KUMUNU 2018 PDE, Dynamical Systems and Applications

Plenary Speakers

Peter Bates, Michigan State University
Multiphase Solutions to the Vector Allen-Cahn Equation: Crystalline and Other Complex Symmetric Structures

I will present a systematic study of entire symmetric solutions u : Rn --> Rm to the vector Allen-Cahn equation where the nonlinear potential function, W(u), is smooth, invariant under a reflection group in Rn, nonnegative and with a finite number of zeros. After introducing a general notion of equivariance with respect to a homomorphism between reflection groups in Rn and Rm, two abstract results are proved concerning the cases of the reflection group being finite or discrete. These give the existence of equivariant solutions for each case. The approach is variational and based on a mapping property of the parabolic vector Allen-Cahn equation and on a pointwise estimate for vector minimizers. This unifies and generalizes results of many authors over the last decade. Joint work with G.Fusco and P. Syrnellis.

Mariana Haragus, University of Bourgogne Franche-Comté, France Bifurcation Theories for a Model from Nonlinear Optics

We consider the Lugiato-Lefever equation, which is a nonlinear Schrödinger-type equation with damping, detuning and driving, derived in nonlinear optics by Lugiato and Lefever (1987). While intensively studied in the physics literature, there are relatively few rigorous mathematical studies of this equation. Of particular interest for the physical problem is the dynamical behavior of periodic and localized steady waves. The underlying mathematical questions concern the existence and the stability of these types of waves. In this talk, I'll show how different tools from bifurcation theory can be used in the mathematical analysis of these questions. The focus will be on periodic waves and their stability.

Kevin Zumbrun, Indiana University
A Stable Manifold Theorem for a Class of Degenerate Evolution Equations

We establish a Stable Manifold Theorem, with consequent exponential decay to equilibrium, for a class of degenerate evolution equations Au'+u=D(u,u) with A bounded, self-adjoint, and one-to-one, but not invertible, and D a bounded, symmetric bilinear map. This is related to a number of other scenarios investigated recently for which the associated linearized ODE Au'+u=0 is ill-posed with respect to the Cauchy problem. The particular case studied here pertains to the steady Boltzmann equation, yielding exponential decay of (tails of) large-amplitude shock and boundary layers.

Invited Speakers

Benjamin Akers, Air Force Institute of Technology
Modulational Instabilities of Traveling Waves

A perturbative framework for predicting modulational instabilities of traveling waves will be presented. Asymptotics of the spectrum of deep water waves on one- and two- dimensional interfaces are calculated. The role of the instabilities in spectrum's analyticity will be discussed. The potential for asymptotic aided numerical computations will be explored.

John Albert, University of Oklahoma
Concentration Compactness and the Stability of KdV Multisolitons

Multisoliton solutions of the Korteweg-de Vries equation are critical points for certain constrained variational problems in which the functionals involved are conserved quantities for the equation. One way to understand their stability properties is to show that they are actually minimizers, in a strong sense: all minimizing sequences converge strongly to the set of minimizing multisoliton profiles. This can be done using a variant of the concentration compactness method known as the profile decomposition or bubble decomposition. We discuss the analysis involved in the case of 2-soliton solutions.

Andrew Comech, Texas A&M University
Spectral Stability of Small Amplitude Solitary Waves in the Nonlinear Dirac Equation

We study the point spectrum of the linearization at a solitary wave solution to the nonlinear Dirac equation with the scalar type self-interaction (known as the Soler model). We focus on the spectral stability, that is, the absence of eigenvalues with positive real part, in the non-relativistic limit (small amplitude solitary waves). We prove the spectral stability for particular nonlinearities (in particular, for pure power nonlinearities between cubic and quintic in one spatial dimension) and also for the "charge-critical" cases. An important part of the stability analysis is the proof of the absence of bifurcations of nonzero-real-part eigenvalues from the embedded threshold points. Our approach is based on constructing a new family of exact bi-frequency solitary wave solutions in the Soler model, on using this family to determine the multiplicity of exact ±2𝜔𝑖 eigenvalues of the linearized operator, and on the analysis of the behavior of “nonlinear eigenvalues” (characteristic roots of holomorphic operator- valued functions). This is a joint work with Nabile Boussaid (Besançon).

Anna Ghazaryan, Miami University 
On the Stability of Planar Fronts

We consider planar fronts in a class of reaction-diffusion systems with the following property: the linearization of the system about the front has no unstable discrete eigenvalues, but its essential spectrum touches the imaginary axis. For perturbations that belong to the intersection of the exponentially weighted space with the original space without a weight, we use a bootstrapping argument to show that initially small perturbations to the front remain bounded in the original norm and decay algebraically in time in the exponentially weighted norm.

Pelin Guven Geredeli, University of Nebraska
Long Time Behavior Properties of Solutions to Evolutionary PDEs

In this talk, we shall discuss our recent research on various classes of linear and nonlinear partial differential equations (PDE's). There will be a focus here on results that deal with long-time behavior of PDE solutions. We will highlight our various approaches to a given problem, including various PDE techniques that are subsequent to our continuous semigroup formulations. In discussing our results, we will feature our work on: (i) our derived global attractor theory for nonlinear parabolic PDE processes, and hyperbolic plate and beam PDE under nonlinear boundary dissipation, (ii) stability properties of various coupled systems of linear PDE of different type- i.e., hyperbolic vs. parabolic.

