Can simple KdV-type equations be derived for shallow water problem with bottom bathymetry?

Commun Nonlinear Sci Numer Simulat 82 (2020) 105073

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Research paper

Can simple KdV-type equations be derived for shallow water problem with bottom bathymetry? Anna Karczewska a, Piotr Rozmej b,∗ a b

Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, Zielona Góra, 65-246, Poland Faculty of Physics and Astronomy, University of Zielona Góra, Szafrana 4a, Zielona Góra, 65-246, Poland

a r t i c l e

i n f o

Article history: Received 9 July 2019 Revised 21 September 2019 Accepted 22 October 2019 Available online 23 October 2019 PACS: 02.60.-x 47.11.+j 47.35+i Keywords: Shallow water waves KdV-Type equations Uneven bottom

a b s t r a c t We give a survey of derivations of KdV-type equations with an uneven bottom for several cases when small (perturbation) parameters α , β , δ are of different orders. Besides usual small parameters α and β , determining nonlinearity and dispersion, respectively, the model introduces the third parameter δ , which is related to bottom variations. Six different cases of such ordering are discussed. Surprisingly, for all these cases the resulting Boussinesq equations can be made compatible only for the particular piecewise linear bottom profiles, and the correction term in the final wave equations has a universal form. For such bottom relief, several new KdV-type wave equations are derived. These equations generalize the KdV, the extended KdV (KdV2), the fifth-order KdV (KdV5) and the Gardner equations. Numerical simulations of the solutions to some of these equations are presented and discussed. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction The Korteveg de Vries equation (KdV in short) [1] belongs to a few most famous equations in mathematical physics. It was originally derived for surface water waves in so-called shallow water wave problem. In the sixties of the last century, rapid development of the theory of nonlinear waves in various physical systems began, which showed that the KdV equation was obtained as the first approximation in the description of many physical phenomena. The range of applications extends, among others, to waves on the surface of liquids, waves in interfaces between various phases of liquids, ion-acoustic waves in plasma, optical impulses in optical fibers and electrical impulses in electrical circuits. There is a vast number of textbooks and monographs referring to studies of these problems, see, e.g. [2–8], to list a few. Wonderful properties of the KdV equation like integrability, a rich variety of analytic solutions and the existence of the infinite number of invariants attracted the attention of physicists, mathematicians, and engineers. KdV and other KdV-type equations are derived under an important assumption, that the bottom of the fluid container is flat. This assumption is not realistic for most of the situations in the real world, in particular, bottoms of rivers, seas, oceans are non-flat. Despite a big number of efforts in studying nonlinear waves in the case of a non-flat bottom the first KdVtype equations in which terms originating from the bottom profile occur appeared only recently. Among the first papers ∗

Corresponding author. E-mail address: [email protected] (P. Rozmej).

https://doi.org/10.1016/j.cnsns.2019.105073 1007-5704/© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

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A. Karczewska and P. Rozmej / Commun Nonlinear Sci Numer Simulat 82 (2020) 105073

treating a slowly varying bottom are papers by Mei and Le Méhauté [9], and Grimshaw [10]. These authors found that for small amplitudes the wave amplitude varies inversely as the depth but they did not obtain any simple KdV-type equation. Djordjevic´ and Redekopp [11] and later Benilow and Howlin [12] studied the motion of packets of surface gravity waves over an uneven bottom using variable coefficient nonlinear Schrödinger equation (NLS). As a result they found fission of an envelope soliton. Some research groups developed approaches combining linear and nonlinear theories [13–15]. The Gardner equation (sometimes called the forced KdV equation) was also extensively investigated [16–19]. Van Groeasen and Pudjaprasetya [20,21] applied a Hamiltonian approach in which they obtained a forced KdV-type equation. Another widely applied method consists in taking an appropriate average of vertical variables which results in the Green–Naghdi equations [22–24]. An interesting numerical study of the propagation of unsteady surface gravity waves above an irregular bottom is done in [25]. Another study of long wave propagation over a submerged 2-dimensional bump was recently presented in [26], although according to linear long-wave theory. Several examples of recent studies on the propagation of solitary waves over a variable topography are given in [27–31]. Several authors derived variable coefficient KdV equation (vcKdV) [32–36] in attempts to describe the evolution of a solitary wave moving onto a shelf. The assumption that bottom is varying slowly permitted to derive vcKdV in the form [34] 7

1

uX + d− 4 uuξ + d 2 uξ ξ ξ = 0.

(1) x

− 12

In (1), the independent variable X = α x (far-field coordinate) and ξ = 0 d (α x )dx − t (appropriate characteristic coordinate), whereas x and t are the original non-dimensional space and time variables, respectively. The function u(ξ , X) represents the elevation of the wave and d = d (X ) is the varying (nondimensional) fluid depth. Within this approach, it was possible to understand the asymptotic fission of the single KdV soliton into two or three solitons for particular depths of the shelf or into oscillatory waves in general. Despite the success of understanding the general features of a solitary wave moving onto a shelf, the vcKdV equation does not apply for cases when the range of bottom obstacles is comparable to the characteristic wavelength. Moreover, solutions of (1) can be hardly translated into original dimensional variables in order to compare theory to observations. Therefore, our aim was to derive KdV-type equation(s) taking directly into account a bottom varying on both short and long scales. In [37], Rosales and Papanicolau considered the propagation of long waves with small amplitudes along a shallow channel with a rough bottom. Article [37] is the only one known to us (apart from our approach) in which the authors introduce besides two small standard parameters, the third one associated with an uneven bottom. This parameter is defined as γ = (L/l )2 where L is the typical length related to bottom changes, and l is the characteristic wavelength of long surface water waves. For periodic-rough bottoms, they obtained the KdV equation (η0 )τ + θ (η0 )2χ + b(η0 )χ χ χ for the principal term of surface elevation η0 (τ , χ ). However, it is difficult to interpret the expressions for the coefficients θ , b since so many intermediate problems are involved. Nachbin and Papanicolau [38] considered the reflection-transmission problem for monochromatic and pulse-shaped surface disturbances for shallow channels with arbitrary rapidly varying bottoms using linear potential theory. Craig et al. [39] extended the results of Rosales and Papanicolau [37] both for two and three-dimensional flows using Hamiltonian long-wave expansions. They considered bottom topography which is periodic in horizontal variables on a short length scale, with the amplitude variation being of the same order as fluid depth. The bottom may also exhibit slow variations at the same length scale as, or longer than, the order of the wavelength of surface waves. The authors obtained effective Boussinesq equations in cases in which the bottom possesses both short and long scale variations. In certain cases, these equations can reduce to unidirectional equations that are similar to the KdV equation. There are also advanced studies of waves in channels of non-uniform cross-sections. One of the early work in this direction is due to Peregrine [40]. Chassagne et al. [41] give a contemporary detailed analysis of undular bore dynamics in channels of a variable cross-section. We do not consider such problems in our paper. One of the most advanced approaches to the problem of varying bathymetry in the shallow water scaling regime is given in [42]. The authors derived a new model system of equations, consisting in the classical shallow water equations coupled with nonlocal evolution equations for a periodic corrector term and, among other results, found a new resonance phenomenon between surface waves and a periodic bottom. Our approach is entirely different. Besides standard small parameters α = a

