Abstracts

Plenary Speakers

Gino Biondini
State University of New York at Buffalo

Nonlinear Schrodinger Equation with Periodic Boundary Conditions: Inverse Scattering, Semiclassical Limits and Exactly Solvable Potentials
The behavior of solutions of integrable nonlinear PDEs with periodic boundary conditions is receiving renewed interest in recent years, due in part to the connection with the emerging subject of soliton and breather gases. In this talk I will review some recent results about the nonlinear Schrodinger equation, including: (i) The characterization of a two-parameter class of exactly solvable elliptic potentials, and (ii) The development of the inverse problem in the inverse scattering transform via a Riemann-Hilbert problem approach, and (iii) The behavior of solutions in the semiclassical limit.

Mark Hoefer
University of Colorado Boulder

When Nonlinear Waves Breathe, They Radiate
Localized waves that exhibit internal oscillations are called breathers. Since their discovery fifty years ago as special solutions of the Sine-Gordon equation, breather solutions have been obtained for other well-known integrable soliton equations and Hamiltonian lattice equations. For continuum equations, emphasis has been placed on solutions that rapidly decay to a constant, homogeneous state. In this talk, a class of breather solutions that rapidly limit to a periodic, inhomogeneous state are described, both mathematically and experimentally. Exact breather solutions of the Korteweg-de Vries (KdV) equation are presented as the nonlinear superposition of a cnoidal wave and a soliton using the spectral theory of the Lam\'e equation and a Darboux transformation. These solutions are then used to describe the transmission and trapping of a soliton and a dispersive shock wave using multiphase Whitham modulation theory.  Finally, experiments in a miscible, viscous core-annular fluid demonstrate the creation, interaction, and properties of breathers including their nonlinear dispersion relation. The experiments are shown to be in qualitative agreement with KdV breather solutions and in quantitative agreement with numerical simulations of a strongly nonlinear, long-wavelength model equation.

Vera Mikyoung Hur
University of Illinois Urbana-Champaign

Stable Undular Bores: Rigorous Analysis and Validated Numerics
I will discuss the ‘global’ nonlinear asymptotic stability of the traveling front solutions to the Korteweg-de Vries–Burgers equation, and other dispersive-dissipative perturbations of the Burgers equation. Earlier works made strong use of the monotonicity of the profile, for relatively weak dispersion effects. We exploit the modulation of the translation parameter, establishing a new stability criterion that does not require monotonicity. Instead, a certain Schroedinger operator in one dimension must have exactly one negative eigenvalue, so that a rank-one perturbation of the operator can be made positive definite. Counting the number of bound states of the Schroedinger equation, we find a sufficient condition in terms of the ’width’ of a front. We analytically verify that our stability criterion is met for an open set in the parameter regime including all monotone fronts. Our numerical experiments, revealing more stable fronts, suggest a computer-assisted proof. Joint with Blake Barker, Jared Bronski, and Zhao Yang.

Invited Speakers

Katheryn Beck
University of Kansas

Steady State Solutions of the Ericksen-Leslie Model
In this talk we consider nematic liquid crystal placed in two parallel plates. We can use the Ericksen-Leslie model with certain boundary conditions to describe this physical system. This model provides us with very rich phenomena. We will first look into steady states in the system and then study the stability of these steady states numerically. The study shows the presence of saddle-node bifurcation and further dynamics of the unstable states shows the evidence of the existence of heteroclinic orbit.

Aikaterini Gkogkou
Tulane University

Solitons and Soliton Interactions in the Complex Coupled Short-Pulse Equation
The complex couplee short-pulse equation (ccSPE) describes the proopagation of ultra-short optical pulses in nonlinear birefringent fibers. The system admits a variety of vector soliton solutions: fundamental solitons, fundamental breathers, composite breathers, and self-symmetric solitons which are special cases of composite breathers. In this talk, we discuss the nature of ccSPE soliton interactions. Using Manakov's method, we describe the interaction between two fundamental solitons, in which case there exists redistribution of energy between the components unless the initial polarization vectors are either parallel or orthogonal. To describe more complicated soliton interactions, we rely on Darboux matrices corresponding to the various types of solitons, combining refactorization problems on generators of certain rational loop groups and long-time-asymptotics of these generators. This leads to the derivation of various Yang-Baxter maps for the polarizations of the solitons, which allows for complete characterization of all types of soliton interactions in the ccSPE.

