Schedule for: 17w5044 - Geometrical Methods, non Self-Adjoint Spectral Problems, and Stability of Periodic Structures

In the 1960s, Benjamin and Feir, and Whitham, discovered that a Stokes wave would be unstable to long wavelength perturbations, provided that the product of the carrier wave number and the undisturbed water depth exceeds $\approx 1.363$. In the 1990s, Bridges and Mielke studied the corresponding spectral instability in a rigorous manner. But it leaves some important issues open, such as the spectrum away from the origin. The governing equations of the water wave problem are complicated. One may resort to simple approximate models to gain insights. I will begin by Whitham's shallow water equation and the modulational instability index for small amplitude and periodic traveling waves, the effects of surface tension and constant vorticity. I will then discuss higher order corrections, extension to bidirectional propagation and two-dimensional surfaces. This is partly based on joint works with Mat Johnson (Kansas) and Ashish Pandey (Illinois).

In this talk, we show some results on the existence and orbital stability of the peak-standing-wave solutions for the cubic-quintic nonlinear Schr\"odinger equation with a point interaction. Via a perturbation method and continuation argument, we obtain stability results in the case of attractive-attractive and attractive-repulsive nonlinearities. In the case of an attractive-attractive case and an focusing interaction we give an complete approach for stability based in the extension theory of symmetric operators.

Spectral stability captures behavior of a solution perturbed by an infinitesimal perturbation. It often determines nonlinear stability but it is limited to the exact form of the dynamics of the system. However, governing equations are often only an approximation of a larger system that models real world situation. We show how are the spectral stability of a solution in the reduced and full (extended) system related, particularly for ODEs in the case of frequently used quasi-steady-state reduction but also in a general case of reduced/extended system. A connection is also drawn with the geometric Krein signature that is shown to naturally characterize spectral properties under such extensions.

A (physically significant) modification of the parabolic Allen-Cahn equation, obtained by substituting the Fick's diffusion law with a relaxation relation of Cattaneo-Maxwell type, is considered. The investigation concentrates on existence and stability of traveling fronts connecting the two stable states of the model, and specifically the nonlinear stability as a consequence of detailed spectral and linearized analyses. The outcome of numerical studies are also presented for determining the propagation speed, in comparison with the parabolic case, and for exploring the dynamics of large perturbations of the front. These results ensue from a collaboration with Corrado Lattanzio (University of L'Aquila), Corrado Mascia (Sapienza University of Roma) and Ramon G. Plaza (National Autonomous University of Mexico).

In continuum models bilayers are homoclinic structures that are generically unstable within second-order systems as the associated translational mode has a single zero. Within the single-component, functionalized Cahn-Hillard (FCH) free energy the unstable mode balances against surface diffusion to generate pearling modes: spatially periodic high-frequency lateral variations in the bilayer width that can be weakly stable or weakly unstable. Almost all biologically relevant lipid bilayers are composed of multiple types of lipids. We present a two-component FCH system constructed from a Geirer-Meinhardt (GM) model that possesses one-parameter families of bilayers with adjustable composition. Tuning the composition induces a real-to-complex eigenvalue bifurcation in the underlying GM system yields robust pearling inhibition (stability) in the full system.

We consider the nonlinear Schrödinger equation in $n$ space dimensions $$iu_t + \Delta u + |u|^{p-1}u = 0, \;x \in \mathbb{R}^n,\; t > 0$$ and study the existence and stability of standing wave solutions of the form $$\begin{cases} e^{iwt}e^{i \sum_{j=1}^k m_j \theta_j}\phi_w(r_1, r_2, \dots, r_k),& n = 2k\\ e^{iwt}e^{i \sum^k_{j=1} m_j \theta_j}\phi_w(r_1, r_2, \dots, r_k, z),& n = 2k + 1 \end{cases}$$ for $n = 2k$, $(r_j,\theta_j)$ are polar coordinates in $\mathbb{R}^2$, $j = 1, 2,\dots, k$; for $n = 2k + 1$, $(r_j,\theta_j)$ are polar coordinates in $\mathbb{R}^2$, $(r_k,\theta_k,z)$ are cylindrical coordinates in $\mathbb{R}^3$, $j = 1, 2,\dots, k-1$. We show the existence of such solutions as minimizers of a constrained functional and conclude from there that such standing waves are stable if $1 < p < 1 + 4/n$.

