss the theoretical, computational, and experimental aspects of this type of extreme wave phenomena.
[05527] Universal rogue wave patterns and their connections with special polynomials
Format : Online Talk on Zoom
Author(s) :
Jianke Yang (University of Vermont)
Abstract : Rogue wave patterns in integrable systems are investigated. We show that universal rogue patterns of various types appear in integrable systems when one of the internal parameters in bilinear expressions of rogue waves gets large, and these universal rogue patterns can be predicted asymptotically by root structures of certain special polynomials, such as the Yablonskii–Vorob’ev polynomial hierarchy and the Okamoto polynomial hierarchies. This is joint work with Dr. Bo Yang of Ningbo University.
[05532] Determinant formula for Rogue waves and the binomial theorem
Format : Talk at Waseda University
Author(s) :
Yasuhiro Ohta (Kobe University)
Abstract : The rogue wave solutions for integrable systems are often given by the determinant formula for tau functions explicitly. The determinant expression is related with the binomial theorem of rational type. We report an observation about the bilinear structure for the tau functions derived through the binomial theorem.
[05549] Two-dimensional rogue waves generated by resonance collision
Format : Online Talk on Zoom
Author(s) :
Jingsong He (Shenzhen University)
Abstract : It is one of important topics to construct rogue waves in two-dimensional integrable systems. In recent years, we have obtained two kinds of rogue wave in few two-dimensional integrable systems by Hirota method. The generating mechanism of them is the resonant collision between different nonlinear waves. In this talk, two kinds of rogue wave of the Davey–Stewartson I equation are discussed with details by analytical and graphic ways. The main
results have been published in two papers: Journal of Nonlinear Science 31(2021) 67 and Letters in Mathematical Physics112(2022)75, co-authored with Jiguang Rao, Athanassios S. Fokas, Yi Cheng.
[04902] Rogue waves of infinite order and their properties, Part 1
Author(s) :
Deniz Bilman (University of Cincinnati)
Abstract : In a study of high-order fundamental rogue wave solutions of the focusing nonlinear Schrödinger equation, a new limiting object termed the rogue wave of infinite order was found in a high-order near-field limit. Subsequently it has been shown that this same limiting object, itself a solution of the focusing nonlinear Schrödinger equation in rescaled variables that also solves differential equations in the Painlevé-III hierarchy, also arises in numerous other settings such as high-order solitons, semiclassical asymptotics, iterated Bäcklund transformations of arbitrary backgrounds, and even other related nonlinear systems. This talk will describe this story and introduce the rogue wave of infinite order and some generalizations of it. This is joint work with Peter D. Miller.
[04895] Rogue waves of infinite order and their properties, Part 2
Author(s) :
Peter David Miller (University of Michigan)
Abstract : The general rogue wave of infinite order is a family of exact solutions of the focusing nonlinear Schr\"odinger equation that also solve ordinary differential equations related to Painlev\'e-III and while being highly-transcendental, nonetheless arise in several natural limits. From their Riemann-Hilbert representation we deduce some elementary properties, detailed asymptotics for large values of the independent variables, and a double-scaling limit. This is joint work with Deniz Bilman.
[05529] Universality and rogue waves in semi-classical sine-Gordon equation
Author(s) :
Bingying Lu (SISSA)
Peter David Miller (University of Michigan)
Abstract : We study the semiclassical sine-Gordon equation with below threshold pure impulse initial data of Klaus-Shaw type. The system exhibits both phase transition and a gradient catastrophe in finite time. Near the gradient catastrophe point, the asymptotics are universally described by the Painlevé I tritronquée solution away from the poles and the rogue wave solutions of sG near the poles; away from the gradient catastrophe, the phase transition exhibits another type of universality.
[05518] Large order breathers of the nonlinear Schodinger equation
Format : Talk at Waseda University
Author(s) :
Xiaoen Zhang (Shandong University of Science and Technology)
Abstract : Multi-soliton and high-order soliton solutions are famous in the integrable focusing nonlinear Schrodinger equation. The dynamics of multi-solitons have been well known to us since the 70s of the last century by the determinant analysis. However, there is little progress in the study of high-order solitons. In this work, we would like to analyze the large-order asymptotics for the high-order breathers, which are special cases of double high-order solitons with the same velocity. To analyze the large order dynamics, we first convert the representation of Darboux transformation into a framework of the Riemann-Hilbert problem. Then we show that there exist five distinct asymptotic regions by the Deift-Zhou nonlinear steepest descent method. More importantly, we first find a novel genus-three asymptotic region, which uncovers that the maximal genus is connected with the number of spectral parameters. All results of the asymptotic analysis are verified by the numerical method.
Derchyi Wu (Institute of Mathematics, Academia Sinica,)
Abstract : Applying the inverse scattering theory, we present an orbital stability theorem of KdV $n$-solitons with explicit phase shifts.
[05530] Rogue waves in the massive Thirring model
Format : Online Talk on Zoom
Author(s) :
Junchao Chen (Lishui University)
Abstract : In this talk, I will talk about general rogue wave solutions in the massive Thirring (MT) model. These rational solutions are derived by using the KP hierarchy reduction method and presented explicitly in terms of determinants whose matrix elements are elementary Schur polynomials. In the reduction process, three reduction conditions including one index- and two dimension-ones are proved to be consistent by only one constraint relation on parameters of tau-functions of the KP-Toda hierarchy. It is found that the rogue wave solutions in the MT model depend on two background parameters, which influence their orientation and duration. Differing from many other coupled integrable systems, the MT model only admits the rogue waves of bright-type, and the higher-order rogue waves represent the superposition of fundamental ones in which the non-reducible parameters determine the arrangement patterns of fundamental rogue waves. Particularly, the super rogue wave at each order can be achieved simply by setting all internal parameters to be zero, resulting in the amplitude of the sole huge peak of order N being 2N + 1 times the background. Finally, rogue wave patterns are discussed when one of the internal parameters is large. Similar to other integrable equations, the patterns are shown to be associated with the root structures of the Yablonskii-Vorob’ev polynomial hierarchy through a linear transformation. This work is joint with Bo Yang and Bao-Feng Feng.
[03606] Resonant breather and rogue wave solutions to a coupled Sasa-Satsuma equation
Format : Talk at Waseda University
Author(s) :
Baofeng Feng (University of Texas Rio Grande Valley )
Chengfa Wu (Shenzhen University)
Abstract : We firstly derive a set of 7 bilinear equations for a coupled Sasa-Sastsuma (CSSI) equation under nonzero boundary conditions and show that they can be reduced from the discrete and continuous KP-Toda hierarchy through a series of reductions such as the CKP-, dimension- and complex conjugate reductions. Then, we derive breather and rogue wave solutions in determinant form. In the last, we show the dynamical behavior of these solutions especially resonant breather solutions.
[03607] Resonant breather and rogue wave solutions to a coupled Sasa-Satsuma equation
Author(s) :
Baofeng Feng (University of Texas Rio Grande Valley )
Chengfa Wu (Shenzhen University)
Abstract : We firstly derive a set of 7 bilinear equations for a coupled Sasa-Sastsuma (CSSI) equation under nonzero boundary conditions and and show they can be reduced from the discrete and continuous KP-Toda hierarchy through a series of reductions such as CKP-, dimension- and complex conjugate reductions. Then, we derive breather and rogue wave solutions in determinant form. In the last, we show the dynamical behavior of these solutions especially resonant breather solutions.