Yifeng Mao
University of Colorado Boulder

Long-time Asymptotics for Time-periodic Linear Boundary Value Problems and Applications
Initial-boundary value problems (IBVPs) with constant initial and time-dependent boundary data, also known as the wavemaker problem, are fundamental in mathematics and physics. In recent work, the IBVP for some linear and integrable nonlinear evolution equations has been solved using the unified transform method. A related approach, called the Q-equation method, has been introduced to derive the Dirichlet-to-Neumann (D-N) map for asymptotically time-periodic boundary conditions. This talk will extend and prove the existence of the unique D-N map for a general third-order wave model. In addition, two representative linear evolution equations with sinusoidal boundary conditions are studied, and asymptotic approximations are obtained for large $t$. Applications of such IBVPs in geophysical fluid dynamics will be introduced.

Jeffrey Oregero (withdrew)
University of Kansas

Recent Developments in the Spectral Theory of Soliton Gases: Periodic Gases
A soliton gas can be viewed as an infinite statistical ensemble of interacting solitons. It represents an important example of integrable turbulence and has recently attracted the attention of the nonlinear waves and integrable systems communities due to its pervasiveness in various physical systems. In this talk I will (i) discuss the spectral theory of soliton gases, (ii) introduce a family of elliptic finite-gap potentials and show how, in certain singular limits, they give rise to so-called deterministic soliton gases, and (iii) discuss some recent developments in the subject.

Wesley Perkins
Lyon College

Modulation Stability for Equations of Whitman Type
In many applications it is natural to observe “locally periodic” patterns. Such patterns appear spatially periodic on local space/time scales while their fundamental wave characteristics (such as amplitude or frequency) may slowly change, i.e., modulate, over large space/time scales. One powerful tool used to study such structures is Whitham’s theory of wave modulations, commonly referred to as Whitham theory. While Whitham theory lacks rigorous justification in general, its predictions match remarkably well with physical and numerical observations, and it has been rigorously justified for a growing number of models and equations.

In the recent work by Binswanger et al., the authors use Whitham modulation theory to analyze a generalized Whitham equation (i.e., a Whitham-type equation with generalized nonlinear flux and linear dispersion relation) and establish a modulational instability criterion. Building on their work, this talk will rigorously connect the modulational instability criterion from the work by Binswanger et al. to the spectral stability of long-wavelength perturbations of periodic traveling wave solutions to the generalized Whitham equation, thereby justifying Whitham modulation theory for the generalized Whitham equation. This provides a succinct justification of Whitham modulation theory for the various equations that can be written in the form of the generalized Whitham equation.

Barbara Prinari
State University of New York at Buffalo

On Maxwell-Bloch Systems with Inhomogeneous Broadening and One-sided Nonzero Background
We present the inverse scattering transform to solve the Maxwell-Bloch system of equations that describes two-level systems with inhomogeneous broadening, in the case of optical pulses that do not vanish at infinity in the future. The inverse problem is formulated in terms of a suitable matrix Riemann-Hilbert problem, and the formulation of the direct scattering problem combines features of the methods with decaying as well as non-decaying fields. We also discuss the asymptotic state of the medium and of the optical pulse.

Abba Ramadan
University of Alabama-Tuscaloosa

On the Stability of Solitary Waves in the NLS System of the Third-harmonic Generation
In this talk, we will consider the NLS system of the third-harmonic generation. Our interest is in solitary wave solutions and their stability properties. The recent work of Oliveira and Pastor discussed global well-posedness, finite time blow-up, as well as other aspects of the dynamics. These authors have also constructed solitary wave solutions, via the method of the Nehari manifold, in an appropriate range of parameters. Specifically, the waves exist only in spatial dimensions $n=1,2,3$. They have also established some stability/instability results for these waves. In this work, we systematically build and study solitary waves for this important model. We construct the waves via the Weinstein functional in the largest possible parameter space, and we provide a complete classification of their spectral stability. Finally, we showed instability by a blow-up, for dimensions 2 and 3, and a more restrictive set of parameters. We used virial identities methods to derive the strong instability, in the spirit of Ohta's approach. This is joint work with Atanas Stefanov.

Milena Stanislavova
University of Alabama-Birmingham

Spectral Stability for Periodic Waves in Some Hamiltonian Systems
A lot of recent work in the theory of partial differential equations has focused on the existence and stability properties of special solutions for Hamiltonian PDE’s. We review some recent works (joint with Hakkaev and Stefanov), for spatially periodic traveling waves and their stability properties. We concentrate on three examples, namely the Benney system, the Zakharov system and the full Klein-Gordon-Zakharov system. We consider several standard explicit solutions, given in terms of Jacobi elliptic functions. We provide explicit and complete description of their stability properties. Our analysis is based on the careful examination of the spectral properties of the linearized operators, combined with recent advances in the Hamiltonian instability index formalism.

