# On the Formulation of Mass, Momentum and Energy Conservation in the KdV Equation

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## Abstract

The Korteweg-de Vries (KdV) equation is widely recognized as a simple model for unidirectional weakly nonlinear dispersive waves on the surface of a shallow body of fluid. While solutions of the KdV equation describe the shape of the free surface, information about the underlying fluid flow is encoded into the derivation of the equation, and the present article focuses on the formulation of mass, momentum and energy balance laws in the context of the KdV approximation. The densities and the associated fluxes appearing in these balance laws are given in terms of the principal unknown variable *η* representing the deflection of the free surface from rest position. The formulae are validated by comparison with previous work on the steady KdV equation. In particular, the mass flux, total head and momentum flux in the current context are compared to the quantities *Q*, *R* and *S* used in the work of Benjamin and Lighthill (Proc. R. Soc. Lond. A 224:448–460, 1954) on cnoidal waves and undular bores.

### Keywords

KdV equation Surface waves Mechanical balance laws Energy conservation Hydraulic head## 1 Introduction

*η*(

*x*,

*t*) represents the excursion of the free surface,

*h*

_{0}is the undisturbed water depth,

*g*denotes the gravitational acceleration, and \(c_{0}=\sqrt{gh_{0}}\) is the limiting long-wave speed.

The equation arises in the so-called Boussinesq scaling regime where wavelength and wave amplitude are balanced in such a way as to allow the formation of traveling-wave solutions. Denoting by *ℓ* a typical wavelength and by *a* a typical amplitude of the wavefield to be described, the number *α*=*a*/*h*_{0} represents the relative amplitude, and \(\beta= h_{0}^{2} / \ell^{2}\) measures the relative wavenumber. The waves fall into the Boussinesq regime if both *α* and *β* are small, and of similar size. In this case, the KdV-equation arises as a simplified asymptotic model describing the wavemotion. In other words, solutions of the full water-wave problem may be approximated on certain time scales by solutions of the KdV equation. This latter fact can be understood in the sense of asymptotics, but has also been established by mathematical proof by Craig [9] and Schneider and Wayne [24]. Incorporated in the arguments of these works are existence results for the water-wave problem in the context of the full Euler equations in the appropriate scaling. Such results are now also available independently (see Lannes [19] and Wu [28, 29]). In particular, Alvarez-Samaniego and Lannes have obtained long-time existence of solutions of the water-wave problem [3] which can be applied to a number of different scaling regimes. A further significant improvement of the results of [9, 24] was achieved by Bona et al. [6] who proved more refined convergence estimates of solutions of the water-wave problem to a family of long-wave systems as well as to one-directional models such as the KdV equations. Further extensions of this method and applications to other systems can be found for instance in the work of Lannes and Bonneton [21].

One of the early drivers of research relating to the KdV equation was the discovery of elastic overtaking collisions of solitons which in some sense seemed to resemble the dynamics of a linear differential equation. The discovery of this elastic solitary-wave interaction subsequently led to the discovery of an infinite number of time-invariant integrals (Miura [23]), and the development of the inverse-scattering method which can be used to provide exact closed form solutions for a broad class of initial data (Ablowitz and Segur [1], Green et al. [13]).

Apart from being a paradigm for the use of the inverse-scattering method, the KdV equation has been used in a large number of studies in the context of wave problems in fluid dynamics. Various dynamical quantities connected with the KdV equation have appeared in the literature. However, it is difficult to find definitive expressions for and derivations of the most important quantities, such as the energy flux. It is our purpose in the present work to present a framework in which mass, momentum and energy fluxes and densities can be expressed in terms of the principal unknown *η* of equation (1.1)

*t*as soon as it is recognized that the KdV equation can be written in the form

The mass balance (1.3) appears in the literature (see [15]), and one may ask whether it is possible to derive expressions for the momentum and energy densities and fluxes which permit the formulation of balance laws similar to (1.3). This problem has been partially solved in the case of steady solutions of the KdV equation. Indeed, Benjamin and Lighthill [4] used the spatial invariance of the mass flux per unit span *Q*, the momentum flux per unit span corrected for the pressure force *S* and the energy per unit mass *R* in steady Euler flow to develop a method for deriving a time-independent KdV equation which contains the quantities *Q*, *R*, and *S* as parameters.

