Abstracts

Bengt Fornberg

Finite Differences and the Trapezoidal Rule in the Complex Plane

With possible exceptions in quantum mechanics, measurable physical quantities do not involve complex numbers. However, most standard and special functions in the applied sciences are analytic functions. Focusing on exact mathematical techniques, a famous quote notes that “The shortest path between two truths in the real domain passes through the complex domain”. It therefore becomes natural to ask if this shortest path also can beneﬁt from finite difference and numerical quadrature methods that are specific to analytic functions. Several new such opportunities have recently been found. While the concepts of complex variables and of analyticalally do not readily generalize beyond 2-D, the new numerical methods can also be adapted to harmonic functions in higher dimensions.

Andre Weideman

Recurrence in the NLS: The view from the complex plane

Uniform wave train solutions of the periodic cubic nonlinear Schrodinger equation can be unstable to perturbations. The result is a periodic solution in time, where periods of growth are followed by periods of decline until the initial condition is reproduced: the so-called recurrence phenomenon. In the complex plane this can be represented as poles that appear from infinity, move towards the real axis (causing growth) and then recedes back (decline) until they return to infinity (recurrence). Numerical results of the pole dynamics will be presented.

Gregory Beylkin

Reduction of Gaussian mixtures for electronic structure calculations

We discuss a girdles approach for solving partial differential and integral equations. For our approach to work, we need: A convergence iteration to obtain the solution, A family of functions (much larger than any possible basis) closed under the necessary operations, A fast reduction algorithm to remove linear dependencies within a set of functions, i.e. for a given accuracy find the “best” linear independent subset of terms to represent all functions in the set.

In our example we look for the solutions as a multivariate Gaussian mixture. While the number of terms at intermediates steps of iteration grows rapidly, most of these terms are linearly dependent. We use a fast reduction algorithm to find “best” linear independent subset of terms to limit the number of terms needed for the solution. As an example we solve the Hartee-Fock equations for two small molecules, Helium Hydride and Lithium Hydride .

This is a joint work with Lucas Monzón and Xinshuo Yang.

Xudan Luo

Obliquely interacting solitary waves and wave wakes in free-surface flows

We investigate the weakly nonlinear isotropic bi-directional Benney-Luke (BL) equation, with a particular focus on soliton dynamics. The associated modulation equations are derived that describe the evolution of soliton amplitude and slope. By analyzing rarefaction waves and shock waves within these modulation equations, we derive the Riemann invariants and modified Rankine--Hugoniot conditions, which help characterize the Mach expansion and Mach reflection phenomena. We also derive analytical formulas for the critical angle and the Mach stem amplitude, showing that as the soliton speed is in the vicinity of unity, the results from the BL equation align closely with those of the Kadomtsev--Petviashvili (KP) equation. Furthermore, as a far-field approximation for the forced BL equation -- which models wave and flow interactions with local topography -- the modulation equations yield a slowly varying similarity solution. This solution indicates that the precursor wavefronts created by topography moving at subcritical or critical speeds take the shape of a circular arc, in contrast to the parabolic wavefronts observed in the forced KP equation.

Igor Romanov

Aspects of integrability

We revisit the question "What is integrability?" in an attempt to understand its deeper meaning beyond existing definitions. The possibility is studied to extend the notion of integrability to some important nonlinear PDEs not known to be integrable. They share with conventional integrable systems rich varieties of special physically relevant solutions. Some of their limits and/or reductions are also conventional integrable equations. At the center stage of our approach there appear linear PDEs with variable coefficients, solutions of which play the role of generating functions of Cole-Hopf-like transformations as well as eigenfunctions of Lax pairs. The important nonlinear PDEs in question are viewed as infinite collections of interacting conventional integrable systems.

Thanasis Fokas

The beauty and power of the complex plane

Three recent developments will be discussed, which involve techniques in complex analysis: the formulation of the x-periodic problem in terms of a Riemann-Hilbert problem with data explicitly defined in terms of the initial datum; the derivation and solution of integrable nonlinear PDEs in three spatial dimensions; unexpected results regarding the large t-asymptotics of the Riemann-zeta function.

Peter Clarkson

Classical solutions of the fifth Painlevé equation

In this talk, I will discuss classical solutions of the fifth Painlevé equation. The general solutions of the Painlevé equations are transcendental in the sense that they cannot be expressed in terms of known elementary functions. However, it is well known that all Painlevé equations except the first equation possess rational solutions, algebraic solutions and solutions expressed in terms of the classical special functions for special values of the parameters. These solutions of the Painlevé equations are often called “classical solutions” and frequently can be expressed in the form of determinants. In the generic case, special function solutions are expressed in terms of Kummer functions and has rational solutions expressed in terms of Laguerre polynomials. I shall discuss properties of the classical solutions, including giving non-unique rational solutions and non-unique modified Bessel function solutions, which are a special case of the Summer function solutions, associated discrete Painlevé equations derived from Bäcklund transformation, and applications.

