## Abstract

The two-dimensional Green–Naghdi (GN) shallow-water model for surface gravity waves is extended to incorporate the arbitrary higher-order dispersive effects. This can be achieved by developing a novel asymptotic analysis applied to the basic nonlinear water wave problem. The linear dispersion relation for the extended GN system is then explored in detail. In particular, we use its characteristics to discuss the well-posedness of the linearized problem. As illustrative examples of approximate model equations, we derive a higher-order model that is accurate to the fourth power of the dispersion parameter in the case of a flat bottom topography, and address the related issues such as the linear dispersion relation, conservation laws and the pressure distribution at the fluid bottom on the basis of this model. The original Green–Naghdi (GN) model is then briefly described in the case of an uneven bottom topography. Subsequently, the extended GN system presented here is shown to have the same Hamiltonian structure as that of the original GN system. Last, we demonstrate that Zakharov's Hamiltonian formulation of surface gravity waves is equivalent to that of the extended GN system by rewriting the former system in terms of the momentum density instead of the velocity potential at the free surface.

## 1. Introduction

In a recent paper [1], we extended the Green–Naghdi (GN) shallow-water model equations to incorporate the arbitrary higher-order dispersive effects while preserving the full nonlinearity. The system of equations thus obtained is the generalization of the model equations derived first by Serre [2] and later by Su & Gardner [3] to describe the one-dimensional propagation of fully nonlinear and weakly dispersive surface gravity waves. We showed that the extended GN system has the same Hamiltonian structure as that of the original GN system, and it is equivalent to Zakharov's Hamiltonian formulation of surface gravity waves. Green & Naghdi [4] were the first to generalize Serre's system to the two-dimensional case while taking into account the effect of an uneven bottom topography. The same model was obtained later by Miles & Salmon [5] through Hamilton's principle. Bazdenkov *et al.* [6] derived the two-dimensional GN system in the study of large-scale flows in planetary atmospheres and oceans. Subsequently, Holm [7] demonstrated that it permits a Hamiltonian formulation by introducing an appropriate non-canonical Lie–Poisson bracket. Recently, Lannes & Bonneton [8] developed a systematic derivation of the various asymptotic model equations including the two-dimensional GN equations. They performed an asymptotic analysis of the Dirichlet–Neumann operator that originates from the formulation of the water wave problem by Zakharov [9] and Craig & Sulem [10]. See also Lannes's monograph [11] which details the derivation of various asymptotic model equations as well as their mathematical analysis. In particular, the well-posedness results of the GN models are presented in section 6 with some references on this topic.

The purpose of this paper is to generalize the one-dimensional extended GN system mentioned above to the two-dimensional system by making use of a novel asymptotic analysis, and show that it has the same Hamiltonian structure as that of the original two-dimensional GN system. There are several new results in the current two-dimensional extension. Among them, a highlight is an analysis of the linear dispersion relation for the extended GN equations which makes it possible to explore the well-posedness of the linearized model equations. From the technical point of view, the one-dimensional analysis can also be applied to the two-dimensional case. However, the computation involved becomes very complicated especially in the vector formulation.

We consider the three-dimensional irrotational flow of an incompressible and inviscid fluid of variable depth. The effect of surface tension is neglected because it has no appreciable influence on the current water wave phenomena. It can be, however, incorporated in our formulation without difficulty. The governing equation of the water wave problem is given in terms of the dimensionless variables by
*ϕ*=*ϕ*(** x**,

*z*,

*t*) is the velocity potential with

**=(**

*x**x*,

*y*) being a vector in the horizontal plane and

*z*the vertical coordinate pointing upwards, ∇=(∂/∂

*x*,∂/∂

*y*) is the two-dimensional gradient operator,

*η*=

*η*(

**,**

*x**t*) is the profile of the free surface,

*b*=

*b*(

**) specifies the bottom topography, and the subscripts**

*x**z*and

*t*appended to

*ϕ*and

*η*denote partial differentiations. The notation ∇

*ϕ*⋅∇

*η*in (1.2) represents the scalar product between the two-dimensional vectors ∇

*ϕ*and ∇

*η*. The equation

*η*=0 represents the still water level. The boundary conditions (1.5) stem from the assumption that the fluid is at rest at infinity. We also assume that the total depth

*h*=

*h*(

**,**

*x**t*) of the fluid given by

*h*=1+

*ϵη*−

*βb*never vanishes. More precisely,

*h*is bounded from below by a positive constant.

