## Abstract

Periodic and localized travelling waves such as wave trains, pulses, fronts and patterns of more complex structure often occur in natural and experimentally built systems. In mathematics, these objects are realized as solutions of nonlinear partial differential equations. The existence, dynamic properties and bifurcations of those solutions are of interest. In particular, their stability is important for applications, as the waves that are observable are usually stable. When the waves are unstable, further investigation is warranted of the way the instability is exhibited, i.e. the nature of the instability, and also coherent structures that appear as a result of an instability of travelling waves. A variety of analytical, numerical and hybrid techniques are used to study travelling waves and their properties.

This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.

Many physical spatially extended systems are modelled by partial differential equations (PDEs) posed on infinite domains. The dynamics on unbounded domains is essentially different from that on bounded domains. In particular, it supports non-stationary, permanent structures, namely periodic and localized travelling waves in the form of wave trains, fronts, pulses, holes, etc.

Travelling waves are solutions of the underlying PDE which preserve their shape while propagating with a certain velocity. They arise and serve as mathematical representations of relevant observable phenomena in applied problems from different fields: optical communication, combustion theory, biomathematics (calcium waves in tissue, nerve conduction, population dynamics, epidemiology, tumour growth), chemistry (autocatalytic reactions such as the chlorite–iodide–malonic acid reaction or the Belousov–Zhabotinsky reaction), to name a few.

Travelling waves are basic coherent structures in PDEs. Information about travelling waves and their properties can be used to study the dynamics of solutions that start near travelling waves; travelling waves often serve as building blocks for more complicated patterns such as, for example, a composition of a front and a periodic pattern or two periodic patterns and a pulse. Therefore, being able to prove the existence of travelling waves is important. Consequently, stability properties are also of interest. Indeed, for a physical system, using an analogy from optical communication, one can think of a travelling wave as a vehicle that transports information from one part of the domain to another. If, despite small inhomogeneities of the medium or slightly changed initial data, the information reaches the destination, and it reaches it in undistorted form, then the wave is stable. Otherwise the wave is said to be unstable.

The stability analysis of travelling waves is a multistep process. The first step involves locating the spectrum of the operator obtained by linearizing the PDE about the travelling wave solution. The spectrum may consist of discrete eigenvalues and a continuous spectrum.

There are several ways to study the spectrum of the operator of the linearization. Among the different techniques available to tackle an eigenvalue problem, the Evans function is a tool that provides a way to locate the spectrum of a linear operator in the complex spectral parameter space. It has been used to study the spectral stability of pulse solutions of many equations such as generalizations of the Korteweg–de Vries equation, the Benjamin–Bona–Mahoney equation and the Boussinesq equation [1]. It is often the case that the Evans function cannot be tackled explicitly or that the solutions of the PDEs are only known numerically. It is thus important to be able to develop tools for studying stability in such a situation. There are several ways to numerically compute the Evans function [2–14]. Barker *et al*. [14] in this issue present a review of the literature on the subject. In addition, the article contains Evans function results for planar strong detonations together with results from recent computations. Their numerical investigations are performed via STABLAB [8], a MATLAB-based numerical library for Evans function computation.

The analytical concept of the Evans function is a prelude to geometric or topological methods in the stability analysis of waves. For singularly perturbed systems, one of the topological methods is based on the construction of the stability index bundle [15] and the association of the number of unstable eigenvalues with its topological invariant, the Chern number. Another example of a topological method is based on calculating the Maslov index. Constructed for Hamiltonian systems, it is a topological invariant that counts intersections of Lagrangian planes in a symplectic vector space [16]. Beck *et al*. [17] herein is devoted to the calculation of the Maslov index related to the number of unstable eigenvalues of a pulse in a reaction–diffusion system with gradient nonlinearity posed on one-dimensional physical space. The authors introduce a new, alternative definition of Maslov index that, under generic assumptions on the system, allows one to obtain a lower bound on the number of unstable eigenvalues, and thus show that the pulses are unstable.

A geometric technique is also the key to the Ricatti approach used by Beck *et al*. [18] in this issue to construct solutions to classes of PDE systems with quadratic, cubic and higher-order odd non-local nonlinearities. The technique is based on the development of Grassmannian flows from appropriate linear subspace flows, which are then associated with the Ricatti equation. The result presented in [18] extends the authors’ previous Ricatti approach applied in [19] to construct solutions to scalar equations with quadratic non-local nonlinearities.