Samuel Walsh
University of Missouri

Gravity Wave-borne Vortices
In this talk, we’ll present some recent work on traveling waves in water that carry vortices in their bulk. We show that for any supercritical Froude number (non-dimensionalized wave speed), there exists a continuous one-parameter family of solitary waves with a submerged point vortex in equilibrium. This family bifurcates from an irrotational laminar flow, and, at least for large Froude numbers, it extends up to the development of a surface singularity. These are the first rigorously constructed gravity wave-borne point vortices without surface tension, and notably our formulation allows the free surface to be overhanging. Through a separate numerical study, we find strong evidence that many of the waves do indeed have an overturned air—water interfaces. Finally, we prove that generically one can perform a desingularization procedure to obtain a solitary wave with a submerged hollow vortex. Physically, these can be thought of as traveling waves carrying spinning bubbles of air in their bulk. This is joint work with Ming Chen, Kristoffer Varholm, and Miles Wheeler.

Weinan Wang
University of Oklahoma

Recent Progress in Some Electro-diffusion Equations
In this talk, we discuss some recent progress in the Nernst-Planck equations describing the time evolution of multiple ionic concentrations in a two-dimensional incompressible fluid. The velocity of the fluid evolves according to either the Euler or the Navier-Stokes equations, both forced nonlinearly by the electric forces generated by the presence of charged ions. More precisely, we will talk about global well-posedness for the 2D Nernst-Planck-Euler equations and the enhanced dissipation phenomenon for the 2D stochastic Nernst-Planck-Navier-Stokes equations with transport noise.

Poster Presentations

Savana Ammons
University of Illinois Urbana-Champaign

The Two-Dimensional Standing Water Wave Problem: Is There a Simpler Solution?
In 2005, Iooss, Plotnikov, and Toland solved the two-dimensional standing water wave problem. By doing so, they proved that standing 2D waves on the surface of an infinitely deep, perfect fluid under gravity exist. The catch? Their proof was over 100 pages long and relied on the notoriously complicated Nash-Moser theorem. However, previous work by Toland and another researcher, Amick, suggests that there may be a simpler way to solve the problem. The goal of this research is to explore this alternative solution method, and to solve the standing wave problem in a more intuitive way.

Brett Ehrman
University of Kansas

Orbital Stability of Smooth Solitary Waves for the Novikov Equation
We show the existence of smooth solitary waves for the Novikov equation. We then perform analysis of the Vakhitov-Kolokolov Condition and arrive at a criterion for orbital stability. Numerical methods then strongly suggest that the stability condition holds true.

Le Gong
Vanderbilt University

Characterization of Frames for Source Recovery from Dynamical Samples
Inspired by environmental monitoring applications for identifying the locations and magnitude of pollution sources, we consider the problem of recovering constant source terms in a discrete dynamical system represented by $x_{n+1} = Ax_n + w$. Here, $x_n$ is the $n$-th state in a Hilbert space $\mathcal{H}$, $A$ is a bounded linear operator in $\mathcal{B}(\mathcal{H})$, and $w$ represents a source term within a closed subspace $W$ of $\mathcal{H}$. In this talk, we present the necessary and sufficient conditions for the reconstruction of $w$ from time-space sample measurements formed through inner products with vectors from a Bessel system $\mathcal{G} \subset \mathcal{H}$, independent of the unknown initial state $x_0$ and applicable to any $w \in W$.

Ting-Yang Hsiao
University of Illinois Urbana-Champaign

Unstable Stokes Waves in Constant Vorticity Flows
In this paper, we delve into the stability of Stokes waves with constant vorticity in two dimensions with finite depth. Using a periodic Evans function approach, we demonstrate the Benjamin-Feir instability for the low-frequency case. We then construct an index function for a Stokes wave of sufficiently small amplitude to obtain an instability relationship diagram of vorticities and wave numbers. On the other hand, we analyze the case when the frequency is high. When the spectrum is away from the origin, the instability happens due to the resonance of order two.

Xiaokai Huo
Iowa State University

Inf-SupNet for Solving High-dimensional PDEs
Solving high dimensional partial differential equations (PDEs) has historically posed a considerable challenge when utilizing conventional numerical methods, such as those involving domain meshes. Recent advancements in the field have seen the emergence of neural PDE solvers, leveraging deep networks to effectively tackle high dimensional PDE problems. This study introduces Inf-SupNet, a model-based unsupervised learning approach designed to acquire solutions for a specific category of elliptic PDEs. The fundamental concept behind Inf-SupNet involves incorporating the inf-sup formulation of the underlying PDE into the loss function. The analysis reveals that the global solution error can be bounded by the sum of three distinct errors: the numerical integration error, the duality gap of the loss function (training error), and the neural network approximation error for functions within Sobolev spaces. To validate the efficacy of the proposed method, numerical experiments conducted in high dimensions demonstrate its stability and accuracy across various boundary conditions, as well as for both semi-linear and nonlinear PDEs.