Joint work with Clare Dunning and Ben Mitchell (University of Kent).

Adrian Nachman

A Nonlinear Plancherel theorem with applications to global well-posedness for a Davey-Stewartson Equation and to the Calderon inverse boundary value problem

We consider a well-studied scattering transform in two dimensions for which a proof of the Plancherel theorem had been a challenging open problem. The talk will explain the background and the main ideas involved in the solution of this problem, as well as in the solution of two other open problems that motivated it: global well-posedness for the defocusing DSII equation in the mass critical case, and global uniqueness for the inverse boundary value problem of Calderon for a class of unbounded conductivities.

All of this is joint work with Idan Regev and Daniel Tataru.

Javier Villarroel

Differential equation of stochastic interest rate models

We provide an overview of the basic interest rate models used in financial mathematics when the instantaneous interest rate evolves via a stochastic differential equation (SDE). The behavior at the boundaries is classified. Feynman-Kac theorem (FKT) shows that the price of zero-coupon bonds solves certain partial differential equations with boundary conditions of terminal type. These ideas can be extended to general financial derivatives that correlate with the level of interest rates, such as swaps or caps.

Jianke Yang

Rogue wave patterns and special polynomials

Rogue waves can show interesting patterns, and those patterns often arise when internal parameters in the rogue wave solutions get large. In this talk, we present some of those patterns in several physically important integrable equations such as the NLS equation, the Manakov system and the Davey-Stewartson equations. More importantly, we reveal a deep connection between those rogue patterns and certain special polynomials such as the Yablonskii–Vorob’ev polynomial hierarchy, the Adler-Moser polynomials, the Okamoto polynomial hierarchies and certain types of double-real-variable polynomials. We show how those rogue patterns can be predicted asymptotically by root structures of those special polynomials.

Yuji Kodama

KP solitons and Riemann surface

The KP solitons may be obtained by pinching (singular limits) smooth Riemann surfaces. In the talk, I will consider a reducible hyper-elliptic Riemann surface, and show that different choices of the homological bases lead to different type of KP solitons.

Paolo M. Santini

Anomalous (rogue) waves in multidimensions

In 1+1 dimensions, like in optical fibers, anomalous (rogue) waves (AWs) are well described by the celebrated integrable nonlinear Schrodinger (NLS) equation, and a large family of analytic solutions is now available, the periodic Cauchy problem has been solved, and a perturbation theory describing the O(1) effects of small perturbations on the AW dynamics has been developed. In d + 1 dimensions (d ≥ 2), like in the ocean and in the nonlinear optics of crystals, the large majority of physically relevant NLS type models are non integrable, and it is not clear yet if the NLS AWs can be really observed. In this talk we explore the existence and novel properties of AWs in integrable and non integrable multidimensional NLS type models, concentrating, in particular, on periodic AWs, on X AWs, and on instantons.

Barbara Prinari

On the discrete Kuznetsov–Ma solutions for the defocusing Ablowitz-Ladik equation with large background amplitude

The focus of this work is on special solutions of the defocusing Ablowitz-Ladik lattice on an arbitrarily large background which are discrete analogs of the Kuznetsov-Ma (KM) breathers of the focusing nonlinear Schrodinger equation. One such solution was obtained in 2019 as a byproduct of the Inverse Scattering Transform, and it was observed that the solution could be regular for certain choices of the soliton parameters, but its regularity was not analyzed in detail. This work provides a systematic investigation of the conditions on the background and on the spectral parameters that guarantee the KM solution to be non-singular on the lattice for all times. Furthermore, a novel KM-type breather solution is presented which is also regular on the lattice under the same conditions. We also employ Darboux transformations to obtain a multi-KM breather solution, and show that parameters choices exist for which a 2-KM breather is regular on the lattice. Finally, numerical results on the existence, stability and spatio-temporal dynamics of these KM breathers are presented.

Constance Schober

Linear stability of damped Stokes waves and downshifting in a viscous higher order nonlinear Schrödinger model

In this talk we examine a higher order nonlinear Schrödinger equation with linear damping and weak viscosity, recently proposed as a model for deep water waves exhibiting frequency downshifting. Through analysis and numerical simulations, we discuss how the viscosity affects the linear stability of the Stokes wave solution, enhances rogue wave formation, and leads to permanent downshift in the spectral peak. The novel results in this work include the analysis of the transition from the initial Benjamin-Feir instability to a predominantly oscillatory behavior, which takes place in a time interval when most rogue wave activity occurs. In addition, we propose new criteria for downshifting in the spectral peak and determine the relation between the time of permanent downshift and the location of the global minimum of the momentum and the magnitude of its second derivative.