a H

and β =

 H 2 l

we introduced the third one

defined as δ = Hh . In these definitions a denotes the wave amplitude, H the average water depth, l the typical wavelength of long surface water waves and ah the amplitude of the variations of the bottom function h(x). In the simplest case (KdV) the parameters α and β specify nonlinearity and dispersion, respectively. The geometry of the considered shallow water problem is presented in Fig. 1. With standard assumptions for incompressible, inviscid fluid and irrotational motion, one obtains the set of Eulerian equations in dimension variables. Next, introduction of the following transformation to dimensionless variables



x˜ = x/l,

z˜ = z/H,

t˜ = t

gH

l

,

η˜ = η/a, φ˜ =

H



la gH

φ

(2)

has made it possible to apply perturbation approach, assuming that appropriate parameters are small. Here and in the following, η(x, t) represents the surface elevation relative to the undisturbed water level and φ (x, z, t) is the velocity potential.

A. Karczewska and P. Rozmej / Commun Nonlinear Sci Numer Simulat 82 (2020) 105073

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Fig. 1. Schematic view of the geometry of the shallow water wave problem for an uneven bottom.

The set of Euler equations, written in nondimensional variables (tildes are now dropped) has the following form

βφxx + φzz = 0, ηt + αφx ηx − 1 2

φt + αφx2 +

1

β

(3)

φz = 0, for z = 1 + αη

1α 2 η2 x = 0, φ + η − τβ 2β z (1 + α 2 βηx2 )3/2

(4) for

z = 1 + αη

(5)

φz − βδ (hx φx ) = 0, for z = δ h(x ).

(6)

Eq. (3) is the Laplace equation valid for the whole volume of the fluid. Eqs. (4) and (5) are so-called kinematic and dynamic boundary conditions at the surface, respectively. The Eq. (6) represents the boundary condition at the non-flat unpenetrable T bottom. In (5), the Bond number τ = gh 2 , where T is the surface tension coefficient. For surface gravity waves this term can be safely neglected, since τ < 10−7 , but it can be important for waves in thin fluid layers. For abbreviation all subscripts ∂ η in (3)-(6) denote the partial derivatives with respect to particular variables, i.e. φt ≡ ∂φ ∂ t , η2x ≡ ∂ x2 , and so on. The velocity potential is seek in the form of power series in the vertical coordinate 2

φ (x, z, t ) =

∞ 

zm φ (m ) (x, t ),

(7)

m=0

where φ (m) (x, t) are yet unknown functions. The Laplace Eq. (3) determines φ in the form which involves only two unknown functions with the lowest m-indexes, f(x, t) := φ (0) (x, t) and F(x, t) := φ (1) (x, t). Hence,

φ (x, z, t ) =

∞  (−1 )m β m ( 2 m )!

m=0

∞ ∂ 2m f (x, t ) 2m  (−1 )m β m+1 ∂ 2m+1 F (x, t ) 2m+1 z + z . 2 m ( 2 m + 1 )! ∂x ∂ x2m+1 m=0

(8)

The explicit form of this velocity potential reads as

1 2

φ = f − β z 2 f 2x +

1 3 6 1 1 2 4 1 3 5 β z f 4x − β z f6x + · · · + β zFx − β 2 z3 F3x + β z F5x + · · · 24 720 6 120

(9)

Next, one applies the perturbation approach, assuming that parameters α , β , δ are small. As pointed in [44] the proper ordering of small parameters is crucial to obtain appropriate final wave equations. Therefore, for each particular case, the perturbation approach has to be performed separately expressing all parameter by only one (called the leading parameter). We will be interested in all possible cases of wave equations obtained in the perturbation approach up to second order. So, we can specify the following cases, see Table 1. Our study extends that done thoroughly by Burde and Sergyeyev [44]. They considered several cases of a different ordering of two small parameters α , β , still for the flat bottom case sometimes going up to third or fourth order. The cases studied in [44] were: β = O(α ), β = O(α 2 ), β = O(α 3 ), α = O(β 2 ) and α = O(β 3 ). The authors showed that different ordering of small parameters implied several kinds of wave equations, previously derived in the literature from different physical assumptions. It is worth to emphasize, that in derivations of the KdV (β = O(α )) in first order perturbation approach, extended KdV (β = O(α )) in second order perturbation approach, fifth-order KdV (α = O(β 2 )) in second order perturbation approach, and Gardner equation (β = O(α 2 )) in second order perturbation approach, given in [44], no additional assumptions are required. The only necessary condition is the compatibility of the Boussinesq equations resulting from the Euler

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A. Karczewska and P. Rozmej / Commun Nonlinear Sci Numer Simulat 82 (2020) 105073 Table 1 Different ordering of small parameters considered in the paper. Case

α

β

1 2 3 4 5 6

O (β ) O (β ) O (β 2 ) O (β 2 ) leading parameter leading parameter

leading leading leading leading O (α 2 ) O (α 2 )