Chartese Jones
University of Missouri

Analysis and Oscillatory Patterns for a Continuous Chimeric Antigen Receptor T (CAR-T) Cells Model
In the performance of multidimensional analysis, a continuous chimeric antigen receptor T cell model shall highlight the dynamical connections as an adoptive immunotherapy practice. We investigate fold bifurcation and Neimark-Sacker bifurcation with an exhibition of stability analysis alongside numerical result. Through trusting the confines of clinical data, laboratory experiments, and mathematical modeling, we shape a virtual laboratory. This virtual laboratory resembles the biological process by utilizing various mathematical formulation of ordinary differential equations.

Chris Mayo
University of Kansas

Well-posedness of the Higher-order Nonlinear Schrödinger Equation on a Finite Interval
The higher-order nonlinear Schrödinger (HNLS) equation is a more accurate alternative to the standard NLS equation when studying wave pulses in the femtosecond regime. It arises in a variety of applications ranging from optics to water waves to plasmas to Bose-Einstein condensates. We establish the local well-posedness of the initial-boundary value problem for HNLS on a finite interval in the case of a power nonlinearity in the sense of Hadamard. More precisely, we prove existence and uniqueness of the solution as well as continuity of the data-to-solution map in the case of Sobolev initial data and boundary data in suitable Sobolev spaces determined by the regularity of the initial data and the HNLS equation. The proof relies on a combination of estimates for the linear problem and nonlinear estimates, which vary depending on the regularity of the data. The linear estimates are established by using the explicit solution formula obtained via the unified transform method of Fokas. This is a joint work with Dionyssis Mantzavinos and Turker Ozsari.

Wesley Perkins
Lyon College

Modulational Stability for Equations of Whitham Type
In many applications it is natural to observe “locally periodic” patterns.  Such patterns appear spatially periodic on local space/time scales while their fundamental wave characteristics (such as amplitude or frequency) may slowly change, i.e., modulate, over large space/time scales. One powerful tool used to study such structures is Whitham's theory of wave modulations, commonly referred to as Whitham theory. While Whitham theory lacks rigorous justification in general, its predictions match remarkably well with physical and numerical observations, and it has been rigorously justified for a growing number of models and equations. In the recent work by Binswanger et al., the authors use Whitham theory to analyze a generalized Whitham equation and establish a modulational instability criterion. Building on their work, this poster will justify Whitham modulation theory for the generalized Whitham equation, thereby providing a succinct justification of Whitham modulation theory for the various equations that can be written in the form of the generalized Whitham equation.

Hewan Shemtaga
Auburn University

Global Existence and Asymptotic Behavior of Chemotaxis Models on a Compact Metric Graph
Chemotaxis phenomena governs the directed movement of micro- organisms in response to chemical stimuli. We investigate a pair of logistic type Keller–Segel systems of reaction-advection-diffusion equations modeling chemotaxis on networks. The distinction between the two systems is driven by the rate of diffusion of chemo-attractant. We prove the global existence of classical solution subject to Neumann-Kirchhoff vertex conditions without any conditions on chemotaxis sensitivity. In addition, we show that solutions with a non-negative and non-zero initial function converge to a globally stable constant solution for relatively small chemotaxis sensitivity. However, as chemotaxis sensitivity increase, we prove the constant solution loses stability and there exist other non-constant steady states bifurcating from the constant solution. This is a joint work with my advisors Dr. Wenxian Shen (Auburn) and Dr. Selim Sukhtaiev (Auburn).

Vaibhava Srivastava
Iowa State University

Exploring Unique Dynamics in a Predator-prey Model with Generalist Predator and Group Defense in Prey
In the current manuscript we consider a predator-prey model where the predator is modeled as a generalist using a modified Leslie-Gower scheme, and the prey exhibits group defense via a generalized response. We show that the model could exhibit finite time blow-up, contrary to the current literature (Eur. Phys. J. Plus 137, 28). We also propose a new concept via which the predator population blows up in finite time while the prey population quenches in finite time. That is the time derivative of the solution to the prey equation will grow to infinitely large values in certain norms, at a finite time, while the solution itself remains bounded. The blow-up and quenching times are proved to be one and the same. Our analysis is complemented by numerical findings. This includes a numerical description of the basin of attraction for large data blow-up solutions, as well as several rich bifurcations leading to multiple limit cycles, both in co-dimension one and two. The group defense exponent $p$ is seen to significantly affect the basin of attraction. Lastly, we posit a delayed version of the model with globally existing solutions for any initial data. Both the ODE model and the spatially explicit PDE models are explored.