In the case of the time-dependent problem it seems that most of the work dedicated to questions such as outlined above has focused on the question of conservation of the total energy. In the full water-wave problem, the total energy of the wave system is given by the Hamiltonian functional first recorded by Zakharov [30]. As it represents the total energy of a closed physical system, this Hamiltonian is invariant with respect to time. It was shown by Craig and Groves [10] that if the derivation of simplified evolution equations such as the Boussinesq system and the KdV equation is based on approximating the Hamiltonian function of the water-wave problem, then it is possible to define the total energy of the wave system in the corresponding approximation. This theory is quite satisfactory if the total energy is sought, but it yields no information about other quantities such as energy flux or momentum flux.

The derivations presented in the present work are formal, and there is no rigorous mathematical proof of the convergence of these approximations as the small parameters *α* and *β* approach zero. The main advancement of the present work is the identification of the expressions which satisfy the balance laws (1.5), (1.6) and (1.7), and the comparison with previous asymptotic results. While a proof of the validity of (1.5), (1.6) and (1.7) might proceed along the lines of the proofs of the validity of the KdV equation as a water-wave model, as shown in [6, 9, 24], such a study is beyond the scope of the present article.

Before we leave the introduction, let us mention some further related work. In the context of steady solutions one may exploit the conservation of mass, momentum and energy in the water-wave problem. Examples are the spatial Hamiltonian approach advocated by Bridges [8], and the work of Longuet-Higgins and Fenton [22] on the solitary wave. A study which is closer to the spirit of the present article is provided by Dutykh and Dias [12] who supplemented a Boussinesq system with an energy equation which yields information about an energy density similar to the quantity appearing in Sect. 5. The present work is related to a recent study of mechanical balance laws in a family of Boussinesq systems by Ali and Kalisch [2], and a much earlier attempt to record similar quantities for the single Boussinesq equation by Keulegan an Patterson [17].

We should also mention the concept of wave action conservation which yields an additional conservation law in the water-wave problem which can be used in the case of non-uniform environments, such as background currents and stratifications. This principle which was pioneered by Whitham [26] and Hayes [16] is based on a Lagrangian description of the problem, and can also be applied in the context of model equations (Grimshaw [14], Whitham [27]).

## 2 Velocity Field and Pressure

The main aim of this section is to establish expressions for the velocity field and pressure in the fluid in terms of the surface excursion *η*. These expressions are well known by-products of the derivation of the KdV equation. Nevertheless, it will be convenient to give a brief review of this derivation in order to fix ideas regarding the geometric setup and the notation.

*x*-axis with the undisturbed free surface, and suppose the fluid domain extends along the entire

*x*-axis. It is assumed that the fluid is inviscid, incompressible and of unit density, the bottom of the channel is flat, and that wave motion transverse to the

*x*-axis can be neglected. The geometric setup is illustrated in Fig. 1. The surface water-wave problem is generally described by the Euler equations with slip conditions at the bottom, and kinematic and dynamic boundary conditions at the free surface. The unknowns are the surface excursion

*η*(

*x*,

*t*), the pressure

*P*(

*x*,

*z*,

*t*), and the horizontal and vertical fluid velocities

*u*

_{1}(

*x*,

*z*,

*t*) and

*u*

_{2}(

*x*,

*z*,

*t*), respectively. With the setup described above, the problem may be posed on a domain \(\{(x,z) \in{\mathbb{R}}^{2} | -h_{0} < z < \eta(x,t) \}\) which extends to infinity in the positive and negative

*x*-direction. On this domain, the two-dimensional Euler equations are

**u**=(

*u*

_{1},

*u*

_{2}) represents the velocity field, and

**g**=(0,−

*g*) is the body forcing. As surface tension effects are neglected, the dynamic free-surface boundary condition calls for the fluid pressure at the surface to be equal to atmospheric pressure. In addition, the kinematic condition requires the normal velocity of the free surface to be equal to the fluid velocity normal to the surface.