The dimensional quantities, with tildes, are related to the corresponding dimensionless ones by the relations *l*, *h*_{0}, *a* and *b*_{0} denote a characteristic wavelength, water depth, wave amplitude and bottom profile, respectively. *g* is the acceleration owing to the gravity, and *ϵ* characterizes the magnitude of nonlinearity, whereas the dispersion parameter *δ* characterizes the dispersion or shallowness, and the parameter *β* measures the variation of the bottom topography. What is meant by ‘full nonlinearity’ is that no restriction is imposed on the magnitude of *ϵ*. Actually, *ϵ* is assumed to be of order 1 in our analysis. On the other hand, we impose the condition *δ*≪1 for the dispersion parameter that features the shallow-water model equations. Note that the scaling *ϵ*/*δ*^{2}=*O*(1) with *ϵ*,*δ*≪1 leads to the classical Boussinesq-type equations [12], whereas the scaling *ϵ*/*δ*=*O*(1) with *ϵ*,*δ*≪1 yields the Camassa–Holm and Degasperis–Procesi equations [13,14]. A large number of works have been devoted to extend the Boussinesq-type equations by including both the higher-order dispersive and nonlinear effects. See the review papers written by Kirby [15] and Madsen & Schäffer [16,17], for example.

This paper is organized as follows. In §2, we reformulate the water wave problem posed by equations (1.1)–(1.5) in terms of the total depth of fluid *h* and the depth-averaged horizontal velocity *h* and *h* and an infinite-order Boussinesq-type equation for *δ*^{2n}, we obtain the extended GN equations which are accurate to *δ*^{2n}, where *n* is an arbitrary positive integer. We call it the *δ*^{2n} model hereafter. The lowest-order approximation *n*=1 yields the GN equations. We then derive the linear dispersion relation for the extended GN system, and investigate its characteristics in detail. We show that the Taylor-series expansion of the linear dispersion relation for the *δ*^{2n} model coincides with that of the exact linearized water wave system up to order *δ*^{2n}. The relevance of the models in practical applications is also discussed while centring on the well-posedness of the linearized system of equations for the *δ*^{2n} model. In §3, we derive, as illustrative examples, various approximate model equations that include the two-dimensional *δ*^{4} model with a flat bottom topography and the two-dimensional *δ*^{2} model (or the GN model) with an uneven bottom topography. For the former model, the associated quantities such as the linear dispersion relation, conservation laws and the pressure distribution at the fluid bottom are presented. Last, the one-dimensional *δ*^{6} model with a flat bottom topography is briefly described. We emphasize that this model is able to avoid the short wave instability which occurs in the *δ*^{4} model. In §4, we show that the extended GN equations can be formulated in a Hamiltonian form by introducing an appropriate Lie–Poisson bracket as well as the momentum density in place of *δ*^{2n} models with odd *n*. Appendix B provides a derivation of the pressure distribution at the fluid bottom, whereas appendix C gives a proof of the formula associated with the variational derivative of the energy functional.

## 2. Derivation of the extended Green–Naghdi equations

### (a) Extended Green–Naghdi system

The GN model is a system of equations for the total depth of fluid *h* and the depth-averaged (or mean) horizontal velocity ** u**=(

*u*,

*v*) and vertical component

*w*of the surface velocity are given, respectively, by

Let us now derive the equations for *h* and ** u**. Because the procedure of their derivation parallels that of the one-dimensional case [1], we summarize the result. First, we multiply (2.1a) by

*h*and then apply the divergence operator to the resultant expression. This leads, after using (1.1) and (1.4), to

*w*from (2.4) into (1.2) now yields the evolution equation for

*h*=

*h*(

**,**

*x**t*):

To obtain the equation of ** u**, we use the relation that follows from the definition of

*u***:**

*u*Now, we introduce the new quantity ** V** by

**, giving**

*V*We point out that a variant of equation (2.9) has been derived by Witting [18] to improve the one-dimensional Boussinesq-type equations, and used by Dommermuth & Yue [19] to develop an efficient numerical method for modelling gravity waves in the two-dimensional setting. The similar approaches have enabled us to improve the accuracy of the linear dispersion characteristics of the two-dimensional Boussinesq-type equations [15–17]. Quite recently, equation (2.9) was generalized to the equation which is applicable to rotational flows [20]. The latter equation was also derived in the framework of the Lagrangian formulation for the water wave problem [21].