While the Evans function and Maslov index count the unstable eigenvalues, and thus are concerned with spectral stability, the ultimate goal is to show the nonlinear stability of the considered waves. The next step in the stability analysis is the transition from the spectral information to the properties of the linearization operator and the semigroup that it generates. These properties are then used to relate the spectral information to the nonlinear stability of the wave. For example, if the linearized operator belongs to a class of sectorial operators, then a spectrally stable wave is also nonlinearly stable [20]. But, generally speaking, it is not always true, or not always easy to see, that the stability properties of the wave in the full nonlinear equation are dominated by the spectral properties of the linearized operator. Situations when difficulties in the analysis arise, for example, are when the linearized operator is not sectorial or when the wave has marginal continuous spectrum. A case when both of these factors are simultaneously present is encountered in a paper in this issue, where Ozbag & Schecter [21] investigate the stability of combustion waves that occur in a model for injection of air into a porous medium that initially contains solid fuel. The system is known to support multiple travelling fronts that are parametrized by their velocity. Not only does the continuous spectrum contain vertical lines, but it also extends all the way to the imaginary axis, so neither of these waves are spectrally stable, and the operator of the linearization is not sectorial. Ozbag & Schecter [21] provide a detailed stability analysis for the family of fast combustion fronts. Using STABLAB [8], the authors demonstrate that there are regimes when there is no unstable discrete spectrum. They deal with the unstable essential spectrum by means of exponential weights and proved nonlinear stability of fronts against a special class of perturbations using results from [22,23]. In general, the stability in an exponentially weighted norm points to a convective nature of the instability [24].

When the system under consideration is Hamiltonian and satisfies certain conditions, the nonlinear stability can be tackled through the use of the Hamiltonian formalism, which provides a way to realize a given solution as the critical point of a Lyapunov functional [25,26]. Hamiltonian formalism can also be used when only linear stability can be proved or disproved. This is because the spectrum of the Lyapunov functional can be related to the spectrum of the operator arising when linearizing the governing PDE [27,28]. In this issue, two contributions [29,30] deal with the topic using the Hamiltonian structure to study spectral stability.

Xu *et al*. [29] consider travelling wave solutions to lattice Hamiltonian systems and study a condition for a pair of eigenvalues associated with the PDE system to cross zero and emerge on the real axis, thus leading to spectral instability. The authors perform their study following two different approaches: one based on Floquet multipliers, and one that is Hamiltonian-based. The article [29] and a previous one from the same authors [31] distinguish themselves from other works on lattice equations by the fact that no tools from integrability are used for the stability analysis. Two examples of lattice dynamical system problems that have solitary wave solutions that change stability are considered. One of the examples involves the *α*-FPU (Fermi–Pasta–Ulam) lattice with exponentially decaying long-range Kac–Baker interactions, while the other example concerns a smooth regularization of the FPU problem. The analytical results for the two examples are complemented by numerical computations.

Feng & Stanislavova [30] study the existence and the stability of vortex solutions of the nonlinear Schrödinger (NLS) equation. Vortex solutions of the NLS are of interest in nonlinear optics and in the theory of Bose–Einstein condensates [32–35]. In their article, Feng & Stanislavova establish the existence of vortex NLS solutions in arbitrary dimensions using a variational approach. Those results constitute a generalization of Mizumachi’s work [36,37], who considered the two-dimensional case only. The argument used by Feng & Stanislavova for the spectral stability is based on the index theory developed in [27,28], and is restricted to perturbations that have the same form as the vortex solutions.

When a wave is unstable, further investigation can give rise to an interesting dynamics of the way the instability behaves, such as is the case in Whitham theory. Herein, Bridges & Ratliff [38] discuss the application of Whitham modulation theory in 2 + 1 dimensions near the Lighthill instability threshold [39], which occurs when the so-called Lighthill determinant vanishes. The authors’ strategy for developing a nonlinear modulation theory near the Lighthill instability is to use an appropriate ansatz to slow down the time scale, go into a moving frame, and slow down the phase, wavenumber and frequency modulation. This procedure results in the derivation of a two-dimensional Boussinesq equation. As a specific example, the authors consider modulations of plane wave solutions to the complex Klein–Gordon equation.

Beyn & Otten [40] in this issue study stability properties of exponentially localized rotating waves that a class of reaction–diffusion systems support. After factoring out the rotation, linearization of the steady-state equation at the exponentially localized component of the wave yields Ornstein–Uhlenbeck operators plus the linearization of the equations of kinetics of the reaction–diffusion system at the wave. The authors study the spectrum of the linearized operator which is related to Ornstein–Uhlenbeck operators. They show that under certain conditions the dispersion relation belongs to the spectrum of the linearized operator and also relate the spectrum of the matrix that describes the rotation of exponentially localized waves to the spectrum of the full linearized operator. Beyn & Otten apply the theory developed in their paper to the spinning soliton solutions of the Ginzburg–Landau equation.