Theodoros Horikis

Extended KdV and mKdV equations: Physical vs. mathematical origins

We discuss the extended/higher order Korteweg–de Vries (KdV) and modified KdV (mKdV) equations. While the higher-order KdV equation is well established, the corresponding mKdV equation is typically derived using integrability properties. In this talk, we present the extended mKdV equation, derived directly from a physical system, offering a fundamentally different form from its integrable counterpart. We explore the connections between the two equations via Miura transformations and analyze their solutions within the framework of asymptotic integrability.

Baofeng Feng

Algebraic structure and various soliton solutions to the Ablowitz-Ladik equation

In this talk, I will first show the connection between the discrete Kadomtsev–Petviashvili (KP) equation, or the Hirota-Miwa equation, with the Ablowitz-Ladik (AL) equation. Then, I will derive the dark and breather solutions to the AL equation starting from the tau functions of the KP hierarchy. Finally, I will derive the general rogue wave solutions to the AL equation by taking a limit on the breather solutions, which was given by Y. Ohta and J. Yang previously.

Ziad Musslimani

Physics in the complex plane and nonlocal integrable systems

Quasi-monochromatic complex reductions of a number of physically significant nonlinear equations are obtained. Starting from the cubic nonlinear Klein–Gordon (NLKG), the Korteweg–de Vries (KdV) and water wave equations, it is shown that the leading order asymptotic approximation can be transformed to the well-known integrable AKNS system associated with second order (in space) nonlinear wave equations. This in turn establishes, for the first time, an important physical connection between the recently discovered nonlocal integrable reductions of the AKNS system and physically interesting equations. Reductions include the parity-time, reverse space-time and reverse time nonlocal nonlinear Schrödinger equations.

Justin Cole

Floquet topological insulators and the spectral localizer

The periodic driving of lattice systems can induce localized edge states that propagate unidirectional around lattice defects. These so-called Floquet insulators are associated with topological invariants such as the Chern number. A discrete model and some interesting physical results will be reviewed. A new approach for diagnosing topology in Floquet insulators will be presented. The approach leverages a spectral localizer to probe genuinely finite and even disordered systems.

Sean Nixon

Spiral waves in dispersive wave equations

Nonlinear dispersive systems support a rich variety of fascinating solutions, such as solitons, dispersive shock waves, rarefaction waves, instantons, to name a few. A far less studied class of solutions exhibit spiral rotations and their dynamics. A class of spiral solutions in energy preserving dispersive wave systems is studied; focusing on a set of nonlinear Dirac and Lieb systems that arise as continuum limits from honeycomb and Lieb optical lattices. In the linear limit, both systems can be reduced to the 2+1 dimensional Klein-Gordon equation, which in turn reduces to the wave equation in a special case. The initial conditions that lead to spiral patterns in the Klein-Gordon equation are derived from the Dirac and Lieb systems. Using stationary phase approximations we find spiral solutions that are algebraically decaying spatially and derive algebraic equations for the structure of the spiral; the form is described by the equation of an ellipse.

Panos Kevrekidis

Teaching an old dog some new tricks: from discrete solitons and vortices, to rogue waves, flat bands and PINNs in variants of the discrete nonlinear Schrodinger model

In this talk we will revisit nonlinear dynamical lattices, starting with an overview of their physical applications in atomic, optical, mechanical and metamaterial systems. Then we will focus most notably on the prototypical discrete nonlinear Schrodinger (DNLS) model, one of the countless areas where Mark Ablowitz's insights and contributions have shaped our understanding and produced numerous (with many still ongoing) research directions. We will explore at first some of its main features, including discrete solitons, discrete vortices and related structures in 1d, 2d and 3d, not only in square, but also in other lattice patterns (hexagonal, honeycomb, etc.).

If time permits, motivated by recent experimental and mathematical developments, we will consider some interesting recent variants on the theme, such as, e.g., what happens in Kagome' lattices. This will motivate a discussion of the notion of so-called flat bands and compactly supported nonlinear states therein. We will discuss simple methods of producing lattices with flat bands, and some experimental implementations thereof in electrical circuits. Compactly supported nonlinear states will also be seen to arise from nonlinearly dispersive variants of the model recently proposed in the context of the mathematical analysis of turbulent cascades. We will also touch upon extreme events and so-called rogue waves in integrable and non-integrable variants of the model. We will end with some machine-learning inspired touches of how to "discover" such lattices from data, using Physics-Informed Neural Networks (PINNs) and how to improve PINNs using the symmetries of the lattice.