Equation (2.9) represents an exact conservation law for the vector ** V**. To interpret the physical meaning of

**, we introduce the velocity potential evaluated at the free surface**

*V**ψ*is found to be

**is equal to the two-dimensional gradient of the velocity potential evaluated at the free surface, and it lies in the (**

*V**x*,

*y*) plane.

We regard any two-dimensional vector function *z* variable. The rotation operator applied to the vector ** A** is then defined by rot

**=rot (**

*A***,0)**

*A*^{T}=(0,0,

*A*

_{2,x}−

*A*

_{1,y})

^{T}, showing that it has only the

*z*component. The similar definition will be used in §3 for the vector products like

**×**

*A***and**

*B***×(**

*A***×**

*B***) with any vectors**

*C***from (2.12) becomes zero identically, i.e.**

*V*The system of equations (2.5) and (2.7) (or (2.9)) is a consequence deduced from the basic Euler system (1.1)–(1.4). The extended GN equations are obtained if one can express the variables ** u**,

*w*in equation (2.7) in terms of

*h*and

*h*and

*δ*

^{2n}, then equation (2.14) yields the evolution equation for

*δ*

^{2n}

*n*=1 of equation (2.15) coupled with equation (2.5) reduces to the original GN equations. In accordance with this fact, we call the system of equations (2.5) and (2.7) (or (2.9), (2.14)) with

*h*and

### (b) Expressions of the velocities *u*,*w* and *V* in terms of *h* and u ¯

*u*

*V*

#### (i) Flat bottom topography

First, we solve the Laplace equation (1.1) in the case of a flat bottom topography, and express the surface velocity ** u**,

*w*and the velocity

**in terms of the variables**

*V**h*and

*δ*

^{2}≪1 which is relevant to the shallow-water models, the solution of equation (1.1) subjected to the boundary condition (1.4) with

*b*=0 can be written explicitly in the form of an infinite series

*f*=

*f*(

**,**

*x**t*) is the velocity potential at the fluid bottom. See Whitam [12], chapter 13 where the similar formula is presented for the one-dimensional problem. We substitute this expression into (2.1a) and perform the integration with respect to

*z*to obtain

^{2}

*f*=∇ ⋅ (∇

*f*), we can rewrite (2.17) in an alternative form

To derive the expansion of ∇*f* in terms of *h* and *δ*^{2}
*δ*^{2n} (*n*=1,2,…) on both sides, we obtain
*F*_{n} can be solved successively with the initial condition (2.20a), the first two of which read
** u**,

*w*and

**can be derived simply by substituting (2.19) with**

*V*

*F*_{n}from (2.20) and (2.21) into (2.2), (2.3) and (2.12), respectively. We write them up to order

*δ*

^{4}for later use:

*h*and

#### (ii) Uneven bottom topography

The effect of an uneven bottom topography on the propagation characteristics of water waves is prominent in the coastal zone. Here, we provide the formulae of ** u**,

*w*and

**in terms of**

*V**b*. In this case, the solution of the Laplace equation (1.1) subjected to the boundary condition (1.5) can be written in the form

*ϕ*

_{n}are to be determined. See, Yoon & Liu [22] for example, where a similar manipulation has been performed. Substituting (2.25) into equation (1.1), we obtain the recursion relation among

*ϕ*

_{n},

*ϕ*

_{n+1}and

*ϕ*

_{n+2}

*ϕ*

_{0}, i.e. the velocity potential at the fluid bottom which corresponds to

*f*in (2.16). For small

*δ*

^{2}, the first three of

*ϕ*

_{n}are found to be

The depth-averaged horizontal velocity ** u**,

*w*) are expressed in terms of

*ϕ*

_{n}by introducing (2.25) into (2.1)–(2.3). Explicitly, they read

*ϕ*

_{0}in terms of

*b*. The approximate expression that retains the terms of order

*δ*

^{2}is given by

**and**

*u**w*

Last, by making use of (2.33) and (2.34), ** V** from (2.8) is shown to have an approximate expression

### (c) Linear dispersion relation for the extended Green–Naghdi system

Here, we show that the exact linear dispersion relation for the current water wave problem can be derived from the extended GN system, and discuss its structure. We consider the flat bottom case for simplicity. Linearization of equations (2.5) and (2.7) about the uniform state *h*=1 and *η* and *η* from the system of equations (2.36) and obtain the linear wave equation for ** u** is a linear function of