Periodic traveling waves are computed on parameterized interfaces, which are not functions of the horizontal coordinate(s). These overturned traveling waves are computed on one and two-dimensional interfaces, on a classic interface between two fluids as well as on boundary formed by a hydroelastic ice sheet. Numerical continuation procedures are coupled with local and global bifurcation theorems. Extreme wave types and bifurcation surfaces are presented. The prospects for stability of overturned traveling waves are discussed.

The Vortex Filament Equation (VFE) is part of an integrable hierarchy of filament equations. Several equations in this hierarchy have been derived to describe vortex filaments in various situations. Inspired by these results, we develop a general framework for studying the existence and the linear stability of closed solutions of the VFE hierarchy. The framework is based on the correspondence between the VFE and the nonlinear Schr\"odinger (NLS) hierarchies. Our results establish a connection between the AKNS Floquet spectrum and the stability properties of the solutions of the filament equations. We apply our machinery to solutions of the filament equation associated to the Hirota equation. We also discuss how our framework applies to soliton solutions.

The spectral instabilities of the stationary periodic solutions of the focusing NLS equation were completely characterized recently. The crux of this characterization was the analysis of the non-self adjoint Lax pair for the focusing NLS equation. Although all solutions are unstable in the class of bounded perturbations, different solutions were found to be spectrally stable with respect to certain classes of periodic perturbations, with period an integer multiple of the solution period. We prove that all solutions that are spectrally stable are also (nonlinearly) orbitally stable, using different Krein signature calculations. Similar, more recent results for the sine-Gordon equation will be shown as well.

In the stability and blowup for traveling or standing waves in nonlinear Hamiltonian dispersive equations, the non-degeneracy of the linearization about such a wave is of paramount importance. That is, one must verify the kernel of the second variation of the Hamiltonian is generated by the continuous symmetries of the PDE. The proof of this property can be far from trivial, especially in cases where the dispersion admits a nonlocal description where shooting arguments, Sturm-Liouville theories, and other ODE methods may not be applicable. In this talk, we discuss the non degeneracy and nonlinear orbital stability of antiperiodic traveling wave solutions to a class of defocusing NLS equations with fractional dispersion. Key to our analysis is the development of a ground state theory and oscillation theory for linear periodic, fractional Schrodinger operators with antiperiodic boundary conditions. This is joint work with Kyle Claassen (KU).

We consider the problem of finding the spectrum of an operator taking the form of a low-rank (rank one or two) non-normal perturbation of a self-adjoint operator. We use a simple idea of classical differential geometry (the envelope of a family of curves) to analyze the spectrum. When the rank of the perturbation is two, this allows us to view the system in a geometric way through a ``phase plan'' in the perturbation strengths. We show how to apply this technique to two problems: a neural network model of the oculomotor integrator (Anastasio and Gad 2007), and a nonlocal model of phase separation (Rubinstein and Sternberg 1992). This is work with Tom Anastasio and Jared Bronski (UIUC).

This talk examines a class of linear hyperbolic systems which generalizes the Goldstein-Kac model to an arbitrary finite number of speeds with transition rates. Under the basic assumptions that the transition matrix is symmetric and irreducible, and the speed differences generate all the space, the system exhibits a large-time behavior described by a parabolic advection-diffusion equation. The main contribution is to determine explicit formulas for the asymptotic drift speed and diffusion matrix in term of the kinetic parameters, establishing a complete connection between microscopic and macroscopic coefficients. It is shown that the drift speed is the arithmetic mean of the velocities. The diffusion matrix has a more complicated representation, based on the graph with vertices the velocities and arcs weighted by the transition rates. The approach is based on an exhaustive analysis of the dispersion relation and on the application of a variant of the Kirchoff's matrix tree theorem from graph theory.

The aim of this talk is to demonstrate effectiveness of extension theory for investigation of stability of standing waves for semi-linear Schrödinger equations with $\delta$- and $\delta'$-interaction on the line and on the star graph. One of our focus topics will be the standing waves with a bump profile.

This talk will consider the stability of time independent solutions to the incompressible, inviscid Euler equations on the torus whose stream functions have the form $\psi = U(\xi) =U(p_1x + p_2y)$ for fixed integers $p_1$ and $p_2$. By an appropriate change of coordinates and separation of variables, the linearised spectral problem is reduced to the study of a Hill's equation with a complex potential. By using Hill determinants, an Evans function of the original linearised Euler equation can be constructed. For certain, well-known shear flows, the form of the Hill determinant makes such an Evans function numerically straightforward to compute.

