## Abstract

The opening article of this issue is intended to provide a review of some relevant topics of the mathematical theory of water waves that have recently received considerable attention in the research literature. We also provide a brief discussion about the content and contribution of the articles that make up this issue.

## 1. Introduction

Of the multitude of fluid wave motions that occur in nature, surface water waves are the most easily observed. Moreover, their great practical importance cannot be overestimated, since roughly two-thirds of the Earth's surface is covered by water waves. Not only are these strong motivations for the study of water waves, attracting the attention of mathematicians across the whole spectrum from applied to pure mathematics, but also a considerable part of the mathematical theory of wave motion over the past 200 years has been pioneered on the basis of studies of water waves. To quote Arnold & Khesin (1998): ‘Hydrodynamics is one of those fundamental areas where progress at any moment can be regarded as a standard to measure the real success of mathematical science’. While the study of water waves draws freely from other disciplines, the development of the subject has proved that one cannot expect the mathematical tools to have been laid out and only be used mechanically. Even at simple levels, problems in hydrodynamics require subtle use of mathematical concepts in order to obtain effective solutions that reflect to some extent the physical reality. Indeed, ‘…hydrodynamical theories stimulated developments in the domains of pure mathematics, such as complex analysis, topology…’ (cf. Arnold & Khesin 1998). Despite the great progress achieved in hydrodynamics in the last two centuries, the subject still offers plenty of fertile ground for further exploration, providing in many fundamental aspects compelling challenges for researchers.

Starting with the work of Stokes and his contemporaries in the nineteenth century (Craik 2004), there have been significant advances in our understanding of water waves. Until the second half of the twentieth century, the study of water waves was dominated by linear theory. This is due to the scarcity of explicit solutions to the governing equations for water waves (see Gerstner (1809) for a deep-water wave with a specific vorticity,1 Crapper (1957) and Kinnersley (1976) for irrotational capillary waves and Constantin (2001*c*) for edge waves with a specific vorticity propagating along a plane beach). There is a large variety of physical water wave phenomena that are outside the range of studies relying on the interpretation of these explicit solutions. For this reason, one has to either rely on mathematical and/or numerical analysis of the governing equations, or perform approximations through perturbation expansions. While linearization is a first indispensable tool for developing an understanding about the complex processes encompassed by the nonlinear governing equations for water waves, the linear framework has limited applicability. Linear theory sometimes gives a good approximation to the wave motion, but its applicability fails in the study of phenomena that are manifestations of genuine nonlinear behaviour. For example, linear water wave theory gives no insight into wave breaking and does not capture the existence of solitary water waves (Stoker 1957). Furthermore, linear theory is not adequate for waves that are not small perturbations of a flat water surface. Future advances in the study of water wave phenomena depend on a fruitful interaction between the conceptual and computational framework, in addition to observation and physical verification via laboratory experiments and field data. With the increased computer power available nowadays, it is possible to carry out computational investigations featuring a considerable degree of detail, once mathematical models for the underlying principles are developed. A successful procedure towards uncovering underlying principles usually comprises several stages, the first consisting of a direct study of the governing equations. The complexity of the system usually prevents a detailed account, but breakthroughs will provide invaluable insight. This forms the basis for the derivation of approximations using simplifying assumptions, such as ‘small amplitude’, ‘long wave’ (or ‘shallow water’, in the sense of a small depth-to-wavelength ratio) and ‘unidirectionality’ within certain regimes (e.g. for irrotational flows or for water flows with constant vorticity), rendering mathematical models of various degrees of sophistication amenable to a more detailed analysis. The obtained simplified model equations are linear to the lowest order of approximation, but higher-order approximations incorporate nonlinear effects. These nonlinear models usually have features, such as exact steady solutions known in closed form, which make them suitable to explain certain observations. Conclusions drawn on the basis of these simpler models sometimes reveal hidden structures of the original problem, thus leading back to a fruitful investigation of the governing equations. Much of the current interest in water waves was stirred by the success of soliton theory, unravelled on the basis of a simpler model—the Korteweg–de Vries (KdV) equation—for the governing equations (Johnson 1997).