*h*=1 that

*f*is given by (2.19). As inspected from (2.20) and (2.21) with

*h*=1, we can put

*F*

_{n}in the form

*α*

_{n}are unknown constants. Substituting (2.39) into (2.20b) with

*h*=1, we obtain the recursion relation for

*α*

_{n}

**in terms of**

*u*In order to examine the linear dispersion characteristics of equation (2.37) with ** u** from (2.41), we assume the solution of the form

**is the two-dimensional wavenumber vector and**

*k**ω*is the angular frequency. To proceed, we first take the divergence of equation (2.37) and then replace

**by (2.41). Last, we substitute**

*u*from which we obtain

The *δ*^{2n} GN model incorporates the dispersive terms of order *δ*^{2n}. Referring to equations (2.5) and (2.15), one can write it in the form
*D* truncated at order (*kδ*)^{2n}. To detail the dispersion characteristics of this model, we introduce the function *D*_{2n}(*κ*) by
*B*_{r} are Bernoulli's numbers defined by
*D*_{2n} read

In view of the fact that *D*_{2n} is a polynomial of order 2*n* in *κ* and coincides with the Taylor-series expansion of the function *δ*^{2n} model (2.48) is represented by

Using the inequality for the Bernoulli numbers, we can show that *D*_{2n} with odd *n* are positive for all *kδ*. More precisely, they have a lower bound 1. As a result, an estimate *ω*/*k*≤1 for *kδ*≥0 follows. We provide a proof of this statement in appendix A. Consequently, the *δ*^{2n} models with odd *n* have a nice property as long as the linear dispersion characteristic is concerned. Actually, they have smooth dispersion relations without any singularities, and possess an important feature that the exact linear dispersion relation has, i.e. *kδ*≥0. On the other hand, *D*_{2n} models with even *n* exhibit single positive zero. For example, the positive zeros of *D*_{4}, *D*_{8} and *D*_{10} are found to be 4.19,3.63 and 3.33, respectively. An asymptotic analysis shows that the zero of *D*_{2n} with even *n* approaches a constant value *π* as *n* tends to infinity. These results imply that *ω* from (2.51) has a singularity and becomes pure imaginary for values of *kδ* exceeding the zero. It turns out that the *δ*^{2n} models with even *n* exhibit an unphysical dispersion characteristic that leads to the ill-posedness result for the linearized systems of equations, and may cause instabilities in short wave solutions in practical numerical computations. In accordance with these observations, the *δ*^{2n} models with odd *n* may be more tractable as the practical model equations than the *δ*^{2n} models with even *n*. Although this important issue should be investigated further from both theoretical and numerical points of view, we leave it for a future work.

## 3. Approximate model equations

The method developed in §2 enables us to extend systematically the GN equations to include the arbitrary order of dispersion. In this section, we derive, as an illustrative example, an extended GN model (*δ*^{4} model) which is accurate to order *δ*^{4} in the case of a flat bottom topography. Subsequently, we obtain the linear dispersion relation and the conservation laws for the model. We also evaluate the pressure distribution at the fluid bottom.

In the case of an uneven bottom topography, the derivation of the *δ*^{4} model can be performed without difficulty. However, because the expression of the *δ*^{4} model is too complicated to write down, we address only the *δ*^{2} model or GN model, and confirm that it reproduces the corresponding model equation derived by different methods.

### (a) The *δ*^{4} model

#### (i) Derivation of the *δ*^{4} model with a flat bottom topography

For the purpose of deriving the *δ*^{4} model with a flat bottom topography, we need only the evolution equation for *h* is already at hand, as indicated by equation (2.5). The procedure for obtaining the equation for

We start from equation (2.9) for ** V**. The second term on the left-hand side of equation (2.9) is expressed in terms of

*h*and

*δ*

^{4}, we obtain

**from (2.24) into the relation (2.13) to give**

*V*Now, we introduce (2.24) and (3.1) into equation (2.9) and use the expression (3.2). The time differentiation *h*_{t} which stems from *V*_{t} is replaced by *R*_{2} is a vector in the (*x*,*y*) plane. It is important that the second term of *R*_{2} multiplied by *h* can be recast in a conservation form. Namely,
*x*_{1}=*x* and *x*_{2}=*y*.

Various reductions are possible for the *δ*^{4} model. Indeed, if we neglect the *δ*^{4} terms in equation (3.3), then it reduces to the two-dimensional GN system when coupled with equation (2.5)
*δ*^{4} model reduces to the classical two-dimensional Boussinesq system
*ϵδ*^{2} and higher-order terms. On the other hand, the one-dimensional forms of equations (2.5) and (3.3) become

#### (ii) Linear dispersion relation

The system of equations (2.5) and (3.3) linearized about the uniform state *h*=1 and *ω* from (3.9) exhibits a singularity at *kδ*≃4.19. As already discussed in §2c, this shortcoming can be overcome if one employs the *δ*^{6} model, for example. See §3c.

