Boussinesq's equations for (2+1)-dimensional gravity-surface waves in an ideal fluid model

We study the problem of gravity surface waves for the ideal fluid model in (2+1)-dimensional case. We apply a systematic procedure for deriving the Boussinesq equations for a prescribed relationship between the orders of four expansion parameters, the amplitude parameter $\alpha$, the long-wavelength parameter $\beta$, the transverse wavelength parameter $\gamma$, and the bottom variation parameter $\delta$. We also take into account surface tension effects when relevant. For all considered cases, the (2+1)-dimensional Boussinesq equations can not be reduced to a single nonlinear wave equation for surface elevation function. On the other hand, they can be reduced to a single, highly nonlinear partial differential equation for an auxiliary function $f(x,y,t)$ which determines the velocity potential but is not directly observed quantity. The solution $f$ of this equation, if known, determines the surface elevation function. We also show that limiting the obtained the Boussinesq equations to (1+1)-dimensions one recovers well-known cases of the KdV, extended KdV, fifth-order KdV, and Gardner equations.


I. INTRODUCTION
an uneven bottom. Section VI deals with the case when α is the leading parameter, β, γ, δ are of the order or α 2 , and derivations are performed up to second order. In this case, limitation to (1+1)-dimensions leads to the Gardner equation extended for an uneven bottom [16], and when δ = 0, that is, when the bottom is flat, reduces to the usual Gardner equation. Section VII contains conclusions.

II. DESCRIPTION OF THE MODEL
Let us consider the inviscid and incompressible fluid model whose motion is irrotational in a container with an impenetrable bottom. In dimensional variables, the set of hydrodynamical equations has the following form φ xx + φ yy + φ zz = 0, in the whole volume, (1) φ z − (η x φ x + η y φ y + η t ) = 0, at the surface, Here φ(x, y, z, t) denotes the velocity potential, η(x, y, t) denotes the surface profile function, g is the gravitational acceleration, ̺ is fluid's density, and p s is additional pressure due to the surface tension [17] p s = −T ∇ · ∇η (1 + |∇η| 2 ) 1/2 = T 1 + η 2 y η xx + (1 + η 2 x ) η yy − 2η x η y η xy 1 + η 2 x + η 2 where T is fluid's surface tension coefficient, and ∇ = (∂ x , ∂ y ) is two-dimensional gradient. The bottom can be non-flat and is described by the function h(x, y). Indexes denote partial derivatives, i.e. φ xx ≡ ∂ 2 φ ∂x 2 , and so on. The next step consists in introducing a standard scaling to dimensionless variables (in general, it could be different in x-, yand z-direction) x =x/L,ỹ = y/l,z = z/H,t = t/(L/ gH),η = η/A,φ = φ/(L A H gH), (6) where A is the amplitude of surface distortions from equilibrium shape (flat surface), H is average fluid depth, L is the average wavelength (in x-direction), and l is a wavelength in y-direction. In general, l should be the same order as L, but not necessarily equal. Then the set (1)-(4) takes in scaled variables the following form (here and next, we omit the tilde signs) Besides standard small parameters α = a H , β = H L 2 and γ = H l 2 , which are sufficient for the flat bottom case, we introduced another one defined as δ = a h H , where a h denotes the amplitude of bottom variations [16,18]. In the perturbation approach, all these parameters, α, β, γ, δ, are assumed to be small but not necessarily of the same order. ST is the term originating from surface tension. The explicit form of this term and its useful approximation up to second order is given below, in (11). For details, see the Appendix A.
The Bond number is defined as τ = T ̺gH 2 . For shallow-water waves, when H 1 m, ST term can be safely neglected, since τ < 10 −7 , but it becomes important for waves in thin fluid layers. In sections III and IV we neglect surface tension terms, in sections V and VI we take them into account.