For Hamiltonian systems of PDEs the stability of periodic waves is encoded by the Hessian of an action integral, as shown in earlier work. This talk will deal with two asymptotic regimes, namely for waves of small amplitude and for waves of long wavelength. In both cases stability criteria can be investigated analytically, thanks to the asymptotic expansions of the Hessian of the action and their special structure. The stability results thus obtained apply to various models of mathematical physics for nonlinear dispersive waves.

We consider the Whitham equation on the whole line. Due to the smoothing nature of the linear operator, the question for existence of traveling wave solutions has been open till recently. In 2012, Ehrnstroem-Groves-Wahlen (EGW) have constructed such waves, but only for values of $c$ slightly bigger than one, even though the admissible range of wave speeds is $c \in (1,2)$. The approach in EGW consists of a tour de force calculus of variations, supplemented by a bifurcation argument from the small KdV waves. Note that the EGW waves are of small amplitude. In this work (joint with M. Ehrnstroem), we construct a one parameter family of such waves, with wave speeds $c \in (1, c_0)$ for some limiting value $c_0$, not necessarily close to $1$. We conjecture that the wave with speed $c_0$ is the maximal amplitude wave (i.e. the highest wave, with amplitude $c/2$) and there are no waves with wave speeds $c \in (c_0,2)$. However, we still find some interesting objects in the interval $(c_0,2)$. The argument uses calculus of variation construction, very different than the one employed by EGW. It is based on constraints on appropriately selected Orlicz spaces. Finally, all our traveling waves are shown to be bell-shaped, confirming the available numerical evidence. I will also speculate a bit about their stability as they are constructed as ground states of the appropriate constrained maximization problems.

In this talk, we explore the simplest equation that exhibits high frequency instabilities, the fifth-order Korteweg-de Vries equation. We show how to derive the necessary condition for an instability of a perturbation of a small amplitude, periodic travelling wave solutions. We proceed by examining how these unstable perturbations change and grow in time as the underlying solution changes. We conclude by commenting on what happens with a different nonlinearity in the underlying equation.

The Krein matrix is a matrix-valued function which can be used to study Hamiltonian spectral problems. Akin to the Evans matrix, it has the property that it is singular when evaluated at an eigenvalue. Unlike the Evans matrix, it is not analytic, but is instead meromorphic. I will briefly go over its construction, and then apply it to the study of spectral stability of small periodic waves for a couple of equations.

In this talk, we will discuss the influence of inhibition on an activity-based neural field model consisting of an excitatory population with a linear adaptation term that directly regulates the activity of the excitatory population. Such a model has been used to replicate cortical traveling wave data as observed in clinical recordings. We will establish conditions for the existence of traveling wave solutions with properties observed in in vivo data. We will also discuss some results concerning the linear stability of traveling wave solutions of this model via the construction of an Evans function.

This is a joint work with M. Beck, G. Cox, C. Jones, R. Marangell, K. McQuighan, A. Sukhtayev, and S. Sukhtaiev. In this talk we discuss some recent results on connections between the Maslov and the Morse indices for differential operators. The Morse index is a spectral quantity defined as the number of negative eigenvalues counting multiplicities while the Maslov index is a geometric characteristic defined as the signed number of intersections of a path in the space of Lagrangian planes with the train of a given plane. The problem of relating these two quantities is rooted in Sturm's Theory and has a long history going back to the classical work by Arnold, Bott, Duistermaat, Smale, and has attracted recent attention of several groups of mathematicians. We will briefly mention how the relation between the two indices helps to prove the conjecture that a pulse in a gradient system of reaction diffusion equations is unstable. We will also discuss a fairly general theorem relating the indices for a broad class of multidimensional elliptic self-adjoint operators.

Consider an infinite chain of masses, each connected to its nearest neighbors by a (nonlinear) spring. This is an FPUT lattice. In the instance where the masses are identical, there is a well-developed theory on the existence, dynamics and stability of solitary waves and the system has come to be one of the paradigmatic examples of a dispersive nonlinear equation. In this talk, I will discuss recent rigorous results of mine (together with T. Faver, A. Hoffman, R. Perline, A. Vainchstein and Y. Starosvetsky) on the existence of traveling waves in the setting where the masses alternate in size. In particular I will address in the limit where the mass ratio tends to zero. The problem is inherently singular and as such the existence theory becomes rather complicated. In particular, we find that the traveling waves are not true solitary waves but rather ``nanopterons", which is to say, waves which asymptotic at spatial infinity to very small amplitude periodic waves. Moreover, we can only find solutions when the mass ratio lies in a certain open set. The difficulties in the problem all revolve around understanding Jost solutions of a nonlocal Schr\"odinger operator in its semi-classical limit.