## 2. Content of the present issue

The last few years have seen a vigorous activity in the mathematical theory of water waves by several independent international research groups. In the autumn of 2005, a large part of the four-month programme ‘Wave Motion’, organized in Stockholm at the Mittag-Leffler Institute of the Swedish Royal Academy of Sciences by A. Constantin (Lund University, Sweden and Trinity College Dublin, Ireland), C. Dafermos (Brown University, USA), H. Holden (NTNU Trondheim, Norway), K. Karlsen (Oslo University, Norway) and W. Strauss (Brown University, USA), focused on water waves. The participation in the programme in water waves was global with representatives attending from Bulgaria, People's Republic of China, France, Germany, Ireland, Italy, Norway, Romania, Russia, South Korea, Switzerland, Sweden, UK and USA. This issue was compiled to present the recent progress in water waves reported during the programme, while other aspects of wave motion that were considered during the programme (e.g. the theory of conservation laws) are not reflected in this issue. Contributions from the following research areas have been included in the present issue:

investigations on water waves with vorticity,

aspects of the theory of vortex sheets,

studies of some integrable model equations for shallow water waves, and

aspects of edge wave theory.

The first paper by Hur (2007) addresses the issue of the symmetry of periodic two-dimensional travelling gravity waves propagating at the free surface of water in a flow with vorticity. The existence of such solutions to the governing equations for an arbitrary smooth vorticity is known (Constantin & Strauss 2004) and, for certain vorticity distributions, it was recently established that if the free surface is monotone between crests and troughs, then the wave must be symmetric about the crest line (Constantin & Escher 2004). The author shows that under mild conditions on the streamline pattern—in a period each streamline should have its minimum below the trough of the wave—symmetry holds for arbitrary smooth vorticity. The paper by Wahlén (2007) deals with the existence of rotational periodic two-dimensional travelling waves with surface tension. Using bifurcation theory, the existence of small-amplitude capillary and capillary–gravity waves of this type can be shown. An interesting comparison between the dispersion relation for gravity waves and that for water waves for which the effects of surface tension are not neglected is made. The third paper in this group (Constantin & Strauss 2007) is devoted to large-amplitude steady rotational gravity water waves. In the irrotational case, a long-standing conjecture of Stokes, proved in Amick *et al.* (1982)—see also the review in Toland (1996)—states that there exists a steady wave of greatest height with a stagnation point at its crest. The existence of steady rotational water waves of large amplitude was recently established (Constantin & Strauss 2004), but concerning waves of greatest height for flows with vorticity, the present knowledge is restricted to numerical simulations (Simmen & Saffman 1985; Teles da Silva & Peregrine 1988; Vanden-Broeck 1996). The paper by Constantin & Strauss (2007) offers some analytical considerations about the possible location of a stagnation point in such waves. The last paper of this issue dealing with rotational water waves is by Henry (2007). Throughout the hydrodynamics literature, it has been quite common to assume that in a periodic two-dimensional travelling water wave, the paths of the particles are closed: almost circular for deep water and of an elliptical shape in shallow water (Lamb 1932; Milne-Thomson 1938; Sommerfeld 1950; Stoker 1957; Lighthill 1978; Crapper 1984; Acheson 1990; Debnath 1994; Johnson 1997). Support for this conclusion is given by the only explicit solution available for gravity water waves: for the wave presented in Gerstner (1809), all particles move in circles. However, rather than being the rule, this feature seems to be the exception. Recently, in Constantin (2006), it was proved that in an irrotational gravity wave solution to the governing equations for water waves propagating over a flat bed, the particle paths in the fluid are never closed if the free surface is not flat.2 It is therefore natural to wonder whether the effects of surface tension could alter this feature of irrotational water flows. The paper by Henry (2007) establishes that for linear capillary and capillary–gravity irrotational water waves, the particle trajectories are not closed if the free surface of water is not flat.

A vortex sheet is the free boundary between two fluids shearing past each other in irrotational flow, the vorticity being measure-valued and supported on the free boundary between the fluids. Linearizing the equation of motion of the vortex sheet one obtains that the motion is linearly stable if surface tension acts, while linear instability holds if the effects of surface tension are neglected—an effect known as the Kelvin–Helmholtz instability (Saffman 1995). The paper by Ambrose (2007) presents a proof of the fact that surface tension regularizes the Kelvin–Helmholtz instability for the full nonlinear equations.