*ϕ*on the unknown time-dependent domain. The surface boundary conditions are then given by

*ϕ*in an asymptotic series is employed. Using the Laplace equation and the boundary condition at the flat bottom shows that the velocity potential takes the form

*A*and

*B*can be found by substituting (2.8) into (2.7). Requiring both equations in (2.7) to yield the same equation for \(\tilde{\eta}\), and using the first-order equivalence

*F*is a polynomial in \(\tilde{\eta}\) and its derivatives, leads to

*α*and

*β*, and reverting to dimensional variables, the KdV equation (1.1) appears in the case of waves propagating mainly to the right. A corresponding equation with different signs appears for waves propagating mainly to the left. It will be convenient later for purposes of comparison to have available the above expressions in the case of a moving reference frame. If the problem is put into a reference frame moving at a velocity

*U*, and the non-dimensionalization \(U = c_{0} \tilde{U}\) is chosen, then the KdV equation and the expression for the horizontal velocity appear as

*U*, and the scaling of

*U*reflects the most relevant case

*U*=±

*c*

_{0}. However, the limiting long-wave speed

*c*

_{0}is far greater than the horizontal velocity of any fluid particle if the amplitude of the surface waves is small. As a consequence, the velocity of the moving reference frame enters the expression (2.14) for the horizontal velocity at the order \(\frac{1}{\alpha}\).

*P*′ can be written with the help of the Bernoulli equation in the form

## 3 Mass Conservation

*U*. Using the incompressibility of the fluid, mass conservation is stated in differential form by (2.2). Using this equation and the kinematic boundary condition (2.3), one can immediately derive the relation

*α*to find

Unlike the KdV equation (2.10) or the formula for the horizontal velocity (2.11), the mass flux contains terms of quadratic order in *α* and *β*. However, these terms are necessary since the mass balance equation (1.5) holds to the same order as the evolution equation (2.10). Note also that the differential mass balance equation (3.1) is the same as the non-dimensional KdV equation (2.13), and if terms of order *α*^{2}, *αβ* and *β*^{2} are disregarded, the KdV equation is a mass balance equation. In other words, in the approximation which leads to the KdV equation, mass is exactly conserved.

## 4 Momentum Balance

*ϕ*as

## 5 Energy Balance

*R*used in Benjamin and Lighthill [4], we record that non-dimensional energy per unit mass is given by

*g*, the energy density at the surface corresponding to the quantity

*R*used in Benjamin and Lighthill [4] appears in dimensional form as

*α*

^{2}can be omitted, and the differential energy balance equation is

One may wonder whether there is a relation between the total energy of the surface wave system and the invariant integrals of the KdV equation (1.2). The formula for *E*^{∗} contains terms that look as though they might combine to a such a conservation law, but the coefficients do not line up in quite the right way. Note that the last computation was done in an fixed frame of reference in order to reach tidier expressions, and to stay in line with the initial requirement that the energy of the quiescent state be zero. As it turns out, it is possible to normalize the potential energy in such a way that the total energy in a reference frame moving at the speed *U*=*c*_{0} is given by a combination of the conservation laws (1.2). This issue will be addressed in Sect. 7.

## 6 Comparison with *Q*, *R* and *S*

*q*

_{M},

*q*

_{I}and

*gH*derived in the previous sections are compared to the corresponding quantities

*Q*,

*S*and

*R*, studied by Benjamin and Lighthill [4]. We take a periodic traveling wave propagating to the left at a speed

*c*>0 in an inertial frame. In a reference frame also moving to the left at the velocity

*U*=−

*c*, the wave becomes steady, and yields a positive mass flux. The surface excursion can be described by a function

*ζ*(

*x*), and equation (1.1) reads

*ζ*′, and then integrating again leads to

*cn*with modulus \(m=\frac{\zeta_{1}-\zeta_{2}}{\zeta_{1}-\zeta_{3}}\). The numbers