The present talk deals with sufficient conditions for orbital stability of periodic waves of a general class of evolution equations supporting nonlinear dispersive waves. Firstly, our main result does not depend on the parametrization of the periodic wave itself. Secondly, motivated by the well known orbital stability criterion for solitary waves, we show that the same criterion holds for periodic waves. In addition, we show that the positiveness of the principal entries of the Hessian matrix related to the ``energy surface function'' are also sufficient to obtain the stability. Consequently, we can establish the orbital stability of periodic waves for several nonlinear dispersive models. We believe our method can be applied in a wide class of evolution equations; in particular it can be extended to regularized dispersive wave equations. This is a joint work with A. Pastor (IMECC/UNICAMP-Brazil) and G. Alves (ULCO-France).

One of the main features of micro magnetic materias is the formation of magnetic domain separated by the so called magnetic walls. These walls are transition layers that move upon the application of external magnetic fields. In bulk material the main type of walls is the Bloch wall. There are two models for micro magnetic precession dynamics with damping. One based in Landau-Lifshitz-Gilbert (LLG) equation and other based on eddy current damping. In this talk we prove some stability results for the Bloch wall under the eddy current damping model and the LLG model.

Models describing waves in anisotropic media or media with imperfections usually have inhomogeneous terms. Examples of such models can be found in many applications, for example in nonlinear optical waveguides, water waves moving over a bottom with topology, currents in nonuniform Josephson junctions, DNA-RNAP interactions etc. Travelling waves in such models tend to interact with the inhomogeneity and get trapped, reflected, or slowed down. In this talk, wave equations with finite length inhomogeneities will be considered, assuming that the spatial domain can be written as the union of disjoint intervals, such that on each interval the wave equation is homogeneous. The underlying Hamiltonian structure allows for a rich family of stationary front solutions and the values of the energy (Hamiltonian) in each intermediate interval provide natural parameters for the family of orbits. It will be shown that changes of stability can only occur at critical points of the length of the inhomogeneity as a function of the energy density inside the inhomogeneity and we give a necessary and sufficient criterion for the change of stability. These results will be illustrated with some examples.

Extending results of Humpherys-Lyng-Zumbrun in the one-dimensional case, we use a combination of asymptotic ODE estimates and numerical Evans-function computations to examine the multidimensional stability of planar Navier-Stokes shocks across the full range of shock amplitudes, including the infinite-amplitude limit, for monatomic or diatomic ideal gas equations of state and viscosity and heat conduction coefficients constant and in the physical ratios predicted by statistical mechanics, with Mach number $M>1.035$. Our results indicate unconditional stability within the parameter range considered, in agreement with the results of Erpenbeck and Majda in the corresponding inviscid case. Notably, this study includes the first successful numerical Evans computation for multi-dimensional stability of a viscous shock wave. This is joint work with J. Humpherys (BYU) and K. Zumbrun (Indiana).

We consider a model of combustion in hydraulically resistant porous media. There are several reductions of this systems that can be used to understand the evolution of the combustion fronts. One reduction is based on the assumption that the ratio of pressure and molecular diffusivities is close to zero, a different reduction is obtained when the Lewis number chosen in a specific way. Fronts exists in both reduced systems. For the stability analysis of the fronts, we, first, consider initial conditions of a specific form, then show that the stability results extend to the fronts in the full system with the same Lewis number. The fronts are either absolutely unstable or convectively unstable.

Mesenchymal migration is a proteolytic and path generating strategy of individual cell motion inside the network of collagen fibres that compose the extracellular matrix of tissues. We analyze the spectral stability of the families of standing and traveling wave solutions of the one-dimensional version of the $M^5$-model, which was proposed by T. Hillen to describe mesenchymal cell movement. Regarding the standing waves, they are spectrally stable and the spectrum of the linearized operator around the waves consists solely of essential spectrum. To prove that in the standing case the point spectrum is empty we use energy estimates together with the integrated-variable technique of Goodman. The panorama is completely different in the traveling case; the wave profiles are spectrally unstable due to the fact that the essential spectrum reaches the closed right-half complex plane. In our pursuit of spectral stability, we have constructed a weighted Sobolev space where the essential spectrum lies inside the open left-half complex plane.