δ parameter parameter parameter parameter

O (β ) O (β 2 ) O (β ) O (β 2 ) O (α ) O (α 2 )

set (3)–(6). The set of Boussinesq equations is obtained from (4) and (5), utilizing (9), (3) and (6). It contains two unknown functions η, w = fx and their derivatives. Fulfilling the compatibility condition means that there exists the function w = w(η, ηx , η2x , . . . ) such that Boussinesq equations become identical and provide nonlinear wave equation for a single unknown function η. The same condition of compatibility of the Boussinesq equations was utilized earlier in [46] for the derivation of the extended KdV (KdV2) equation. Therefore, in our model the proper ordering of small parameters and the compatibility condition plays an essential role in the consitent derivation of final KdV-type wave equations. We emphasize that from mathematical point of view no additional assumptions are needed. The form of velocity potential (9), determined by (7) and the Laplace Eq. (3), is the same for all considered cases 1–6. The boundary condition at the bottom (6) implies different forms of the F function, depending on the particular ordering of small parameters α , β , δ . It is worth noticing the important difference between the cases related to a flat bottom and those when the bottom is not even. In the former ones F = 0, due to δ = 0 in (6). Therefore, when the Boussinesq equations are used to determine correction terms Q one can always utilize the condition Qt = −Qx . These facts ensure to derive KdV-type equations up to arbitrary order. This is not possible for an uneven bottom. In the latter case the boundary condition (6) imposes a differential equation on f and F which can be solved to obtain F(f(x, t), h(x)) only in some low orders depending on ordering relations between small parameters. For higher orders that equation cannot be solved. In the following sections, we discuss derivations of KdV-type wave equations in cases 1–6. Some examples of numerical simulations illustrating soliton motion over a linearly sloped bottom are presented in Section 8. The last Section 9 contains conclusions. In this paper, we focused on derivations of wave equations which include terms from the uneven bottom. We leave the broader numerical studies of derived equations to the next article. 2. Case 1: α = O(β ), δ = O(β ) Due to the velocity potential formula (9) the natural leading parameter is β . In order to consider perturbation expansion in only one small parameter, we can set

α = Aβ , δ = Dβ ,

(10)

where the constants A, D are of the order of 1. Substitution of (9) into (6) gives (with z = Dβ h(x )) the following nontrivial relation between the functions Fx and f

Fx − Dβ (h fx )x −

1 2 3 2 1 1 4 6 4 D β (h F2x )x + D3 β 4 (h3 f3x )x + D β (h F4x )x + . . . = 0. 2 6 24

(11)

Keeping only terms lower than third order leaves

Fx = Dβ (h fx )x ,

(12)

which allows us to express the x-dependence of the velocity potential through f, h and their x-derivatives up to third order. With higher order terms in (11) it is impossible. The Eq. (12) determines Fx up to second order. Since this term enters (9) with the factor β z, the velocity potential is determined correctly up to third order in β . It is worth to emphasize that due to the presence of the term − β1 φz in (4), the Boussinesq equations resulting from the substitution of (9) into (4) and

(5) are correct up to second order. Therefore, the boundary condition at the uneven bottom implies the limit on the order of theory in which the Boussinesq equations can be derived. For the case α = O(β ), δ = O(β ) this is second order. It is easy to see that the form of (11), when considered to higher orders does not allow us for obtaining an explicit expression of Fx through f, h and their x-derivatives. Substituting (12) into (9) and retaining terms up to third order in β gives the velocity potential as

1 2

φ = f − β z 2 f 2x +

1 3 6 1 1 2 4 β z f 4x − β z f6x + +β 2 zD(h fx )x − β 3 z3 D(h fx )3x . 24 720 6

(13)

Due to the term β1 φz in the Eq. (4), to obtain equations up to second order one has to keep the velocity potential up to third order.

A. Karczewska and P. Rozmej / Commun Nonlinear Sci Numer Simulat 82 (2020) 105073

5

Inserting (13) into (4) and (5), with z = 1 + Aβη, and retaining terms up to second order yields the set of the Boussinesq equations in the following form (with usual notation w = f x )

  1   1 1 ηt + wx + β A(ηw )x − w3x − D(hw )x + β 2 −A (ηw2x )x + w5x + D(hw )3x = 0, 6

2

120

(14)





1 w2xt − τ η3x 2   1

1 1 + β 2 A −(ηwxt )x + wx w2x − ww3x + w4xt + D(hwt )2x = 0. 2 2 24

wt + ηx + β Awwx −

(15)

In the lowest (zero) order the Boussinesq set reduces to

ηt + wx = 0, wt + ηx = 0, implying w = η, ηt = −ηx , wt = −wx .

(16)

In the first order the Boussinesq set reduces to

  1 ηt + wx + β A(ηw )x − w3x − D(hw )x = 0,

(17)

6



wt + ηx + β Awwx −



1 w2xt − τ η3x = 0. 2

(18)

Note that terms originating from an uneven bottom appear in (17) but not in (18). This is the reason why in first order the Boussinesq Eqs. (17) and (18) can be made compatible only for the particular case of the bottom function h(x). Let us remind the general method of derivation of KdV-type wave equation from the Boussinesq set (here (14) and (15)). For simplicity, consider  the case ofthe flat bottom (D=0). Looking for compatibility of first order Boussinesq equations we assume w = η + β AQ (1 ) + Q (2 ) . Inserting this w into (14) and (15) and next solving the resulting differential equations yields Q (1 ) = − 14 η2 and Q (2 ) = 16 (1 − 3τ )η2x . With this w both Boussinesq equations become identical with the   KdV equation ηt + ηx + β 32 Aηηx + 16 (1 − 3τ )η3x = 0. On the way to this result, one replaces t-derivatives by x-derivatives determined in previous order solution (here one replaces ηt by −ηx from (16)). Looking for compatibility of higher order Boussinesq equations one proceeds in an analogous way utilizing previous order solutions, and so on. To solve first order problem with an uneven bottom we assume the function w in the form



1 1 w = η + β − A η2 + (2 − 3τ )η2x + DQ 4 6



(19)

where the first two terms in the correction function assure the KdV equation in the case of the flat bottom. Then inserting (19) into (17) and (18) gives in first order

ηt + ηx + β ηt + ηx + β

3 2

3 2



Aηηx +

1 (1 − 3τ )η3x − D(hη )x + DQx = 0 6

Aηηx +

1 (1 − 3τ )η3x + DQt 6



and

= 0,

(20)

(21)

where in (21) we already replaced ηt by −ηx (from zeroth order). The Eqs. (20) and (21) become compatible when

Qx − Qt = (hη )x .

(22)

2.1. Consequences of the compatibility condition (22) The same compatibility condition (22) appears in all cases of the ordering of small parameters considered in this paper. Now, we discuss the consequences implied by this universal condition. All known KdV-type equations, for instance the KdV, extended KdV, fifh-order KdV, mKdV, and Gardner equations, have the general form ηt = F (η, ηx , η2x , . . . , ηnx ) which contains both linear and nonlinear terms. The right hand side of the condition (22) implies that the appropriate correction term Q has to contain, besides η and some of its x derivatives, terms with h and its x derivative(s). A term ∫η dx is admissible, as well, since its x- and t-derivatives are still expressed by local function η. Let us seek for Q in the following general form (a, b to be determined)

Q = a hη + b hx



η dx.

