## Abstract

The dynamics of a nonlinear and dispersive long surface capillary-gravity wave model equation is studied analytically in its short-wave limit. We exhibit its Lax pair and some non-trivial conserved quantities. Through a change of functions, an unexpected connection between this classical surface water-wave model and the sine-Gordon (or sinh-Gordon) equation is established. Numerical and analytical studies show that in spite of integrability their solutions can develop singularities and multivaluedness in finite time. These features can be traced to the fact that the surface tension term in the energy involves second-order derivatives. It would be interesting to see in an experiment whether such singularities actually appear, for which surface tension would be specifically responsible.

## 1. Introduction

One finds several different types of linear- and nonlinear-wave phenomena in almost any branch of physics. As a few examples, we have surface waves and internal waves in fluids (Pedlosky 1979; Mei 1989), acoustic waves (Bleistein 1984), electromagnetic and magnetohydrodynamic waves (Jeffrey & Taniuti 1964). In all these fields, wave propagation is a very important issue. Linear and dispersive wave propagation is well understood and there exist analytical and numerical methods to solve these problems. On the other hand, the study of nonlinear and dispersive-wave propagation is a very difficult task and, in general, analytically intractable. Consequently, a great deal of effort has been made to establish approximate model equations. Most of them are one-dimensional in space and asymptotic in time and space (typically long-wave asymptotics). The general idea behind them is that these approximations model the more general and complete initial problem.

In addition, some of these models have very interesting mathematical properties. Their solutions can be nonlinear structures that propagate without dispersion (solitary waves) or nonlinear structures that can interact strongly and retain their identity (solitons), some of them are completely integrable systems with a Lagrangian formulation, they may be associated with a hierarchy of higher-order models with a Hamiltonian or a bi-Hamiltonian structure, etc.

The oldest and most widely known equation bringing together these two aspects is the classical Korteweg–de Vries (KdV) equation: KdV models many nonlinear and dispersive systems and possesses some of the above-mentioned mathematical properties.

This double aspect of some asymptotic models has prompted the present work which concerns *short-wave dynamics*. Nonlinear asymptotic dynamics of *monochromatic* short waves has hardly been studied and only a few results are known. Hence, this work has two purposes: in the first part, we carry out a nonlinear and theoretical analysis of short-wave dynamics in a special reduction of the Euler equation and in the second part, we study its mathematical properties.

Apart from the conclusion, the rest of this paper is divided into two main sections: §§2 and 3. These sections are structured as follows. In §2*a*, we introduce the important issue of short-wave propagation in some long-wave model equations. In §§2*b*(i) and 2*b*(ii), we derive an integrated form of the Euler equations with surface tension. It should be valid only for long waves. Nevertheless, even if the model is derived in a limit where large scales dominate, the existence of short waves related to the local nature of the surface tension can be neither eliminated nor ignored. From the analysis of the linear dispersion relation, we detected that the model can propagate short waves. On this ground, we attack the problem of nonlinear short surface waves via a multiple-time perturbative method and in §2*c* we derive an asymptotic nonlinear model equation for short capillary-gravity waves.

Section 3 is mainly concerned with some mathematical aspects of the integrated Euler equations and the asymptotic model. Section 3*a* studies the solitary-wave solution of the integrated Euler equations. In §3*b*, we examine the asymptotic model analytically and we show that it is completely integrable. A connection between our model and other completely integrable models appearing in rather different physical and mathematical contexts is established. We make a simple numerical study of some particular solutions. It shows that in spite of integrability these solutions develop singularities and multivaluedness in finite time.

## 2. Modelling short-wave dynamics

### (a) Short-wave propagation in long-wave model equations

The propagation of surface waves in an ideal incompressible fluid is a classical subject of investigation in fluid dynamics. The complete surface water-wave problem cannot be solved analytically. A widely known approximation is the *shallow water theory*, which has produced several long waves in shallow water model equations: the KdV (Whitham 1974) and the modified KdV equations (Dodd *et al*. 1982); the various versions of the Boussinesq equations (Infeld & Rowlands 1990); the Benjamin–Bona–Mahony–Peregrine equation (Peregrine 1967; Benjamin *et al*. 1972); the regularized wave equation (Peregrine 1966); the Camassa–Holm (Fuchssteiner & Fokas 1981; Camassa & Holm 1993); the Degasperis–Procesi equation (Degasperis & Procesi 1999); and many others.