Joel Been

The fractional integrable Toda lattice

We introduce a fractional extension of the integrable Toda lattice defined on the real line with decaying initial data. By establishing a completeness relation for squared eigenfunctions of the scattering equation (a real tridiagonal eigenvalue problem), we write the nonlinear evolution equation associated with the Toda lattice in terms of underlying squared eigenfunctions and define the fractional Toda lattice equation. The methods are discrete counterparts of continuous ones recently used to find fractional integrable extensions of the Korteweg-deVries and nonlinear Schrodinger equations. Inverse scattering transform methods are used to linearize and find explicit soliton solutions to the integrable fractional Toda lattice equation. The methodology can also be used to find and solve fractional extensions of a Toda lattice hierarchy.

Christopher W. Curtis

Following information flow in multiscale systems

In this work, we quantify the timescales and information flow associated with multiscale energy transfer in a weakly turbulent system through a novel interpretation of transfer entropy. Our goal is to provide a detailed understanding of the nature of complex energy transfer in nonlinear dispersive systems driven by wave mixing. Further, we present a modal decomposition method based on the empirical wavelet transform that produces a relatively small number of nearly decorrelated, scale-separated modes. Using our method, we are able to track multiscale energy transfer using only scalar time series measurements of a weakly turbulent system. This points to our approach being of broader applicability in real-world data coming from chaotic or turbulent dynamical systems.

Mark Hoefer

Wave-mean field interaction in integrable turbulence

The kinetic theory of soliton gases for the Korteweg-de Vries (KdV) equation is used to develop a solvable model for wave-mean field interaction in integrable turbulence. The waves are soliton ensembles that interact with a critically dense soliton gas or soliton condensate, which models the mean field. A key finding is the existence of an induced mean field by the soliton gas. For the case of a "polychromatic" soliton gas, the obtained hydrodynamic equations are diagonalized in Riemann invariants and admit exact self-similar solutions predicting soliton gas filtering (trapping/transmission) by the mean field, the spontaneous formation of vacuum states, and contact shocks. The obtained soliton gas statistical moments agree with ensemble averages of numerical simulations. The developed approach for the KdV equation readily generalizes to other integrable systems, with applications in fluids, nonlinear optics and condensed matter.

Gino Biondini

Integrable turbulence and breather gas fission from semiclassical potentials in self-focusing media

I will present an analytical model of integrable turbulence in the focusing nonlinear Schrödinger (fNLS) equation generated by various semiclassical potentials, including (but not limited to) a two-parameter family of elliptic potentials. I will show that the spectrum of these potentials exhibits a thermodynamic band/gap scaling compatible with that of soliton and breather gases depending on the value of the elliptic parameter m of the potential. I will then demonstrate that, upon augmenting the potential by a small random noise (inevitably present in real physical systems), the solution of the fNLS equation evolves into a fully randomized, spatially homogeneous breather gas, a phenomenon termed breather gas fission. Moreover, I will show that: (i) the statistical properties of the breather gas at large times are determined by the spectral density of states generated by the unperturbed initial potential; (ii) the kurtosis of the breather gas can be computed as a function of the elliptic parameter m, demonstrating that it is greater than 2 for all non-zero m, implying non-Gaussian statistics, and (iii) the theoretical predictions are validated by comparison with direct numerical simulations of the fNLS equation and are confirmed by recent experiments in recirculating optical fiber loops. These results establish a link between semiclassical limits of integrable systems and the statistical characterization of their soliton and breather gases.

Rod Halburd

Singularity and Integrability: beyond the Painlevé property

For ordinary differential equations, the Painlevé property, that all solutions are single-valued about all movable singularities, is closely associated with integrability. We will study equations for which analytic continuation of any solution around curves not enclosing fixed singularities results in only a finite number of possible values at any point. This provides a more flexible approach to integrability detection that is less sensitive to changes of variables. We will also show how singularity structure can be used to identify all meromorphic solutions of certain ODEs, even when the general solution is not meromorphic and the equation is only partially integrable. A spin off of this work is the appearance of the Painlevé equations in the solution of some natural problems from complex function theory.

Sarbarish Chakravarty

The Darboux-Halphen system

An overview of the Darboux-Halphen and related differential equations will be presented. These equations admit special solutions in terms of automorphic functions and their general solutions possess a movable natural barrier which is dense set of essential singularities in the complex plane across which the solution can not be analytically continued.

James Keiser

Reminiscences: Forty years of friendship

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