#### (iii) Conservation laws

The *δ*^{4} model derived here exhibits the following four conservation laws:
*F* decreasing rapidly at infinity. The factors *ϵ*^{2} and *ϵ* attached in front of the integral sign in *H* and ** L**, respectively, are only for convenience. The quantities

*M*,

**and**

*P**H*represent the conservation of the mass, momentum and total energy, respectively, which can be confirmed directly by using equations (2.5) and (3.3). The fourth conservation law

**follows from (2.9) and (2.24). The geometrical interpretation of**

*L***has been discussed in detail in the one-dimensional case. See**

*L**Remark 6 of Matsuno [1]. This quantity will appear in §4 in developing the Hamiltonian formulation of the two-dimensional extended GN system.*

#### (iv) Pressure distribution at the fluid bottom

The pressure distribution at the flat bottom in the context of nonlinear shallow-water waves has attracted considerable attention. See Constantin *et al.* [24] and Deconinck *et al.* [25]. Recently, the calculation of the bottom pressure owing to the passage of a large amplitude solitary wave was carried out in the framework of the one-dimensional GN model, and the results were compared with those obtained by the linear theory [26,27]. Here, we present an explicit formula for the bottom pressure *P*_{b} in terms of *h* and

Now, the bottom pressure can be expressed in the form

Various simplifications are possible for the above formula. Among them, we consider the two special cases. The first example is the bottom pressure caused by the passage of a travelling wave of the form *σ*=** k**⋅

**−**

*x**ωt*being the phase variable. Substituting these forms into equation (2.5) and integrating it with respect to

*σ*under the boundary conditions

*P*

_{b}from (3.14) in terms of the total fluid depth

*h*. Actually, replacing the time derivative ∂/∂

*t*and the gradient operator ∇ by −

*ωd*/

*dσ*and

*k**d*/

*dσ*, respectively, and then using (3.15), we find a compact expression for

*P*

_{b}

*σ*.

The second example is the one-dimensional reduction of (3.14). The corresponding formula can be written simply in the form

### (b) The Green–Naghdi model with an uneven bottom topography

In accordance with the method developed in §2, let us derive the GN model which takes into account an uneven bottom topography. Since its derivation is almost parallel to that of the flat bottom case, we describe only the outline.

The expression corresponding to (3.1) follows by using (2.33)–(2.35), giving

Next, introducing (2.35) into the relation rot(** V**)=

**0**from (2.13), we obtain

Last, we substitute (2.35) and (3.18) into equation (2.9) and use the formula (3.19) whereby we replace *h*_{t} by *et al.* [6] and Lannes & Bonneton [8].

### (c) Remark

As already demonstrated in §2c, the *δ*^{2n} models with even *n* have singularities in their linear dispersion relations, although the dispersion characteristics for small values of the dispersion parameter have been improved considerably when compared with those of the original GN model. For example, the dispersion relation for the *δ*^{4} model exhibits a singularity at *kδ*≃4.19, and this feature may limit the range of applicability of the model. The simplest extended GN model which avoids this undesirable behaviour in higher wavenumber is the one-dimensional *δ*^{6} model with a flat bottom topography. Its derivation can be made straightforwardly by means of the procedure developed in this section. Therefore, without entering into the detailed analysis, we describe only the final result.

The evolution equation for *δ*^{6} can now be written in the form
*D*_{6} from (2.50). Obviously, the singularity does not occur in *ω* for arbitrary values of *kδ*, as opposed to the *δ*^{4} model. This ensures the well-posedness of the system of linearized equations for the model. Another way to improve the linear dispersion characteristics of the *δ*^{4} model is to replace the depth-averaged velocity by the fluid velocity evaluated at a certain depth. This procedure, however, leads to the violation of the conservation of total energy of the *δ*^{4} model. See Madsen & Schäffer [17], for example. Refer also to Bona *et al.* [28] as for a mathematical analysis on the linear properties of higher-order shallow-water models. In any case, it is an interesting issue to explore various features of the *δ*^{6} model in comparison with those of the *δ*^{4} model and its modified versions, as well as those of the *δ*^{2} (or GN) model.