Then

Qx = a(hx η + hηx ) + b(h2x

(23) 

η dx + hx η )

and

(24)

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A. Karczewska and P. Rozmej / Commun Nonlinear Sci Numer Simulat 82 (2020) 105073

Qt = ahηt + bhx





ηt dx = −ahηx + bhx (−ηx ) dx = −ahηx − bhx η.

(25)

In (25) we replaced ηt by −ηx from zeroth order relation. So, the condition (22) for Q in the form (23) is expressed by the formula



(a + 2b − 1 )hx η + (2a − 1 ) hηx + bh2x η dx = 0.

(26)

The Eq. (26) is valid only when simultaneously



(2a − 1 ) and bh2x η dx = 0,

a + 2b − 1 = 0,

This permits fulfilling the condition (22), but only for h2x = 0, with

a=

1 2

and then b =

1 . 4

This important result was first indicated in [45], for cases 1 and 2 of ordering of small parameters. The conclusion in [45] stated that the compatibily condition (22) can be satisfied only for linear bottom function h(x ) = k x, where k is a  constant. For this linear bottom function the correction term has the form Q = 14 (2kxη + k ηdx ). The linear function h = kx suffers, however, a significant drawback. For sufficiently large |x| it violates the assumption that δ is small. This disadvantage is removed by allowing that h(x) is an arbitrary piecewise linear function. Then

Q=



1 2hη + hx 4



 η dx , with h2x = 0.

(27)

In next sections we will show that this is a universal feature, necessary for compatibility of the Boussinesq equations for any ordering of small parameters. One has to remember that all newly derived wave equations are valid only for the bottom given by a piecewise linear function. 2.2. Generalization of KdV for the piecewise linear bottom function The correction function (27) makes the Eqs. (20) and (21) compatible and the resulting first order KdV-type equation has the following form

ηt + ηx + β

3 2

Aηηx +



1 1 − 3τ η3x − D(2 hηx + hx η ) = 0. 6 4

(28)

In original notations for small parameters, this equation reads as

3 2

ηt + ηx + αηηx +

1 1 − 3τ βη3x − δ (2 hηx + hx η ) = 0. 6 4

(29)

For the case of the flat bottom (D = δ = 0), (29) reduces to the usual KdV equation

3 2

ηt + ηx + αηηx +

1 − 3τ βη3x = 0. 6

Therefore, the Eq. (29) generalizes KdV to a case of a piecewise linear bottom profile. When h(x) is a bounded arbitrary function the Eqs. (20) and (21) cannot be made compatible. Attempts to continue derivation of a wave equation to second order in small parameters show that Eqs. (14) and (15) cannot be made compatible [45]. Let us confirm this fact. Assume





1 2 − 3τ 1 w = η + β − A η2 + η2x + D 2hη + hx 4 6 4



+ β 2 A2



ηdx



1 3 3 + 7τ 2 2+τ 12 − 20τ − 15τ 2 η +A ηx + A ηη2x + η4 x 8 16 4 120

+ β 2 DQ

(30)

In (30) we use the form of first order correction (28) and the part of second order correction which is appropriate when D = 0. Therefore Q in (30) is responsible only for this part which depends on the bottom relief. An attempt to make Eqs. (14) and (15) compatible leads to the condition on Qx − Qt which gives no hope for obtaining second order wave equation expressed by local variables η, h and their derivatives. Let us remind that in the case of the flat bottom (with surface tension neglected) the second order wave equation reduces to well known the so-called extended KdV equation

3 2

1 6

3 8

ηt + ηx + αηηx + βη3x − α 2 η2 ηx + αβ

 23 24

ηx η2 x +



5 19 2 ηη3x + β η5 x = 0 12 360

(31)

derived for the first time by Marchant and Smyth [46] and sometimes called KdV2. This equation is nonintegrable. Despite 2 this fact, we with our co-workers found several forms of analytic solutions to KdV2: soliton solutions (∼ sech [B(x − vt )]) √ in [43], cnoidal solutions (∼ cn2 [B(x − vt )]) in [47] and superposition cnoidal solutions (∼ dn2 [B(x − vt )] ± m cn[B(x − vt )] dn[B(x − vt )]) in [48,49].

A. Karczewska and P. Rozmej / Commun Nonlinear Sci Numer Simulat 82 (2020) 105073

3. Case 2:

7

α = O ( β ), δ = O ( β 2 )

In this case we set

α = Aβ , δ = Dβ 2 .

(32)

Now, we insert the general form of velocity potential (9) into the bottom boundary condition (6) which in this case is

φz − Dβ 3 (hx φx ) = 0, for z = Dβ 2 h(x )

(33)

obtaining relation similar to (11)

Fx − Dβ 2 (h fx )x −

1 2 5 2 D β (h F2x )x + O(β 7 ) = 0. 2

(34)

From (34) we have

Fx = Dβ 2 (h fx )x ,

(35)

valid up to fourth order in β which inserted into (9) gives the velocity potential valid up to fourth order

1 2

φ = f − β z 2 f 2x +

1 3 6 1 2 4 1 β z f 4x − β z f6x + Dβ 3 z(h fx )x + β 4 z 8 f 8x + O ( β 5 ) . 24 720 40320

(36)

Therefore, the Boussinesq equations can be consistently derived up to third order (remember term β1 φx in (4)). However, we will proceed to second order, only. Substituting the velocity potential (36) into (4) and (5) and retaining terms up to second order supplies the Boussinesq equations in the following form

  1   1 1 ηt + wx + β A(ηw )x − w3x + β 2 −A (ηw2x )x + w5x − D (hw )x = 0, 6



wt + ηx + β Awwx −



2

120



1 1 1 1 w2xt − τ η3x + β 2 −A(ηwxt )x + A wx w2x − A ww3x + w4xt 2 2 2 24

(37)



= 0.