Practically, all these models possess a linear dispersive mechanism such that long waves propagate very well but short waves do not propagate at all. For example, the equations belonging to the KdV or to the modified KdV families have linear dispersion relations which amplify short waves. Some of the Boussinesq-type equations have complex phase velocities for short waves.

Nevertheless, some of these models may also propagate short waves at least at the linear level. Why does this phenomenon occur? Surface water waves are the final result of an enormous number of propagating wave components, including short, intermediate and long wavelengths. However, standard methods of derivation of these long-wave models, from the Euler equation, do not guarantee the complete elimination of short waves. For example, a version of the Boussinesq equations, the Benjamin–Bona–Mahony–Peregrine, the regularized wave equations, the Camassa–Holm equation and the Degasperis–Procesi equation are typical long-wave models capable of propagate short waves.

If short waves can propagate linearly, then they may propagate nonlinearly too. So suppose we perform numerical simulations of long-wave dynamics in a model also capable of propagating short waves. If short waves are excited and propagated the final dynamics must be a nonlinear superposition of short and long scales. Moreover, if short waves are unstable the entire model must be contaminated, and the final dynamics will generically show instability.

For this reason, this problem was widely studied (Dingemans 1973; Broer *et al*. 1976; Van Der Houwen *et al*. 1991; Katopodes *et al*. 1998). Particularly, the Boussinesq-type equations and the regularized wave equation were studied with great details in Katopodes *et al*. (1998). However, the numerical experiments performed with these models were only compared with linear theoretical estimates.

### (b) The integrated Euler equations

In this section, we introduce a long-wave reduction of the Euler equations which allows short-wave propagation. We exhibit two derivations: a direct approach *à la Green–Naghdi* (Green *et al*. 1974; Green & Naghdi 1976) and a variational approach in §2*b*(ii).

#### (i) Direct approach

We consider the rotational Euler equations. Let the particles of the fluid medium be identified by a fixed rectangular Cartesian system of centre *O* and axes (*x*, *y*, *z*) with *Oz* the upward vertical direction. We assume translational symmetry in *y* and we will only consider a sheet of fluid in the *xz* plane. This fluid sheet is moving in a domain with a rigid bottom at *z*=0 and an upper free surface at *z*=*S*(*x*, *t*). The velocity vector is (*u*, *w*). The continuity equation is(2.1)and the Newton equations (in the flow domain) are(2.2)(2.3)where *p*^{*}(*x*, *z*, *t*) is the pressure, *σ* is the density and *g* is the gravitation. The kinematic and dynamic boundary conditions are(2.4)(2.5)(2.6)where *T* is the surface tension.

Shallow water theories make asymptotic expansions directly in these Euler equations of motion. So, the velocities *u* and *w*, the pressure *p*, etc. … are handled perturbatively. In our approach, instead of studying the entire problem via a perturbation theory, we will study the nonlinear evolution of a given initial Ansatz for the velocity field. We assume that *u* is independent of *z*, so(2.7)This *a priori* given velocity profile can be justified from linear theoretical arguments or better still from direct visualization of the particle trajectories of a plane periodic wave in reasonably shallow water (Van Dyke 1997). This procedure is known as the *columnar-flow Ansatz* and has been introduced long ago by Green & Naghdi, not in this form but in the rather different framework of Cosserat surface theory (Green *et al*. 1974; Green & Naghdi 1976).