## 4. Hamiltonian structure

### (a) Hamiltonian

In this section, we show that the two-dimensional extended GN system derived in §2 can be formulated as a Hamiltonian form. First, recall that the basic Euler system of equations (1.1)–(1.4) conserves the total energy (or Hamiltonian) *H* which is the sum of the kinetic energy *K* and the potential energy *U*:
*H* can be confirmed by direct computation using (1.1)–(1.4) coupled with the boundary conditions (1.5). The integrand of *K* is then modified, after using (1.1) and (1.4), as well as the definitions (2.2), (2.3) and (2.10), as
*w* from (2.4) into (4.2), the expression of the kinetic energy *K* reduces, after performing the integration with respect to ** x** under the boundary condition (1.5), to

*η*by (

*h*−1+

*βb*)/

*ϵ*in the expression of the potential energy. The quantity ∇

*ψ*(=

**) expressed in terms of**

*V**h*and

**up to order**

*V**δ*

^{2}. Inserting this into (4.4), we obtain a series expansion of

*H*in powers of

*δ*

^{2}

*H*

_{n}being given by

### (b) Momentum density

In formulating the extended GN system as a Hamiltonian form, it is crucial to introduce the momentum density ** m**. It is given by the following relation

*K*is quadratic in

*K*obeys the scaling law

*U*is independent of

**from (4.6),**

*m**H*is expressed compactly as

*ψ*with the momentum density

**:**

*m*We can compute the momentum density in accordance with the definition (4.6). For example, the approximate expression of ** m** which takes into account the

*δ*

^{2}terms follows from (4.5). Explicitly,

*h*and

**which, substituted into (4.10), yields the Hamiltonian as a functional of**

*m**h*and

**. Note that the kinetic energy obeys the scaling law**

*m**K*(

*ϵ*

**,**

*m**h*,

*b*)=

*ϵ*

^{2}

*K*(

**,**

*m**h*,

*b*), and hence

### (c) Evolution equation for the momentum density

To derive the evolution equation for the momentum density ** m**, we first establish the following formula that provides the variational derivative of

*H*with respect to

*h*:

Now, we proceed to derive the evolution equation for ** m**. We start from the evolution equation for

**from (2.9). First, we multiply (2.9) by**

*V**h*and use equation (2.5) to recast it into the form

**=**

*m**ϵh*

**from (4.11), we obtain the evolution equation for**

*V*

*m***=**

*V***/(**

*m**ϵh*). This leads to

*h*and use (2.5), we can put it in the form of local conservation law

As already discussed in a sentence below equation (4.12), the velocity *h* and ** m**. Thus, the resulting evolution equation, when coupled with equation (2.5) for

*h*, constitutes a closed system of equations for

*h*and

**. This system is equivalent to the extended GN system and will be used for establishing the Hamiltonian formulation of the latter system.**

*m*### (d) Hamiltonian formulation

In this section, we demonstrate that the two-dimensional extended GN system can be formulated as a Hamiltonian system. To this end, we introduce the non-canonical Lie–Poisson bracket between any pair of smooth functional *F* and *G*
** m**=(

*m*

_{1},

*m*

_{2}) and ∂

_{1}=∂/∂

*x*,∂

_{2}=∂/∂

*y*. Note that the partial derivatives ∂

_{i}(

*i*=1,2) operate on all terms they multiply to the right. Then, our main result is given by the theorem 4.1.

### Theorem 4.1

*The two-dimensional extended GN system (2.5) and (2.9) (or equivalently, (4.18)) can be written in the form of Hamilton's equations
**and
*

### Proof.

The identification *F*=*h* and *G*=*H* in (4.20) gives {*h*,*H*}=−∇⋅ (*h*(*δH*/*δ*** m**)). By virtue of (4.13), this becomes

The equation of motion for *m*_{i} (4.21b) with (4.20) can be written explicitly as
** m**=

*ϵh*∇

*ψ*=

*ϵh*

**, we deduce**

*V***evolves according to the equation**

*m*We recall that the bracket (4.20) has been introduced by Holm [7] to formulate the two-dimensional GN equations as a Hamiltonian system. Combining this fact with theorem 1, we conclude that the extended GN system has the same Hamiltonian structure as that of the GN system. Hence, its truncated version like the *δ*^{2n} model shares the same property. As for the Hamiltonian formulation of the GN system, see also Camassa *et al.* [29] for the two-dimensional case, and Constantin [30] and Li [31] for the one-dimensional case.