(38)

The Boussinesq Eqs. (37) and (38) for the Case 2 and Case 1 are identical when δ = D = 0. Since, in (37) the term −D (hw )x appears only in second order, the first order solutions are those of the KdV, with



w = η + β −A and



1 2 1 η + ( 2 − 3 τ ) η2 x , 4 6

  3 1 ηt + ηx + β A ηηx + (1 − 3τ )η3x = 0 ηt

2 6 3 1 + ηx + α ηηx + β (1 − 3τ )η3x = 0 2 6

(39)

or (40)

in original variables. Now, we aim to satisfy the Boussinesq system (37) and (38) with the terms of the second order included. Then, we set (the first term with β 2 is known from the flat bottom case)



w = η + β −A



1 2 1 η + ( 2 − 3 τ ) η2 x 4 6

+ β 2 A2



1 3 3 + 7τ 2 2+τ 12 − 20τ − 15τ 2 η +A ηx + A ηη2x + η4 x 8 16 4 120

(41)

+ β 2 DQ.

Next, we insert the trial function (41) into (37) and (38) and retain terms up to second order in β . Proceeding analogously as in the case of first order we find that compatibility of the Boussinesq Eqs. (37) and (38) requires the following condition for the correction function Q

Qx − Qt = (hη )x , the same as the condition (22). Note, that in order to replace t-derivatives by x-derivatives one has to use the properties of the first order Eq. (40), that τ is, ηt = −ηx − β A 23 ηηx + 1−3 6 η3x and its derivatives. Using universal formula for the correction functions (27), for a piecewise linear bottom, we obtained in this case, α = O(β ), δ = O(β 2 ), the equation

  23 + 15τ 3 3 1 − 3τ 5 − 3τ ηt + ηx + αηηx + βη3x − α 2 η2 ηx + αβ ηx η2 x + ηη3x 2 6 8 24 12

2 1 19 − 30 τ − 45 τ + β2 η5x − δ (2 hηx + hx η ) = 0 360

4

which generalizes the extended KdV (KdV2) Eq. (31).

(42)

8

A. Karczewska and P. Rozmej / Commun Nonlinear Sci Numer Simulat 82 (2020) 105073

4. Case 3:

α = O ( β 2 ), δ = O ( β )

In this case we set

α = Aβ 2 , δ = Dβ .

(43)

Since δ is of the same order as β the formulas (11)–(13) expressing the velocity potential hold. Now we substitute the velocity potential (13) into the kinematic and dynamic boundary conditions at the unknown surface which in this case are

1

ηt + Aβ 2 φx ηx −

β

1 2

φz = 0, for z = 1 + Aβ 2 η,

1 2

φt + Aβ 2 φx2 + Aβφz2 + η − τ β

(44)

η2 x = 0, for z = 1 + Aβ 2 η. (1 + A2 β 5 ηx2 )3/2

(45)

Next, we neglect all terms of orders higher than β 2 . The result consists in the following Boussinesq equations (in the meantime the second equation was differentiated by x)

    1 1 1 ηt + wx − β D (hw )x + w3x + β 2 A(ηw )x + D(hw )3x + w 5x = 0 , 6

wt + ηx − β



2

120

   1 1 τ η3x + w2xt + β 2 Awwx + D(hwt )2x + w4xt = 0, 2

24

(46) (47)

where the usual notation w = fx is used. For the flat bottom case (D = 0) the Eqs. (46) and (47) can be made compatible up to any order. Some of them are derived from different physical models. Below we cite this equations keeping terms up to second order in β (see, [44, Eqs. (A.1) and (A.2)])



w=η+β

2 − 3τ 1 12 − 20τ − 15τ 2 η2x + β 2 −A η2 + η4 x , 6 4 120



ηt + ηx + β

1 − 3τ 3 19 − 30τ − 45τ 2 η3x + β 2 A ηηx + η5 x 6 2 360

(48)

= 0.

(49)

This result is equivalent to the well known fifth-order KdV equation derived by Hunter and Sheurle [50] as a model equation for gravity-capillary shallow water waves of small amplitude. Neglecting surface tension (reasonable for shallow water problem) and changing variables by



x˜ =

3α ( x − t ), 2β

t˜ =

1 4



3α 3 t, 2β

one reduces the Eq. (49) to

ηt˜ + 6ηηx˜ + η3x˜ + P η5x˜ = 0,

P=

19 , 40

which is the fifth-order KdV equation obtained in [50] with P defined in a different way. This equation is known to have a rich structure of solitary wave solutions, see, e.g., Grimshaw et al. [51]. As pointed out in [44] the wave equation obtained in third order belongs to the type K(m, n) introduced by Rosenau and Hyman [54] with m = 4 and n = 1 which in some range of wave velocities admits soliton-like traveling wave solutions. For the case D = 0 limitation of Eqs. (46) and (47) to first order yields

  1 ηt + wx − β D (hw )x − w3x = 0 6

and

  1 ηx + wt − β τ η3x + w2xt = 0. 2

(50) (51)

Since in zeroth order η = w, ηt = −ηx , wt = −wx , one assumes that in the first order

w=η+β

 2 − 3τ 6

 η2x + DQ ,

(52)

where the first part of the correction term is alraedy known from (48) and Q is responsibe for first order correction related to the bottom term in (50). Then, substitute (52) into Eqs. (50) and (51) and retain terms only to the first order. This yields

ηt + ηx + β

 1 − 3τ 6

 η3x + DQx − D (hη )x = 0

(53)

A. Karczewska and P. Rozmej / Commun Nonlinear Sci Numer Simulat 82 (2020) 105073

and

ηt + ηx + β

 1 − 3τ

9

 η3x + DQt = 0.

6

(54)

Compatibility of these equations requires

Qx − Qt = (hη ).

(55)

This is the same condition as (22) which cannot be satisfied for general form of h(x) but can be satisfied for the particular case of piecewise linear function. In this particular case Q given by (27) makes the Eqs. (53) compatible. So, with

w=η+β

 2 − 3τ 6

   1 η2x + D 2hη + hx η dx

(56)

4

we obtain the resulting first order KdV-type equation in the following form

ηt + ηx + β

 1 − 3τ 6

 1 η3x − D(2 hηx + hx η ) = 0.