From equation (2.1), the Ansatz equation (2.7) and the boundary condition (2.4) at the bottom, we have(2.8)The integration of (2.2) over *z* from 0 to *S*(*x*, *t*) using equation (2.6) gives(2.9)where the integrated pressure *p*(*x*, *t*) is given by(2.10)Now we multiply equation (2.3) by *z* and integrate over *z* to obtain(2.11)

The elimination of *p* between equations (2.9) and (2.11) gives(2.12)Finally, equation (2.5) using equation (2.8) yields to(2.13)Equations (2.12) and (2.13) are the Green–Naghdi equations with surface tension. They are a reduction of the Euler equations in a thin domain.

#### (ii) Variational approach

At this point, it may be worth exhibiting another derivation of these equations using a variational approach for the Lagrangian of inviscid incompressible flow. In the bulk, this Lagrangian is simply given by the kinetic energy of the fluid, together with a Lagrange multiplier term to enforce incompressibility(2.14)while at the surface, it is just the gravitational potential energy, the surface tension energy and another Lagrange multiplier term to enforce the surface dynamic boundary condition(2.15)with *h* the average depth.

It is straightforward (Kim *et al*. 2001) to verify that the equations of motion derived from this Lagrangian are equivalent to the system (2.1)–(2.3), (2.13) and (2.6), and that making the variational Ansatz (2.7) and varying the resulting Lagrangian gives back the system (2.13) and (2.12). The advantage of this approach lies in the power of the variational method, which tends to give excellent numerical results. For example, we have tried the Ansatz(2.16)which generalizes equation (2.7). When this Ansatz is used with the Lagrangian (2.14) and (2.15), one obtains generalized Green–Nagdhi equations by varying with respect to *u*_{n}(*x*, *t*). In the linearized approximation, these equations replace the hyperbolic tangent in the well-known exact dispersion law(2.17)with the (*N*, *N*+1) Padé approximant reproducing the power series expansion of the hyperbolic tangent up to order *k*^{4N+2} included. This should imply remarkable numerical convergence properties also in the nonlinear regime as *N* becomes large, the study of which goes beyond the scope of this paper.

### (c) The asymptotic model

The system (2.12) and (2.13) for *S*(*x*, *t*) and *u*(*x*, *t*) is the result of a first reduction of the Euler equations via the columnar-flow hypothesis. In order to carry out the perturbative theory, it is best to consider the associated non-dimensional system. We adimensionalize the original variables *S*, *u*, *x* and *t* according to(2.18)Hence, the system (2.13) and (2.12) becomes(2.19)(2.20)where *θ*=(*T*/*σh*^{2}*g*) is the dimensionless Bond number. The system (2.19) and (2.20) incorporates finite dispersion both in the long-wave and in the short-wave limits. Indeed, the linear dispersion relation(2.21)is well behaved for *k*→0 (long waves) or *k*→∞ (short waves). In the long-wave case, we obtain the KdV equation. Here we are looking for a short-wave limit. To define a short wave (wavelength *l*, wave-number *k*=2*π*/*l*), one needs to compare *l* to an underlying space scale. We use the unperturbed depth *h* as the natural reference, and thus consider(2.22)where *ϵ*≪1 will be used as the parameter of the asymptotic expansions. Therefore, we need to introduce two appropriate variables: one space variable *ζ* describing a local pattern and a time variable *τ* measuring a large time. These are worked out by means of the definition(2.23)(2.24)Thereby *ζ* and *τ* are order one when *x* is very small and *t* is very large, so that they are appropriate variables to describe short waves asymptotically in time.

The definitions (2.23) and (2.24) will have a meaning if they can be introduced as a change of variables compatible with the progressive-wave solution of frequency given by equation (2.21) in the short-wave limit *k*→∞. This is the case because we have for *k*=1/*ϵ*→∞(2.25)Now the new variables (2.23) and (2.24) define the operators(2.26)(2.27)The nonlinear dynamics of small short capillary-gravity waves can be isolated from the system (2.19) and (2.20) by perturbation in *ϵ* and using equations (2.26) and (2.27). To do this, let us consider the expansions (every term *u*_{n}, *H*_{n}, *n*=0, 1, … goes to 0 for *ζ*→∞)(2.28)(2.29)We obtain at orders 1/*ϵ*^{3} and 1/*ϵ* from equations (2.19) and (2.20) the system(2.30)(2.31)The solution is(2.32)At orders 1/*ϵ* and *ϵ*^{0}, we obtain the system(2.33)

(2.34)

Using (2.30), (2.31), (2.33) and (2.34), we finally obtain(2.35)In the laboratory variables, it reads(2.36)with *v*_{p}=(3*T*/*σh*)^{1/2} and *u*(*x*, *t*) the fluid velocity at the surface.