*ζ*

_{1},

*ζ*

_{2}and

*ζ*

_{3}are the three roots of the cubic polynomial appearing in (6.1), arranged in the order

*ζ*

_{3}<

*ζ*

_{2}<

*ζ*

_{1}. The constants of integration in (6.1) can be written in terms of the roots as \(\mathcal{A}= g(\zeta_{1}\zeta_{2} + \zeta_{1}\zeta_{3} +\zeta_{1}\zeta _{2})\) and \(\mathcal{B}=g \zeta_{1} \zeta_{2} \zeta_{3}\). The wavespeed is given by

*K*(

*m*) is the complete elliptic integral of the first kind. In the current setup,

*ζ*

_{1}represents the wave crest,

*ζ*

_{2}is the wave trough, and

*ζ*

_{3}is a parameter which has influence only on the wavelength

*λ*and wavespeed

*c*. In the traveling reference frame, the quantities

*M*,

*q*

_{M},

*I*,

*q*

_{I},

*E*and

*q*

_{E}can now be computed as functions of

*x*, and these are plotted in Fig. 2 for a particular case. Note that the mass flux

*q*

_{M}is constant since the KdV equation features exact mass conservation. Moreover, the momentum density

*I*is also constant since it is equal to the mass flux. The momentum flux is nearly constant, but features small variations which are visible when plotted at a finer scale.

*q*

_{M},

*q*

_{I}and

*gH*defined for the evolution problem with the corresponding quantities

*Q*,

*S*and

*R*defined for the steady problem. In order to facilitate the comparison, let us briefly recall the development presented by Benjamin and Lighthill [4]. Steady periodic traveling waves are considered in a moving reference frame, in which the mass flux

*Q*is positive, and the momentum flux

*S*and energy per unit mass

*R*are also given. In this case, the steady KdV equation appears as

*ξ*is the total flow depth. Using the same method as above, the solution is found as

*Q*,

*R*and

*S*are given by

*ξ*

_{3}<

*ξ*

_{2}<

*ξ*

_{1}are the roots of the cubic polynomial

*gξ*

^{3}−2

*Rξ*

^{2}+2

*Sξ*−

*Q*

^{2}.

*ζ*represents the deflection of the fluid surface from rest while

*ξ*is the total flow depth, the solutions of (6.1) and (6.5) must be related by

*h*

_{0}=(

*Q*

^{2}/

*g*)

^{1/3}, and the total head

*R*can be expressed in terms of the wavespeed as

*ξ*

_{1},

*ξ*

_{2}, and

*ξ*

_{3}and calculate

*Q*,

*S*and

*R*from (6.6), and then use (6.7) and (6.2) to compute the corresponding values of

*q*

_{M}(

*ζ*),

*q*

_{I}(

*ζ*) and

*gH*(

*ζ*) as defined in the previous sections. For example, in Fig. 3,

*ξ*

_{1}=1.4 m ,

*ξ*

_{2}=1 m and

*ξ*

_{3}=0.95 m are chosen which give

*h*

_{0}=1.1 m,

*ζ*

_{1}=0.3 m,

*ζ*

_{2}=−0.1 m and

*ζ*

_{3}=−0.15 m. The wavelength is

*λ*=10.04 m, and the wave amplitude is

*a*=0.2 m. Besides showing the wave profile, Fig. 3 features a comparison of the quantities

*q*

_{M},

*q*

_{I}and

*gH*as defined by (3.2), (4.1) and (5.1), respectively with the corresponding parameters

*Q*,

*S*and

*R*. As can be seen, the difference between

*Q*and

*q*

_{M}(

*ζ*), the difference between

*S*and

*q*

_{I}(

*ζ*), and also the difference between

*R*and

*gH*(

*ζ*) are all reasonably small. In order to further quantify these differences, waves with various combinations of the parameters