The existence and qualitative properties of spatial and spatio-temporal patterns is of great current research interest in the literature. Patterns such as stripes, hexagons, spirals and target patterns are of universal nature. They appear in complex pattern forming systems such as chemical reactions, physical and biological systems and often these patterns are realized as solutions of PDEs. In this issue, various aspects of stripe patterns, which are also known as roll solutions in the literature, are studied as solutions of the Swift–Hohenberg equation by two groups of researchers. Their contributions [41,42] are described below.

Ercolani *et al*. [41] study the properties of stripe solutions and line defects that appear as the boundary between stripes that have different orientations. These structures are studied as solutions of the Swift–Hohenberg equation and associated phase-diffusion equation which is the regularized Cross–Newell equation derived within this context in [43,44]. In [41], the authors establish a relationship between grain boundary solutions of the Swift–Hohenberg and Cross–Newell equations and shed light on the instability mechanism in these pattern forming systems that yield the appearance of dislocations at the core of the line defects when the angle between the stripes that are on both sides of the line defect is decreased.

Scheel & Weinburd [42] show that the Swift–Hohenberg equation, where the control parameter enters in the equation as a space-dependent, piecewise constant function with jump discontinuity, supports a family of complex spatial patterns that are manifested as a transition from a constant state to a striped pattern state. The existence of such patterns is shown by using dynamical systems methods such as spatial dynamics, equivariant bifurcation theory and normal forms, where the control parameter that parametrizes these solutions is used in the analysis as a bifurcation parameter. The spatial transition patterns considered in [42] are constructed as transverse heteroclinic connections between a point and a periodic orbit, where the point corresponds to the constant state and the periodic orbit corresponds to the striped pattern. A delicate analysis combined with detailed numerical calculations shed light on qualitative and quantitative characteristics of these patterns along with their stability properties.

Delcey & Haragus [45] study herein the existence and stability of small periodic solutions of the Lugiato–Lefever equation which is derived within the context of nonlinear optics. The Lugiato–Lefever equation has been the subject of active research for various types of coherent structures and their stability properties. Delcey & Haragus show the existence of periodic waves by considering it a bifurcation problem of a constant state at the onset of the Turing instability. Within the regime under consideration, by exploiting the symmetries of the underlying equations, they use a unified approach that is anchored in equivariant bifurcation theory and central manifold reduction to obtain results on the existence, stability and instability properties of periodic waves that the Lugiato–Lefever equation supports for a certain range of parameters.

Doelman *et al*. [46] in this issue in a rather general setting study the existence, stability and dynamic properties of pulses in a coupled system of linear reaction–diffusion equations, forced with a family of spatially localized impulse functions that account for spatially localized impurities. The system considered in [46] has multi-scale structure that stems from the nature of the problem. The authors develop a framework under the umbrella of geometric singular perturbation theory which is an effective technique for studying the existence, geometric structure of steady single- and multi-pulse solutions, their stability properties and bifurcations when multiple scales are present due to strong spatially localized impurities.

In last two decades or so, there have been multiple socio-physical attempts to understand mechanisms of opinion formation in a population (e.g. [47]). Recent failure of social and media structures to predict the outcome of a process of opinion formation justifies an increased and renewed interest in this topic. For the case when there are only three options for holding an opinion (opinion A, not A or undecided), a model of opinion formation was proposed in [47]. It is a mean-field model, i.e. the proportions of the total population that are in one of the mentioned states are tracked down. Bujalski *et al*. [48] herein propose an extension of the system from [47]. In particular, conditions under which consensus and pluralism occur and the influence of zealots in the dynamics of forming an opinion are understood. Furthermore, the authors extend the model for one city to a network of interacting cities by considering an ordinary differential equation system on a directed graph. For two interacting cities, it is shown that the presence of sufficiently many zealots holding an opinion in one city leads in both cities to a consensus on that opinion. In the network of many cities (with and without a hub), cycle graphs are used. In the case with a hub, it is shown that with zealots in the hub their opinion spreads over the whole network. In a network without a hub, the clustering phenomenon, i.e. the formation of clusters of cities that share an opinion, and its association to a saddle-node bifurcation were captured. While the techniques in the paper are classical, the topic and the conclusions of the analysis are immensely timely and relevant.

To summarize, the goal of this theme issue is to present a selection of the most current results on the subject of periodic and localized travelling waves, especially in the area of the stability analysis of waves and, more generally, pattern formation, as well as examples of applications of the methods and techniques.

## Data accessibility

This article has no additional data.

## Competing interests

We declare we have no competing interests.

## Funding

S.L. and V.M. were each supported by a Simons Foundation Collaboration Grant for Mathematicians. A.G. was supported, in part, by the NSF grant no. DMS-1311313 while working on this theme issue.

## Footnotes

One contribution of 14 to a theme issue ‘Stability of nonlinear waves and patterns and related topics’.

- Accepted January 4, 2018.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.