## 3. Mathematical properties

### (a) Solitary-wave solution of the integrated Euler equations

As we will see below, equation (2.36), as confirmed by the equivalence with the sine-Gordon equation, exhibits solitons and solutions for which large enough amplitudes become singular. Therefore, it would be interesting to investigate the existence of, presumably, not solitons, but solitary waves for the full system (2.12) and (2.13) with surface tension and determine whether there is a domain in the space of parameters where they become singular. To do so, we first have to consider solutions propagating at a constant velocity *v*, which we choose to be positive; that is, solutions depending on the variable *x*−*vt* only. Using ∂_{t}=−*v∂*_{x} and setting *t*=0, one can write equations (2.13) and (2.12) in terms of *x*(3.1)

(3.2)

Next, expressing *u*−*v* in terms of *S* with equation (3.2) with the boundary condition *S*=*h* at infinity, replacing it in equation (3.1) and integrating twice, we obtain an equation for *S* and *S*_{x} which reads(3.3)

We look for solutions that tend to *h* exponentially at infinity, so we assume *S*∼*h*e^{a(x)}, *a*→0 and check for what values of the parameters the function *a* is real. Expanding *S* as *h*(1+*a*(*x*)+(1/2)*a*^{2}(*x*)) equation (3.3) becomes(3.4)

The condition for *a* to be real is, therefore,(3.5)

Eventually, we deduce that there can exist solitary-wave type solutions for(3.6)

The next point is whether some of these solutions are singular and, if so, in which subset of equation (3.6). Considering equation (3.3) as a sort of complicated anharmonic oscillator equation for *S* with *x* as the time variable, we can plot the potential energy *E*_{p}, defined by , for a set of values of *θ* and *v* and study the behaviour of the corresponding solutions. The equation to plot is(3.7)

The graphs corresponding to the domains (i)–(iv) can be found in appendix B, together with a discussion of the shapes of the corresponding solitary waves. We note that the *depression* solitary wave of case (i) (‘strong’ surface tension) has been observed experimentally in mercury (Falcon *et al*. 2002).

### (b) Lax pair and finite-time singularities

Let us consider now the asymptotic model (2.36). After appropriate rescalings of the variables, one can bring equation (2.36) into the form(3.8)where *γ* being expressed in terms of the physical parameters of equation (2.36). The corresponding Lagrangian is(3.9)

Equation (3.8) is integrable with Lax pair (in usual notations)(3.10)(3.11)where *σ* are the usual Pauli matrices, *λ* the ‘eigenvalue’ and(3.12)One of its most remarkable properties is that with *F*, one builds the first non-trivial conserved quantity for all *γ*(3.13)and through the change of function from *u*(*x*, *t*) to(3.14)with(3.15)one finds that *v* satisfies the sinh-Gordon equation in light-cone coordinates(3.16)

We give more details on this important transformation in appendix A. This is valid for *γ*<1 and for small *u*_{xx} so that *F* is real. If *u*_{xx} is large, a similar change leads to the cosh-Gordon equation, and if *γ*>1 one obtains the sine-Gordon equation in light-cone coordinates(3.17)

Whatever the value of *γ*, it follows from the change of variables (*x*, *t*)↔(*y*, *t*) that a regular *v*(*y*, *t*) can give back a *singular*, *multivalued u*(*y*, *t*) if the change from *y* back to *x* is not one-to-one. This happens when |*v*| is large enough, forcing *u*_{xx} to infinity and a change of sign of *F* in the equation for *y*. We give an example of this in figure 1, where two solutions for *u* are plotted from breather solutions of the sine-Gordon equation, one (dashed curve) with an amplitude just below the singularity threshold, so that *u* is still regular and single-valued, the other (solid curve) with an amplitude above the singularity threshold, which displays a swallowtail behaviour.