(57)

4

Note, that the Eq. (57), in the case of the flat bottom (D = 0), is reduced to the linear dispersive one. Therefore the Eq. (57) has no soliton solutions. Can we derive a reasonable second order equation? Let us assume

w=η+β

 2 −3τ 6



1 η2x + D 2hη + hx 4



η dx





+β

2

1 12 −20τ −15τ 2 −A η2 + η4x + DQ . 4 120

(58)

In (58) we already used the form of the first order correction (56) and second order correction for flat bottom case (48). Now, we seek for Q which is the second order correction originating only from the bottom terms in the Boussinesq Eqs. (46) and (47). Inserting (58) into (46) and (47) and replacing t-derivatives by x derivatives according to (57) we obtain the following condition for the second order correction Q (below we used h(x ) = kx)

  11  1+τ 3 ( η + xη x ) + ( η 2 x + xη 3 x ) + k 2 D xη + x 2 η x 4 2 16 8  5   1 2 +k D ηdx + xηx dx .

Qx − Qt = − k

1

32

(59)

16

This condition seems to have no solutions which could supply the second order wave equation of similar form as the first order Eq. (57), that is, ηt = −[ηx + F (η, ηx , . . . , x )]. 5. Case 4:

α = O ( β 2 ), δ = O ( β 2 )

In this case we set

α = Aβ 2 , δ = Dβ 2 .

(60)

O ( β 2 ),

Since δ = the forms of the function F and the velocity potential are given by (35) and (36). The Boussinesq set receives in this case the following form

  1 1 ηt + wx − β w3x + β 2 A(wη )x + w5x − D(hw )x = 0, 6

wt + ηx − β

(61)

120

1 2





w2xt + τ η3x + β 2 Awwx +

1 w4xt 24



= 0.

(62)

In first order w in the form

w=η+β

2 − 3τ η2 x 6

(63)

makes the Eqs. (61) and (62) compatible, with the result which is the linear equation

ηt + ηx + β

1 − 3τ η3 x = 0 . 6

(64)

In second order we look for w in the form



w=η+β

1 2 − 3τ 12 − 20τ − 15τ 2 η2 x + β 2 − A η 2 + η4 x 6 4 120

+ β 2 DQ.

(65)

In (65) we already used the part of second order correction term known to make compatible Boussinesq’s set for the flat bottom (see, e.g., [44, Eqs. (A.8) and (A.9)]. Substitution of (65) into (61) and (62) gives

ηt

1 − 3τ + ηx + β η3 x + β 2 6



19 − 30τ − 45τ 2 η5 x 360

+ β 2 (DQx − D(hη )x ) = 0

(66)

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and

ηt + ηx + β

1 − 3τ η3 x + β 2 6



3 19 − 30τ − 45τ 2 Aηηx + η5 x 2 360

+ β 2 DQt = 0,

(67)

respectively. (In (67) t-derivatives are already properly replaced by x-derivatives from first order Eq. (64).) Compatibility of Eqs. (66) and (67) requires the same condition as (22)

Qx − Qt = (hη )x . Using the correction Q given by (27) we obtain the wave equation (in original parameters)

3 2

ηt + ηx + αηηx + β

1 1 − 3τ 19 − 30τ − 45τ 2 η3 x + β 2 η5x − δ (2 hηx + hx η ) = 0. 6 360 4

(68)

This equation is the generalization of the fifth-order KdV Eq. (49) for the case of uneven bottom. However, the Eq. (68) is correct only for a piecewise linear bottom profile. 6. Case 5:

β = O ( α2 ) , δ = O ( α )

In this case the leading parameter is α . We set

β = Bα 2 , δ = Dα .

(69)

Now, we have to express all perturbation equations with respect to parameter α . Then the velocity potential (9) can be rewritten as

1 2

1 3 6 6 1 2 4 4 B α z f 4x − B α z f 6x + · · · 24 720 1 1 3 6 5 + Bα 2 zFx − B2 α 4 z3 F3x + B α z F5x + · · · . 6 120

φ = f − Bα 2 z 2 f 2 x +

(70)

The boundary condition at the bottom (6) takes now the following form

φz − BDα 3 (hx φx ) = 0 for z = Dα h(x ).

(71)

Applying this equation to φ given by (70) implies

1 Fx = α D(h fx )x + α 4 BD2 (h2 F2x )x − α 5 BD3 (h3 f3x )x + O(α 8 ). 6

(72)

This equation allows us to express Fx through h, f and their derivatives only when terms of the fourth and higher orders are neglected

Fx = α D(h fx )x + O(α 4 ).

(73)

This formula allows us to express φ through only one unknown function f and its derivatives. Note that next terms in Fx would enter in φ in sixth order. Therefore we can express the velocity potential containing terms from the bottom function only up to fifth order in α

1 2

φ = f − Bα 2 z 2 f 2 x +

1 1 2 4 4 B α z f4x + BDα 3 z(h fx )x − B2 Dα 5 z3 (h fx )3x + O(α 6 ). 24 6

This form of the velocity potential implies the Boussinesq set as

1 6

ηt + wx + α ((ηw )x − D(hw )x ) − α 2 B w3x + α 3 B wt + ηx + α wwx − α 2 B







1 D(hw )3x − (ηw2x )x = 0, 2

   1 1 1 τ η3x + w2xt + α 3 B −(ηwxt )x + wx w2x − ww3x + D(hwt )2x = 0, 2

(74)

2

2

(75)

(76)

where terms up to α 3 are retained. Note that for the uneven bottom (δ , D = 0) consistent Boussinesq’s set cannot be derived in orders higher than α 3 . Due to the term φz /β = φz /(Bα 2 ) in the kinematic boundary condition at the surface (4) and relations (72) and (73) we cannot extend Eqs. (75) and (76) with the potential (74) to higher orders. When the bottom is flat, there is no such limitation. Begin with first order Boussinesq’s set. Assuming w = η + α (− 41 η2 ) + α DQ one obtains from (75) and (76)

ηt + ηx + α ηt + ηx + α

3 2

3 2

 ηηx + DQx − D(hη )x = 0,

(77)

 ηηx + DQt = 0.

(78)

A. Karczewska and P. Rozmej / Commun Nonlinear Sci Numer Simulat 82 (2020) 105073

11

Eqs. (75) and (76) become compatible in first order when the condition (22) holds, that is

Qx − Qt = (hη )x . Using the correction Q given by (27) we obtain first order equation for the piecewise linear bottom relief as

1 4

3 2

ηt + ηx + αηηx − δ (2 hηx + hx η ) = 0.

(79)

This equation does not contain the dispersive term η3x . An attempt to derive second order wave equation fails. We can assume

 1

w=η+α −

4

η2 + D



1 2hηx + hx 4



η dx



+ α2

 2 − 3τ 6

 η2 x + D Q .