*ξ*

_{1},

*ξ*

_{2}, and

*ξ*

_{3}are computed, and the differences in the above quantities are plotted as functions of the two small parameters \(\alpha= \frac{a}{h_{0}}\) and \(\beta = \frac {h_{0}^{2}}{\lambda^{2}}\). For comparison of different values of

*α*and

*β*, it appears most convenient to keep the water depth

*h*

_{0}constant. Then the mass flux

*Q*=

*c*

_{0}

*h*

_{0}is also constant. The first equation in (6.6) implies that \(\xi_{3}= \frac {h_{0}^{3}}{\xi_{1} \xi_{2}}\) so the problem now depends only on the wave crest

*ξ*

_{1}and the wave trough

*ξ*

_{2}. The condition

*ξ*

_{3}<

*ξ*

_{2}<

*ξ*

_{1}leads to

*ξ*

_{1}>

*h*

_{0}) and the wave trough is bounded below by \(\xi_{2} > h_{0}\sqrt{\frac{h_{0}}{\xi_{1}}}\). Consequently, the total head

*R*will be restricted as \(\frac{3g}{2} \frac{h_{0}^{3}}{\xi_{1}^{2}} < R < \frac{3g}{2} \xi_{1}\), while the momentum flux will be bounded as \(\frac{3g}{2} \frac{h_{0}^{6}}{\xi_{1}^{4}} < S < \frac{3g}{2} \xi_{1}^{2}\).

*q*

_{M}and

*Q*. The center panels of Fig. 4 show the error between

*q*

_{I}and

*S*. The lower panels of Fig. 4 show the error between

*gH*and

*R*. These errors are plotted as level curves with respect to the small parameters

*α*and

*β*with

*Q*held fixed. It can be seen clearly in all cases that the error diminishes with decreasing values of

*α*and

*β*.

## 7 Exact Conservation

In the following, we address exact conservation of mass, momentum and energy in KdV evolution. It is assumed that solutions are smooth, and that the wave motion is localized in the sense that the function *η*(*x*,*t*) describing the free surface decays rapidly enough as *x*→±∞ so that all integrals appearing here are defined. First of all, total mass can only be defined on a finite interval, so that the most preferable form to state mass conservation is (1.3). However, one may define excess mass by \(\int_{-\infty}^{\infty} \eta\, dx\), and it clearly follows from (1.3) that this quantity is constant with respect to *t*.

*U*=0, and for a localized surface disturbance, the total horizontal momentum

*U*=

*c*

_{0}, so that \(\tilde{U}= 1\). In this particular case, the energy density is given in dimensional form by

*c*

_{0}is

## 8 Conclusion

In this article, expressions for mass, momentum and energy densities and fluxes which are valid in the KdV approximation have been found. The quantities have been compared to the quantities *Q*, *R* and *S* which were previously derived by Benjamin and Lighthill in the steady case [4]. It has also been shown that exact conservation of total mass, momentum and energy holds in special cases. For the exact conservation, the mathematical formulations of the first three conservation laws (1.2) have been used. The main result of the paper is the identification of the quantities *M*, *q*_{M}, *I*, *q*_{I}, *E* and *q*_{E} in the context of the KdV approximation. However, as already mentioned in the introduction, the method used in this paper is a formal one, and the results presented here should be understood as a first step towards a mathematical procedure which will give a definite proof that the balances (1.5), (1.6) and (1.7) are valid to the same order and over the same time scales as the KdV equation (1.4) itself. In order to provide such a proof one might follow the procedure pioneered by Craig [9], and recently refined by Bona et al. [6]. In this latter work, the Hamiltonian formulation of Zakharov and a careful analysis of the Dirichlet-Neumann operator, such as defined by Craig and Sulem [11] play a prominent role. The general procedure has been further extended and applied to the justification of a variety of simplified model equations and systems, and the recent monograph by Lannes [20] contains a large variety of different cases.

While the methods for a mathematical justification of the derivation of many model equations are available, it is not entirely clear how to apply them to the justification of the associated balance laws treated in the present article. Such a study will be an interesting topic for future work.

## Notes

### Acknowledgements

This research was supported in part by the Research Council of Norway.

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