### (e) Remark

For the *δ*^{4} model discussed in §3, the approximate Hamiltonian is given by (3.12). It follows from (4.6) that
*H* with respect to *h* under fixed ** m** is found to be as

**is also computed from (2.24) with the aid of the formula**

*m***=**

*m**ϵh*

**whereas the expression of**

*V,**δH*/

*δh*in terms of

*h*and

*b*=0. Introducing (4.25) and (4.26) into equation (4.21b) and using equation (4.21a) (or (2.5)) to replace the time derivative of

*h*by

**from (4.25) is used in this integral, it yields the conserved quantity**

*m***given by (3.13).**

*L*For the two-dimensional GN model with an uneven bottom topography, one needs the Hamiltonian truncated at order *δ*^{2}, which is available in (4.5). Introducing this expression into (4.21), the equations of motion for *h* and ** m** are found to coincide with the GN system (2.5) and (3.20), as observed for the first time by Holm [7].

In the one-dimensional case, the evolution equation for ** m** from (4.21b) (or (4.18)) simplifies to

*δ*

^{6}can be expressed as (Matsuno [1])

*m*computed from the one-dimensional version of (4.6) with

*H*from (4.28) is

*δH*/

*δh*with

*m*being fixed is found to be

*h*

_{t}by

## 5. Relation to Zakharov's Hamiltonian formulation

### (a) Zakharov's formulation

Zakharov [9] (see also [32]) showed that the water wave problem (1.1)–(1.5) permits a canonical Hamiltonian formulation in terms of the canonical variables *η* and *ψ*. If we change the variables from (*η*,*ψ*) to (*h*,∇*ψ*), then the equations of motion for the variables *h* and ∇*ψ* are written in the form
*H* is given by (4.1). If we define the Poisson bracket between any pair of smooth functionals *F* and *G* by

### (b) Transformation of the Zakharov system to the extended Green–Naghdi system

Here, we establish the theorem 5.1.

### Theorem 5.1

*Zakharov's system of equations (5.3) is equivalent to the extended GN system (4.21).*

### Proof.

We change the variable ∇*ψ* to the momentum density ** m**, whereas

*h*remains the common variable for both systems, and take the variation of any smooth functional

*F*in two alternative ways. This gives

**is found to be as**

*m**δ*

**=**

*m**ϵ*∇

*ψδh*+

*ϵhδ*∇

*ψ*. We substitute this expression into the right-hand side of (5.4) and then compare the coefficients of

*δh*and

*δ*∇

*ψ*on both sides. It follows from the coefficient of

*δh*that

*δ*∇

*ψ*yields

*F*=

*H*in (5.5) and (5.6) and use (4.11), then we obtain the relations

To proceed, we introduce the second relation in (5.7) into the first equation in (5.1) and see that it coincides with equation (2.5), or equivalently equation (4.21a). On the other hand, substitution of ∇*ψ* from (4.11) and the first relation in (5.7) into the second equation in (5.1) recasts it into the form

Last, we show that the bracket (5.2) transforms to the bracket (4.20) under the change of variables *m*_{i}=*ϵh*∂*ψ*/∂*x*_{i} from (4.11), we can modify the third term in the integrand on the right-hand side of (5.10) as

### (c) Remark

The Lie–Poisson bracket defined by (4.20) has a skew-symmetry {*F*,*G*}=−{*G*,*F*}, and satisfies the Jacobi identity {*F*,{*G*,*H*}}+{*G*,{*H*,*F*}}+{*H*,{*F*,*G*}}=0 for any smooth functionals *F*,*G* and *H*. The skew-symmetric nature follows simply by means of the integration by parts. The Lie–Poisson bracket (4.20) stems naturally from the canonical Poisson bracket through a sequence of variable transformations

## 6. Conclusion

In this paper, we have developed a systematic procedure for extending the two-dimensional GN model to include higher-order dispersive effects while preserving full nonlinearity of the original GN model, and presented various model equations for both flat and uneven bottom topographies. We have also shown that these models permit Hamiltonian formulation. When compared with a previous work [1], the novelties of this study are summarized as follows: (i) the general formulation of the water wave problem in the three-dimensional setting which takes into account the effect of uneven bottom topography, (ii) the analysis of the linear dispersion relation for the extended GN system which features the linearized GN system, (iii) the derivation of the one-dimensional *δ*^{6} model which is the second example of the GN models with non-singular linear dispersion relation, the first one being the original GN model, (iv) computation of the pressure distribution at the fluid bottom in the framework of the two-dimensional *δ*^{4} model, (v) a simple derivation of the key relation between the momentum density and gradient of the surface potential which plays the central role in establishing the Hamiltonian structure of the GN system.