The standard perturbation approach to the system of Euler's equations (7)-(10) consists of the following steps. First, the velocity potential is sought in the form of power series in the vertical coordinate where φ (m) (x, y, t) are yet unknown functions. The Laplace equation (7) determines φ in the form which involves only two unknown functions with the lowest m-indexes, f (x, y, t) := φ (0) (x, y, t) and F (x, y, t) := φ (1) (x, y, t) and their space derivatives. Hence, The explicit form of this velocity potential reads as Next, the boundary condition at the bottom (10) is utilized. For the flat bottom case, it implies F = 0, simplifying substantially next steps. In particular, F = 0 makes it possible to derive the Boussinesq equations up to arbitrary order. For an uneven bottom, the equation (10) determines a differential equation relating F to f . This differential equation can be resolved to obtain F (f, f x , f xx , h, h x ) but this solution can be obtained only up to some particular order in leading small parameter. Then, the velocity potential is substituted into kinematic and dynamic boundary conditions at the unknown surface (8)- (9). Retaining only terms up to a given order, one obtains the Boussinesq system of two equations for unknown functions η, f valid only up to a given order in small parameters. The resulting equations, however, depend substantially on the ordering of small parameters.
In 2013, Burde and Sergyeyev [15] demonstrated that for the case of (1+1)-dimensional and the flat bottom, the KdV, the extended KdV, fifth-order KdV, and Gardner equations can be derived from the same set of Euler's equations (7)- (10). Different final equations result from the different ordering of small parameters and consistent perturbation approach up to first or second order in small parameters.
In 2020, we extended their results to cases with an uneven bottom in [16], but still in (1+1)dimensional theory. We showed that the terms originating from the bottom have the same universal form for all these four nonlinear equations. However, the validity of the obtained generalised wave equations is limited to the cases when the bottom functions are piecewise linear. On the other hand, the corresponding sets of the Boussinesq equations are valid for the arbitrary form of the bottom functions.
In the present paper there are four small parameters, α, β, γ, δ. In order to make calculations easier, we will follow the idea from [15,16], relating all small parameters to a single one, called leading parametr. This method allows us for easier control of the order of different terms, but the final forms of the obtained equations are presented in original parameters α, β, γ, δ. We discuss several cases, which are listed in Table I. The table does not contain all possible second-order cases, but only those that lead to well-known KdV-type and Gardner equations when reduced to (1+1)-dimensions.
Let us begin with the assumption that all small parameters are of the same order. For easier control of the orders of different terms let us denote where A, G, D are arbitrary but close to 1.
Insertion the velocity potential (14) into the boundary condition at the bottom (10) imposes the following relation Equation (15), when only terms up to second order in small parameters are retained, that is allows us to express F by f and reduce the set of unknown functions to only η and f . In higher order approxiation equation (15) cannot be resolved for F . Then, retaining only terms up to second order, one obtains the velocity potential, valid up to second-order approximation, expressed in terms of a single unknown function f (x, y, t) as Now, we substitute the velocity potential (17) into (8) and (9), keeping terms only up to first order. Due to the term 1 β φ z in (8) and only second order valid expression for the potential (17) we can obtain valid Boussinesq's equations only in first order in α, β, γ, δ.
From (8) and (9) we get respectively Equations (18)- (19) are the first order Boussinesq equations for (2+1)-dimensional shallow water problem with an uneven bottom. Let us check whether the reduction of the equations (18)- (19) to (1+1)-dimensions leads to correct results. When we reduce these equations to (1+1)-dimensional (assuming translational invariance in y direction, that is, setting all y-derivatives equal zero), then we arrive at Denoting w = f x and taking x-derivative of (21) one gets this set of Boussinesq's equation in the following form identical to the equations (17)-(18) derived by us in [16]. In [16], we also demonstrated that these two equations could be made compatible and reduced to the KdV equation generalised for uneven bottom [16,Eq. (29)], when the bottom profile h(x) is a piecewise linear function. This equation has the following form For the flat bottom, δ = 0, the equation (24) reduces to the usual Korteweg-de Vries equation.
In [13], the authors presented the derivation of two new (2+1)-dimensional third-and fifthorder nonlinear evolution equations when α, β, γ are of the same order and the bottom is flat (δ = 0). However, as we proved in [14], all results shown in [13] are false since the derivation is inconsistent and violates the fundamental property of the velocity potential. When the method used by the authors is applied consistently, the problem is reduced to well known KdV equation. For details, see [14].