Mark Hoefer, University of Colorado
Whitham Theory Applied to Modulated Solitary Waves

Whitham modulation theory is a well-known asymptotic technique to describe the slow modulations of nonlinear, periodic, traveling wave solutions of nonlinear dispersive partial differential equations. The resultant first order, quasi-linear Whitham modulation equations can be obtained by averaging conservation laws for the governing partial differential equation over a period of the periodic wave family. This approach has been successfully utilized for the problem of stability of periodic waves and the propagation of dispersive shock waves with many applications. This talk will focus upon the little- studied, singular, solitary wave (zero wavenumber) limit of the Whitham modulation equations. These equations describe a slowly varying mean flow that is decoupled from the solitary wave modulation. Interestingly, the solitary wave modulation is described by an amplitude field that is spatially defined everywhere; solitary wave trajectories are characteristics of the modulation system. Applications include modulated solitary waves and the propagation of solitary waves through other nonlinear waves such as dispersive shock waves and rarefaction waves. Multiple governing partial differential equations and physical examples will be highlighted.

Peter Howard, Texas A&M University
The Maslov Index for Linear Hamiltonian Systems on [0,1] and Applications to Periodic Waves

Working with general linear Hamiltonian systems on [0,1], and with a wide range of self-adjoint boundary conditions, including both separated and coupled, I will discuss a general framework for relating the Maslov index to spectral counts. As an example of the general framework, I will analyze the spectrum of linear operators obtained when Allen-Cahn equations and systems are linearized about stationary periodic solutions.

Stéphane Lafortune, College of Charleston
Stability of Traveling Waves in a Model for a Thin Liquid Film Flow

We consider a model for the flow of a thin liquid film down an inclined plane in the presence of a surfactant. The model is known to possess various families of traveling wave solutions. We use a combination of analytical and numerical methods to study the stability of the traveling waves. We show that for at least some of these waves the spectra of the linearization of the system about them are within the closed left-half complex plane.

Hung Le, University of Missouri
Elliptic Equations with Transmission and Wentzell Boundary Conditions and an Application to Steady Water Waves in the Presence of Wind

In this talk, we present results about the existence and uniqueness of solutions of elliptic equations with transmission and Wentzell boundary conditions. We provide Schauder estimates and existence results in Holder spaces. As an application, we develop an existence theory for small-amplitude two-dimensional traveling waves in an air-water system with surface tension. The water region is assumed to be irrotational and of finite depth, and we permit a general distribution of vorticity in the atmosphere.

Greg Lyng, University of Wyoming
Coordinates for Multidimensional Evans-Function Computations

The Evans function has become a standard tool in the mathematical study of the stability of nonlinear waves. In particular, computation of its zero set gives a convenient numerical method for determining the point spectrum of the associated linear operator (and thus the spectral stability of the wave in question). In this talk, I will describe the central (and perhaps unexpected) role that coordinate choices play in making Evans- function computations for multidimensional viscous shock waves viable. This is joint work with Blake Barker, Jeff Humpherys, and Kevin Zumbrun.

Yulia Meshkova, St. Petersburg State University
Operator Error Estimates for Homogenization of Periodic Hyperbolic Systems

The talk is devoted to homogenization for solutions of periodic hyperbolic systems with rapidly oscillating coefficients. We wish to approximate solutions in the L2- and H1- norms. To obtain estimate in the energy norm we assume that the initial data for the solution is zero. But the initial data for the time derivative of the solution is non-zero. So, the solution can be represented as the corresponding operator sine acting on the non-zero initial data. We obtain principal term of approximation for the solution and approximation in the Sobolev class H1 with the correction term taken into account. The results can be written as approximations of the operator sine in the uniform operator topology with the precise order error estimates. We use the spectral approach to homogenization problems developed by M. Sh. Birman and T. A. Suslina. The method is based on the scaling transformation, the Floquet-Bloch theory and analytic perturbation theory. It turns out that homogenization is a spectral threshold effect at the bottom of the spectrum. More details: arXiv:1705.02531. The research was supported by project of Russian Science Foundation no. 17-11-01069.

Peter Miller, University of Michigan
Extreme Superposition: Rogue Waves of Infinite Order

Using a recently-obtained analytical representation of the rogue-wave solutions of the focusing nonlinear Schroedinger equation on a nonzero background, we study the asymptotic behavior of rogue waves of increasingly high order. We identify a new solution of the focusing nonlinear Schroedinger equation that we call the rogue wave of infinite order, and show that it approximates high-order rogue waves near the amplitude peak, under suitable rescaling. The rogue wave of infinite order also satisfies ordinary differential equations in space and time related to the Painleve-III hierarchy, and has nontrivial asymptotics with high oscillation and algebraic decay for large x and t. This is joint work with Deniz Bilman and Liming Ling.

Yannan Shen, California State University, Northridge Nonlinear Waves in Lattices

In this talk we will give some examples of nonlinear waves in lattice dynamical systems, including nonlinear optics, nonlinear metamaterials and granular crystals. We will show we derived from an intrinsically discrete physical system, through a long- wavelength oscillations to a continuous model that can support solitary waves. We will discuss possible connections between continuum and anti-continuum limit and possible future directions of research.