There are a number of interesting problems associated with the extended GN equations that are worthy of further study. In conclusion, we list some of them.

(i) The identification of physically relevant models among various extended GN equations.

(ii) The effect of higher-order dispersion on the wave characteristics in comparison with that predicted by other asymptotic models like Boussinesq equations.

(iii) Numerical computations of the initial value problems as well as solitary and periodic wave solutions.

(iv) The justification of the asymptotic models by means of the rigorous mathematical analysis.

## Data accessibility

There are no accompanying data.

## Competing interests

I have no competing interests.

## Funding

There is no funding for this work.

## Acknowledgements

The author thanks the reviewers for their useful comments and suggestions.

## Appendix A. Positivity of *D*_{2n} with odd *n*

We show that the function *D*_{2n}(*κ*) defined by (2.49) is positive definite for odd *n*. More precisely, it is bounded from below by 1. To prove this statement, we begin with the expression of *D*_{2n} with *n* replaced by 2*n*+1:
*B*_{r} [23]
*λ*=(*κ*/*π*)^{2}. If we introduce the function *g*=*g*(*λ*) by
*D*_{2(2n+1)}≥1+2*λg*(*λ*). The inequality *D*_{2(2n+1)}≥1 follows provided *g*(*λ*)≥0 for *λ*≥0, which we shall now demonstrate.

For *λ* in the interval 0≤*λ*≤1, we multiply the first term of *g* by *λ* to obtain the inequality
*r*=*n*+1 in the second sum of (A 5) and replacing the summation index *r* by *r*+1, we can derive the inequality
*g*(*λ*)≥0 is true for 0≤*λ*≤1.

For *λ*>1, we modify *g* from (A 4) in the form
*g*(*λ*)>0 for *λ* in the interval *λ*=1 in parentheses of (A 7) and take into account (A 8) to show that *g* is bounded from below by the inequality
^{−4r−2})(1−2^{−4r+1})≥(1−2^{−6})(1−2^{−3}) holds for *r*≥1, we can simplify (A 9) as
*g*:
*g*(*λ*)>0 for *g*(*λ*)>0 holds for *λ*>1, as well as for 0≤*λ*≤1. This completes the proof of the positivity of *D*_{2n} with odd *n*. It is important that the present proof establishes the more stringent result *D*_{2(2n+1)}(*kδ*)≥1. Using this inequality in (2.51), we obtain the estimate *kδ*≥0.

## Appendix B. Derivation of (3.14)

In accordance with Bernoulli's law, the dimensionless pressure *P* in the fluid which is scaled by the quantity *ρgh*_{0} (*ρ*: constant fluid density) is given by
*z*=−1, *ϕ*_{z}=0 from (1.4), and *ϕ*_{t}=*f*_{t}, ∇*ϕ*=∇*f* by (2.16). Inserting these relations into (B 1), we find that the bottom pressure *P*_{b}=*P*(** x**,−1,

*t*) has a simple expression in terms of

*f*:

**from (2.12) into equation (2.9) and integrate it under the boundary conditions**

*V**ψ*

**,**

*u**w*and

**in terms of**

*V**f*are already given by (2.22)–(2.24), respectively, whereas the expression of

*ψ*follows from (2.10) and (2.16). Substituting these expressions into (B 3), we obtain the following formula correct up to order

*δ*

^{4}:

*h*=1+

*ϵη*, and we have put

*r*=∇

^{2}

*f*for simplicity. The approximate expression of ∇

*f*in terms of

*h*and

*δ*

^{2}can be derived from (2.19) and (2.20a). It reads

*h*

_{t}by

*δ*

^{6}terms. After some straightforward calculations, we obtain a lengthy expression of the right-hand side of (B 4) in terms of

*h*and

*P*

_{b}.

## Appendix C. Proof of (4.14)

It now follows by taking the variational derivative of *H* from (4.1) with respect to *h* that
** u** and

*w*from (2.2) and (2.3), respectively. The relation below follows from (2.2) and (2.10) coupled with the formula

*δh*(

**,**

*x**t*)/

*δh*(

*x*^{′},

*t*)=

*δ*(

**−**

*x*

*x*^{′}), where

*δ*(

**−**

*x*

*x*^{′}) is the two-dimensional delta function:

*w*from (2.4) and (C 3) into the second term of (C 2) and then perform the integration by parts with respect to

**. By virtue of the formula**

*x*- Received February 19, 2016.
- Accepted May 6, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.