(80)

In w given by (80) three first terms ensure compatibility of the Boussinesq Eqs. (75) and (76) in first order, the fourth one makes terms with α 2 B = β compatible. Then the only unknown part of correction gives a condition on Qx − Qt which does not give any hope for determining Q through η, ηx , . . . , h, hx and then deriving second order wave equation. 7. Case 6:

β = O ( α2 ) , δ = O ( α2 )

The leading parameter is α . We set

β = Bα 2 , δ = Dα 2 .

(81)

The velocity potential is expressed, as in the Section 6, by (70), but the boundary condition at the bottom (6) takes now the form

φz − BDα 4 (hx φx ) = 0, for z = Dα 2 h(x ).

(82)

So, from (70) and (82) one gets

Fx = α 2 D(h fx )x +

1 1 6 2 2 α BD (h F2x )x − α 8 BD3 (h3 f3x )x + O(α 12 ) = 0. 2 6

Neglecting higher order terms we can use

Fx = α 2 D(h fx )x + O(α 6 ),

(83)

which ensures the expression of φ through only one unknown function f and its derivatives. Note that next terms in Fx would enter in φ in α 8 order. Then we can express the velocity potential as

1 2

φ = f − Bα 2 z 2 f 2 x +

1 3 6 6 1 1 2 4 4 B α z f 4x − B α z f6x + BDα 4 z(h fx )x − B2 Dα 6 z3 (h fx )3x + O(α 8 ). 24 720 6

(84)

With potential given by (84) we obtain from (4) and (5) the Boussinesq set

ηt + wx + α (ηw )x − α 2 1

1



1 Bw3x + D(hw )x − α 3 B(ηw2x )x 6 2



1 1 2 B ( η 2 w 2x ) x + B w 5x = 0 , 2 2 120     1 1 1 wt + ηx + α wwx − α 2 B τ η3x + w2xt + α 3 B −(ηwxt )x + wx w2x − ww3x 2 2 2   1 2 1 4 + α B D(hwt )2x + wx (ηwx )x − w(ηw2x )x − (η wxt )x + Bw4xt = 0, 2 24

α4

(85)

BD(hw )3x −

(86)

respectively, where terms up to α 4 are retained. In first order w = η + α (− 41 η2 ) makes (85) and (86) compatible giving

3 2

ηt + ηx + αηηx = 0.

(87)

In second order one assumes w in the form

w=η−





1 1 2 − 3τ αη2 + α 2 η3 + η2x + α 2 DQ, 4 8 6

(88)

which substituted to (85) and (86) gives (after neglection of terms of higher orders)

 3  3 1 − 3τ ηt + ηx + αηηx + α 2 − η2 ηx + Bη3x + α 2 (DQx − D(hη )x ) = 0 2

8

6

(89)

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from (85) and

 3  3 1 − 3τ ηt + ηx + αηηx + α 2 − η2 ηx + Bη3x + DQt = 0 2

8

(90)

6

from (86). Compatibility of Eqs. (89) and (90) requires the same condition as (22)

Qx − Qt = (hη )x . For a piecewise linear bottom profile, due to the universal result (27) given in Subsection 2.1, we obtain

 3  1 − 3τ 1 3 ηt + ηx + αηηx + α 2 − η2 ηx + βη3x − δ (2 hηx + hx η ) = 0. 2

8

6

4

(91)

Note that for the flat bottom (D = 0), w given by (88) makes the Boussinesq Eqs. (85) and (86) compatible and gives the well known Gardner equation

 3  1 − 3τ 3 ηt + ηx + αηηx + α 2 − η2 ηx + βη3x = 0. 2

8

(92)

6

Therefore, the Eq. (91) is the generalization of the Gardner equation for the case of the uneven bottom, valid when the bottom is given by the piecewise linear function. 8. Examples of numerical solutions In this section, we show some examples of motion of solitons, initialy moving over an even bottom when they enter the region where the bottom is no longer even. These results are obtained by numerical time evolution according to wave equations derived in previous sections. In order to be able to compare the influence of the bottom on the soliton movement, in all cases presented in this section, we assumed the same bottom shape in the form of a piecewise linear function. We discuss here only three cases: Case 1, where KdV solitons exist, Case 2, where KdV2 solitons were discovered by us in [43] and Case 4, where KdV5 solitons are known [52,53], as well. In the calculations we use our finite difference code described in detail in [43]. The cases 1 and 2 in which α = O(β ) are appropriate for shallow water waves where surface tension can be safely neglected. This is because when average water depth is of the order of some meters then τ < 10−7 . Therefore in the simulations presented in Subsections 8.1 and 8.2 we set τ = 0 in the corresponding wave equations. 8.1. Case 1 In this case α = O(β ), δ = O(β ), we performed calculations according to the first order Eq. (28). The piecewise linear bottom function is taken in the form



h (x ) =

1 X

0 x 1

for for for

x ≤ 0, 0X

(93)

with X = 15 for all cases considered. The shape of the bottom is displayed in presented figures below the snapshots of soliton’s motion, not in the same scale. In Fig. 2, we dispalyed the results of numerical simulation obtained for the following parameters: α = 0.2424, β = 0.3, δ = 0.15. As the initial condition we took the KdV soliton

η (x, t ) = A sech2 [B(x − x0 − vt )], where  3α α B= A and v = 1 + A, 4β 2

(94)

with x0 = −5, t = 0 and the amplitude A = 1 (to compare with the Case 2). If we come back to dimension variables then the soliton’s amplitude in meters is obtained by multiplying its nondimenional value by H. For instance, if H = 2 m then the initial soliton’s height (above the undisturbed wate level) is Hα ≈ 0.485 m. The horizontal coordinates have to be multiplied by l = √H ≈ 3.65 m. So, the range x ∈ [0, 15] in the disβ

played figures corresponds to the interval (approximately) [0,54] in meters. The time increment between the consecutive profiles in presented Figs. 2 and 3 is dt = 1.25 (in dimensionless units) what corresponds to approximately 0.56 s. Note that v0 ≈ 4.96 m/s. 8.2. Case 2 In this case, α = O(β ), δ = O(β 2 ), the appropriate wave equation is the Eq. (42). It is worth to remind that when δ = 0, the Eq. (42) reduces to the extended KdV (KdV2). Since KdV2 possesses exact soliton solution [43], we use this solution as the initial condition in the example presented in Fig. 3.