In the present work, we intend to explore a variant of the $\phi^6$ model originally proposed in \textit{Phys. Rev. D} \textbf{12}, 1606 (1975) as a prototypical, so-called, ``bag'' model where domain walls play the role of quarks within hadrons. We examine the prototypical steady state of the model, namely an apparent bound state of two kink structures. We explore its linearization and find that as a function of a prototypical parameter controlling the curvature of the potential an effectively arbitrary number of internal modes may arise in the point spectrum of the linearized analysis. We \textit{intend to} use Evans function analysis to predict the bifurcation points of the relevant internal modes and confirm these theoretical predictions numerically. Finally, given the remarkable flexibility of the model in possessing different numbers of internal modes we \textit{once again intend to} explore the dynamics of multi-bound-state collisions to identify the role of the additional internal modes in enhancing the complexity of the observed scattering scenarios. I. Christov, P. G. Kevrekidis, A. Saxena, and R. Decker are my collaborators.

Recently, partly motivated by applications to surface waves, rapid progresses on the stability theory of periodic waves have been obtained. In particular, for parabolic systems --- including those encoding the shallow water description of viscous roll-waves --- an essentially complete theory is now available. We shall expound here some first contributions to a dispersive theory, still to come.

Some PDEs and ODEs admit a Lax pair (a pair of linear operators) to be completely solve the equation. One of these operators defines a spectral problem. For some equations (as for the Korteweg-deVries, KdV, equation) this operator is self-adjoint and, consequently, its discrete spectrum is real. However, for some other equations, this operator is non-selfadjoint, such as for the nonlinear Schrödinger (NLS) equation or the Ablowitz-Ladik equation. In 2001, M. Klaus and J. K. Shaw found symmetries and conditions on the potentials for the Zakharov-Shabat system (spectral problem for the NLS equation) for the eigenvalues to lie on the imaginary axis. In this talk, I will show which would be an equivalent to the Klaus-Shaw theorem for the Ablowitz-Ladik lattice. This is a work in progress. This is a joint work with P. Shipman (Colorado State University) and S. Shipman (Louisiana State University).

Roll-waves are well known hydrodynamic instabilities appearing in open channel flows driven by gravity. A classical model to describe such flows is given by the shallow water equations with bottom friction. Periodic travelling waves of this system are necessarily discontinuous and shocks are driven by classical Rankine-Hugoniot conditions. In order to regularize these solutions, viscous effects can be taken into account: in this framework, a complete stability theory is established. Less is known in the context of discontinuous periodic waves. It is the purpose of this talk to present a framework to study the stability of discontinuous periodic waves. I will then provide some partial analytical results and numerical results on the stability of discontinuous roll-waves.

In this talk, we discuss the stability of periodic traveling wave solutions describing the interface between two fluids of varying density and vorticity trapped between two rigid lids. Using a generalization of a non-local formulation of the water wave problem due to Ablowitz, et al., and Ashton & Fokas, we determine the spectral stability for the periodic traveling wave solution by extending Fourier-Floquet analysis to apply to this non-local problem. We develop a numerical scheme to determine traveling wave solutions by exploiting the bifurcation structure of the non-trivial periodic solutions. Next, we determine numerically the spectral stability for the periodic traveling wave solution by extending Fourier-Floquet analysis to apply to the non-local problem. We can generate the full spectra for all traveling wave solutions. We discuss Kelvin-Helmholtz and Benjamin-Feir instabilities, as well as explore the suppression or amplification of such instabilities as a function of shear strength, density stratification, and the ratio of depths between the fluids.

The Maslov index is a powerful and well known tool in the study of Hamiltonian systems, providing a generalization of Sturm-Liouville theory to systems of equations. For non-Hamiltonian systems, one no longer has the symplectic structure needed to define the Maslov index. In this talk I will describe a recent construction of a "generalized Maslov index" for a very broad class of differential equations. The key observation is that the manifold of Lagrangian planes can be enlarged considerably without altering its topological structure, and in particular its fundamental group. This is joint work with Paul Cornwell, Chris Jones and Robert Marangell.

We construct several families of symmetric localized standing waves (breathers) to the one-, two-, and three-dimensional discrete nonlinear Schrödinger equation (DNLS) with cubic nonlinearity using bifurcation methods about the continuum limit. Such waves and their energy differences play a role in the propagation of localized states of DNLS across the lattice. The energy differences, which we prove to exponentially small in a natural parameter, are related to the Peierls-Nabarro Barrier in discrete systems, first investigated by M. Peyrard and M. D. Kruskal (1984). These results may be generalized to different lattice geometries and inter-site coupling parameters. Finally, we discuss the local stability properties of these bound states. This is joint work with Michael I. Weinstein.