So, f t = −η and f tt = −η t = f xx + γ β f yy . Then we have and the first-order term receives the same form like the term with the factor 1 6 in front. Then equation (25) receives the simpler form If the solution f (x, y, t) to (27) is known, the equation (19) supplies the surface profile function A preliminary idea how to look for solution to the equation (27) is presented in the appendix B.
In this case, the boundary condition at the bottom (10) imposes the following relation which gives F = β 3 D [(hf x ) x + G(hf y ) y ] valid up to fourth order. Therefore, in this case, it is possible to obtain the Boussinesq equations valid up to third order in small parameters. From (8) we get (up to second order) whereas the result from (9), up to second order, is Note that the term with δ in (30), originating from the uneven bottom, has identical form as such term in (18) although now it is of second order. Let us rewrite (31) in the form Multiplying both sides by (1 + αβf xxt + αγf yyt ) and retaining only terms up to second order we obtain Substituting η defined by (33) into (30) and keeping only the terms up to second order allows us to obtain a nonlinear (2+1)-dimensional differential equation for the function f (x, y, t). This equation has the form (34). In (34) we didn't come back to original parameters α, β, γ, δ for easier recognition of first order and second order terms. In principle, if the solution f (x, y, t) to (34) is known, the equation (33) supplies the surface profile function. The complexity of equations (33) and (34) may be the reason why there are so many small amplitude wrinkles and ripples that are observed on the surface of seas and oceans. However, the complexity of the equation (34) gives a little hope for finding the solution.
Now, the boundary condition at the bottom (10) imposes the following relation and consequently the velocity potential valid up to fourth order. Therefore, in this case it is possible to obtain the Boussinesq equations valid up to third order in small parameters. We will proceed up to second order which is enough complicated. From (8) we obtain whereas the result from (9) is In (39), we didn't take into account terms from surface tension (setting τ = 0 in (9)). This omission is fully justified when water depth is on the order of meters. Now, η determined from (39) is substituted into (38). Limiting this equation up to second order one obtains Equation (40) This set of equations, for δ = 0, is identical to the equations (29)-30) in [15], limited to second order. Eliminating by the standard method the function w, one obtains from them the well-known fifth-order For thin fluid layers, surface tension may introduce significant changes in the dynamics. Using surface tension term in the form (11) in dynamical boundary condition at the surface (9), that is we obtain instead (39) more complicated equation In this case, because we can not express η trough only f and its derivatives, we can not reduce the Boussinesq equations (38)-(44) to a single equation for the function f . However, limiting to (1+1)-dimensions (as usual setting γ = 0, w = f x and differentiating (44) over x) we obtain Denote now β = B α 2 , γ = G α 2 , δ = D α 2 .
Now, the boundary condition at the bottom (10) imposes the following relation which gives F = α 4 (BD(hf y ) y + DG(hf y ) y ) and consequently the velocity potential valid up to fourth order in the leading parameter. Therefore, in this case, it is possible to obtain the Boussinesq equations valid up to third order in small parameters. We will proceed up to the second order, which is complicated enough. Then we can safely neglect higher order terms in surface tension term (11) and use the dynamic boundary condition in the form (43). From kinematic boundary condition at the surface (8), we obtain whereas, from the dynamic boundary condition at the surface (43), the result is Equations (48)-(49) are the Boussinesq equations for the case when β ≈ γ ≈ δ = O(α 2 ). When τ is of the order of 1, what occurs for thin fluid layers, equation (49) can not be resolved to obtain η = G(f, f x , f y , f t , . . .), that is, as a function of f and its derivatives only. Therefore, in this case, the Boussinesq set (48)-(49) can not be reduced to a single equation for f function.
Limiting equations (48)-(49) to (1+1)-dimnesions by setting all y-derivatives equal to zero (alternatively setting γ = 0) we obtain from (48) From (49), after setting γ = 0 and differentiating over x we have Equations (50)-(51) are identical with equations (85)-(86) from [16], when the latter are limited to second order. Therefore, one can make them compatible and derive a single wave equation for wave profile function η(x, t). Such procedure leads to the equation identical with the equation (91) from [16], which is the Gardner equation generalised for an uneven bottom. For δ = 0, that is for the flat bottom, equation (50)