My name is Wesley Perkins. I recently obtained my Ph.D. in Mathematics from the University of Kansas, and I will be starting my postdoc as the C.C.-Hsiung Visiting Assistant Professor at Lehigh University this coming August. To mention a little bit about myself before I dive into my research, I am passionate about investing in students and helping them succeed inside and outside the classroom. I enjoy reading, hiking, kayaking, playing games, spending time with friends, and most things related to Star Wars.

Research

My research focuses on the existence, stability, and dynamics of nonlinear wave solutions to partial differential equations (PDEs) arising from physical applications. In particular, I am interested in studying nonlinear waves and coherent structures that are motivated by experimental or numerical observations. These physically observable patterns can often be studied through a PDE that models the physical system. Several questions naturally arise when using a PDE to study observed patterns:

• Does an idealized version of the observed pattern exist as a solution to the PDE?
• Is such a solution stable, i.e., is it able to persist in the presence of a small perturbation?
• What are the local dynamics (i.e., the dynamics near such a solution) within the PDE as time evolves?
• Can the PDE predict new observations?
• Can the PDE explain why we see some patterns and not others?

The first three questions endeavor to determine, among other things, whether or not the PDE is a good model for the physical system, and their answers can gauge the strengths and weaknesses of the PDE and inform what (if anything) needs to change in the model. The last two questions seek to understand what the mathematics can teach us about the underlying physical phenomena, and the answers to these questions can motivate new experimental and numerical research.

The stability of a particular class of solutions is of fundamental practical importance as solutions which are unstable do not naturally arise in applications except possibly as transient phenomena. In particular, understanding why some patterns are stable and others are unstable can motivate methods of stabilizing unstable patterns via the addition of controls to the physical system. The ability to stabilize unstable patterns is key in many applications where there exist unstable, yet desirable, structures.

There has historically been significant interest in the stability theory of asymptotically constant structures, such as pulses or fronts. More recently, there has been a growing community of mathematicians interested in the stability of spatially-periodic structures. These are often idealized versions of physically-observable patterns which are almost spatially periodic, in the sense that their internal wavelength is much smaller than the size of the physical domain; hence, they may be modeled as exact periodic solutions on unbounded domains to eliminate the influence of far away boundaries. Applications where such patterns exist are numerous and include surface and internal water wave propagation, optical signal propagation, plasma and astrophysics, and inclined thin film flow.

One powerful tool used to study such periodic structures is known as Whitham's theory of wave modulations, sometimes referred to as Whitham theory. Whitham theory is a formal, physically-motivated theory used to understand the stability and dynamics of periodic waves when in the presence of perturbations that modulate their fundamental characteristics, such as amplitude or frequency. Whitham theory lacks rigorous justification in general, leading to the open research problem of establishing such rigorous justification. Nevertheless, Whitham theory is commonly used by applied mathematicians and physicists, and its predictions are nearly universally accepted.

To study the stability of periodic structures using rigorous mathematics, as opposed to the formal asymptotic methods used in Whitham theory, one must start by choosing an appropriate function space to encode the class of perturbations being considered. There are two important and widely studied classes of perturbations that arise naturally in applications. If the nonlinear wave or coherent structure is \(T\)-periodic, then one may consider subharmonic perturbations, i.e., \(NT\)-periodic perturbations for some positive integer \(N\), or localized perturbations, i.e., perturbations which are integrable on the line. Previous results concerning the class of subharmonic perturbations are non-uniform in \(N\), in the sense that they are degenerate in the limit \(N\to\infty\). In particular, the allowable size of the perturbation decreases as \(N\to\infty\), which is undesirable in situations where (arbitrarily) large values of \(N\) are of physical interest.

It might appear, at first glance, that Whitham theory, subharmonic perturbations, and localized perturbations have nothing to do with each other. However, I have shown that the study of localized perturbations is fundamental to the rigorous justification of Whitham theory [5, 1]. Moreover, I have developed new methodologies that establish a deep connection between localized and subharmonic perturbations [2, 6, 3, 4]. Indeed, these new methodologies establish subharmonic stability results with a domain of attraction and decay rate that are uniform in \(N\). We can actually go further and provide a uniform domain of attraction for which perturbations eventually exhibit the faster (\(N\)-dependent) rates of decay.

I would like to thank Dr. Robby Marangell for inviting me to contribute to this issue of DS-Web.

References

[1]   W. A. Clarke, R. Marangell, and W. R. Perkins, Modulational stability for generalized Whitham equations, In preparation, 2021.

[2]   M. Haragus, M. A. Johnson, and W. R. Perkins, Linear modulational and subharmonic dynamics of spectrally stable Lugiato-Lefever periodic waves, Journal of Differential Equations, 280 (2021), pp. 315-354.

[3]   M. Haragus, M. A. Johnson, W. R. Perkins, and B. de Rijk, Nonlinear modulational dynamics of spectrally stable Lugiato-Lefever periodic waves, arXiv:2106.01910 [math.AP], 2021.

[4]   M. Haragus, M. A. Johnson, W. R. Perkins, and B. de Rijk, Nonlinear subharmonic dynamics of spectrally stable Lugiato-Lefever periodic waves, In preparation, 2021.

[5]   M. A. Johnson and W. R. Perkins, Modulational instability of viscous fluid conduit periodic waves, SIAM Journal on Mathematical Analysis, 52(1), 2020, pp. 277-305.

[6]   M. A. Johnson and W. R. Perkins, Subharmonic dynamics of wave trains in reaction–diffusion systems, Physica D: Nonlinear Phenomena, 422 (2021), 132891.