Furthermore, this singularity and multivaluedness can be reached in finite time from a regular solution. In particular, in the sine-Gordon case, one can start from a solution *v*(*y*, *t*) consisting of two breathers far enough from each other and each weak enough so that the change *y*→*x* is single valued, but strong enough so that when they overlap, |*v*| becomes large enough for the singularity to appear. In the case *γ*=1, and even in the sinh-Gordon case, one can replace the breathers with wave packets and make them collide and give rise to this singularity before they disperse. In all these cases, the singularity in *u* is a swallowtail just like in figure 1.

The (singular, multivalued) *u*(*x*, *t*) corresponding to the sine-Gordon soliton for *v*(*y*, *t*) is also interesting. It is displayed on figure 2 for the particular value *γ*=10/9.

For *γ*=0, equation (3.8) was already discussed in Cewen (1990), Hunter & Zheng (1994), Alber *et al*. (1995, 1999, 2001) and it contains peakons. These peakons, which are solitons, and their scattering are qualitatively very easy to study *via* the change of variables to the sinh-Gordon equation, where they correspond to singular solutions obtained by simple analytic continuations of the sine-Gordon multisoliton solutions. We note that a connection between the high-frequency limit of the Camassa–Holm equation and the sinh-Gordon equation was established in Dai & Pavlov (1998)

## 4. Conclusion

The starting point of this paper was to investigate the dynamics of short waves in a long-wave model. Our study has shown that if in a given initial perturbation of the system (2.12) and (2.13) short waves are present, they must propagate in a way governed by the integrable equation (2.36), which in some cases show singular and multivalued solutions. Simultaneity of complete integrability and formation of singularities in finite time are rather unusual and not well understood (Hunter & Zheng 1994). This study was carried out independently of the long-wave modes even if we know that the integrated Euler equations (2.12) and (2.13) lead to KdV in the long-wave limit. At present, we lack the tools needed to take into account both scales together. This is an important open problem and some progress was made in Manna & Merle (1998) and Kraenkel *et al*. (1999).

In a future paper we will analyse more thoroughly the implication of the Lorentz invariance of equation (2.36) and its relationship with sine- and sinh-Gordon together with a rather wide class of generalizations which all display similar types of singular behaviours and intriguing equivalences with more standard models.

The system (2.12) and (2.13), and consequently the asymptotic model (2.36) were derived without taking into account viscosity. However, viscosity cannot always be neglected in real fluids and this certainly affects short-wave dynamics and alters not only the singular behaviour of the solutions but also integrability. Viscosity, implying second-order spatial derivatives like those appearing in our short-wave equations, might tend to suppress the formation of the singularities we have found. However, these singularities can be traced back to the fact that the capillary parts of the equation of motion involve higher derivatives also, independently of the fact that we have taken a short-wave limit in an equation initially designed for long waves. Hence, it may be not unreasonable to conjecture that similar singularities will also appear in a more accurate treatment (which would go beyond the scope of this paper), and even in an actual experimental situation where capillary effects are strong enough like in Falcon *et al*. (2002).

## Acknowledgments

This work was supported in part by FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) and Research Training Network grant of the European Commission contract number HPRN-CT-2002-00325. M.A.M. wishes to thank R.A. Kraenkel and J. Léon for stimulating discussions and the Instituto de Física Teórica-UNESP in São Paulo for its hospitality. A.N. is grateful to A.V. Mikhailov and V.I. Zakharov for discussions and to the Newton Institute in Cambridge and the Centre de Recherches sur les Très Basses Températures in Grenoble for their hospitality.

## Footnotes

- Received February 12, 2007.
- Accepted May 4, 2007.

- © 2007 The Royal Society