Several papers of this issue are devoted to qualitative studies of two recently derived integrable nonlinear models for the propagation of shallow water waves: the CH equation (Camassa & Holm 1993; Johnson 2002) and the DP equation (Degasperis & Procesi 1999). Both nonlinear PDEs share with the classical KdV model the properties that they have a bi-Hamiltonian structure and are formally integrable3 (Fokas & Fuchssteiner 1981; Camassa & Holm 1993; Degasperis & Procesi 1999). In the case of CH, an inverse scattering/inverse spectral analysis can be performed and one can prove that the equation is completely integrable: its flow is equivalent to the flow of a (mostly) infinite set of parameters moving linearly at constant speed (Constantin & McKean 1999; Constantin 2001*a*; Constantin *et al.* 2006). However, while all smooth solutions of the KdV equation exist for all times, both CH and DP admit global solutions as well as breaking waves: the solution remains bounded, but its slope becomes unbounded in finite time (Constantin & Escher 1998; Constantin 2000; Liu & Yin 2006). Another interesting aspect of these recent models is the presence of peaked solitary waves, called *peakons*. The CH peakons are known to have a stable shape (in the sense that they are orbitally stable); therefore, they represent physically recognizable patterns (Constantin & Strauss 2000). Moreover, they are solitons, retaining their shape and speed after interacting with other peakons (Beals *et al.* 1999). Whitham (1980) emphasized the need for shallow water models that exhibit soliton interaction, the existence of peaked waves, and allow for breaking waves. After this brief motivation for the study of the CH and DP equations in the context of water waves, let us proceed with presenting the content of the papers in this issue related to the aspects of these models. The paper by Ivanov (2007) presents the derivation of both CH and DP as approximations to the governing equations for water waves. In Escher (2007), the existence of weak solutions to the DP equations in the periodic case is of interest, special emphasis being placed also upon the wave-breaking phenomenon. The paper by Lenells (2007) is devoted to a detailed classification of the travelling waves for both equations. It turns out that, in addition to peakons, both equations admit also travelling waves with cusps, as well as a multitude of peculiar waves obtained by combining peaked and cusped wave segments into new fractal-like travelling wave solutions. The contribution by Beals *et al.* (2007) deals with aspects relating the string density problem to the integrability of the CH equation, while El Dika & Molinet (2007) analyses the stability of multi-soliton solutions to the CH equation. Finally, the paper by Kolev (2007) discusses the connection between the CH equation and the geodesic flow on the diffeomorphism group of the circle. This geometric aspect is deeply connected with the fact that the least action principle holds for the CH equation (cf. Constantin & Kolev 2003) and explains from a geometric perspective the bi-Hamiltonian structure of the equation.

The paper by Johnson (2007) presents a field survey of edge waves and offers several ideas about future directions for research on these intriguing water waves. These waves were originally considered to be a curiosity (Lamb 1932), but are now recognized to play a significant role in near-shore hydrodynamics—especially in relation to sediment transport (Komar 1998). Edge waves are waves that progress along the shoreline, with an amplitude that is maximal at the shoreline and decays rapidly offshore. This manifestation of water waves offers a plethora of very intriguing phenomena awaiting further studies.

## Footnotes

One contribution of 13 to a Theme Issue ‘Water waves’.

↵This solution was rediscovered by Rankine in 1863. A recent detailed description of this wave is given in Constantin (2001

*b*).↵A qualitative description of the particle trajectory is given, showing that all particles above the flat bed describe a path similar to the apparently special type photographed in Longuet-Higgins (1986). On the flat bed, the particles have a similar,

*albeit*one-dimensional, oscillatory motion with a forward drift.↵In other words, each equation has two expressions as a Hamiltonian evolutionary equation, with the two Hamiltonian operators compatible (i.e. any linear combination is again a Hamiltonian operator). The bi-Hamiltonian structure ensures for both equations the existence of infinitely many first integrals (conservation laws) that are functionally independent. This last feature is the characteristic of formally integrable equations, being reminiscent of finite-dimensional, classical integrable systems. For both equations, the existence of infinitely many conservation laws can be alternatively obtained from the associated isospectral problem via a

*Lax pair*formulation, due to the time invariance of some spectral data under the flow (see Drazin & Johnson (1989)).- © 2007 The Royal Society