A. Karczewska and P. Rozmej / Commun Nonlinear Sci Numer Simulat 82 (2020) 105073

13

Fig. 2. Time evolution of the KdV soliton entering the region with the piecewise linear bottom function (93) obtained in numerical inegration of the Eq. (29) with parameters α = 0.2424, β = 0.3, δ = 0.15.

Fig. 3. Time evolution of the KdV2 soliton entering the region with the piecewise linear bottom function (93) obtained in numerical inegration of the Eq. (42) with parameters α = 0.2424, β = 0.3, δ = 0.15.

Contrary to the KdV equation which leaves one parameter freedom for the coefficients of the exact solutions (therefore KdV permits for solitons of different amplitudes), the parameters α , β of the KdV2 equation fix the coefficients of the unique soliton solution.So, for the evolution shown in Fig. 3 the initial condition has the same form (94) but with coefficients: A≈

0.2424

α

, B≈

0.6 βα A and v ≈ 1.11455. The parameter α = 0.2424 assures the amplitude equal one.

For comparison of the KdV2 soliton motion according to the Eq. (42) to the KdV soliton motion according to the Eq. (28) we used the same values of the parameters α , β , δ . Using δ = 0.15 = D β 2 with D ≈ 1.67, not much different from unity, does not contradict to the assumption δ = O(β 2 ).

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A. Karczewska and P. Rozmej / Commun Nonlinear Sci Numer Simulat 82 (2020) 105073

Fig. 4. Time evolution of the fifth order KdV soliton entering the region with the piecewise linear bottom function (93) obtained in numerical inegration of the Eq. (68) with parameters α = 0.2424, β = 0.3, δ = 0.15.

Fig. 5. Solitons’ maxima versus their positions from Fig. 2 - red symbols, from Fig. 3 - green symbols and from Fig. 4 - blue symbols. In the case of KdV5 displayed are the rescaled absolute values. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

8.3. Case 4 In this case, α = O(β 2 ), δ = O(β 2 ), the appropriate wave equation is the Eq. (68). It is worth to remind that when δ = 0, the Eq. (68) reduces to (49), the fifth-order KdV or KdV5 [50–53]. The fifth-order KdV has exact soliton solution [52,53], so we use this solution as the initial condition in the example presented in Fig. 4. The explicit form of this solution is

η (x, t ) = A Sech4 [B(x − vt )],

(95)

where A, B, v are functions of the coefficients of the 5th-order KdV Eq. (49). For illustration we chose τ = 0.35 > 13 which assures B ∈ R. The numerical evolution of the soliton (95) according to the Eq. (68) for δ = 0 (flat bottom) confirms that it moves with the constant shape and constant velocity.

A. Karczewska and P. Rozmej / Commun Nonlinear Sci Numer Simulat 82 (2020) 105073

15

Snapshots of numerical evolution of the fifth order KdV soliton (95) according to the Eq. (68) for δ = 0.15, α = 0.2424, β = 0.3 are displayed in Fig. 4. In dimensionless coordinates this setup is the same as in previously discussed examples in Subsections 8.1 and 8.2. Coming back to dimension variables we realize that it is the  system of completely different scale. First, taking T = 72 mN/m as water surface tension one obtains water depth h =

T gτ ≈ 0.0046 m. Next, with

β = 0.3 the dimensionless interval x ∈ [0, 15] corresponds to [0, 0.123 m]. Indeed, the Eq. (68) describes soliton motion in capilary-gravity case with the uneven bottom. In general, the properties of this motion in dimensionless variables are similar to those observed in Subsections 8.1 and 8.2 for the cases 1 and 2. 8.4. Brief comparison In all three cases displayed in Figs. 2–4 the solitons move initially over the flat bottom with undisturbed shapes and constant initial velocities. Next, moving over the slope all solitons experience an amplitude increase and a corresponding decrease of velocity. The deformation of the profile, that is a lowering of the water level behind the soliton when it reaches the flat region is very small in Case 1 and 2. In Case 3 it is almost impercetible. Qualitatively, the properties of soliton motion calculated in nondimensional variables are very similar in all these three cases. Particularly similar are cases 1 and 2. The differences are seen in detail in Fig. 5 in which the ratios of solitons maxima to their initial values max[η]/η0 versus their positions (read from the calculated data) are plotted. Since in KdV5 case the amplitudes are negative we plot their absolute values multiplied by 1/|A(t = 0 )| to compare the changes with respect to initial values. More detail studies of numerical evolution according to the derived equations are planned in a near future. 9. Conclusions In attempts to derive KdV-type equations for the case of the uneven bottom, we found two universal features. First, the boundary condition at the uneven bottom implies the limit on the order of theory in which the Boussinesq equations can be derived. It usually does not exceed the second order, whereas for the flat bottom one can proceed to arbitrary order. Second, regardless of the different ordering of small parameters α , β , δ , the condition of compatibility of Boussinesq’s equations containing the bottom terms is always the same in the lowest order and given by (22). This condition cannot be satisfied for an arbitrary bottom profile h(x). However, the condition (22) is satisfied when the bottom relief is given by an arbitrary piecewise linear function. For such bottom profiles, we derived four KdV-type equations which generalize equations known for the flat bottom case. The Eq. (29) generalizes the usual KdV equation. In (29) terms originating from the bottom are of first order. The Eq. (42) generalizes the extended KdV (KdV2) equation. The Eq. (68) generalizes the fifth-order KdV (KdV5) equation whereas (91) generalizes the Gardner equation. In the last three cases, terms induced by the piecewise linear bottom are of second order. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments P.R. thanks for the financial support from the program of the Polish Minister of Science and Higher Education under the name “Regional Initiative of Excellence” in 2019–2022, project no. 003/RID/2018/19, funding amount 11 936 596.10 PLN. This paper is dedicated to the memory of our friend Professor Eryk Infeld, who recently passed away. References [1] Korteweg DJ, de Vries G. On the change of form of the long waves advancing in a rectangular canal, and on a new type of stationary waves. Phil Mag 1985;39(5):422–43. [2] Whitham GB. Linear and Nonlinear Waves. New York: John Wiley & Sons; 1974. [3] Drazin PG, Johnson RS. Solitons: an Introduction. Cambridge University Press; 1989. [4] Ablowitz MJ, Clarkson PA. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press; 1991. [5] Hirota R. The Direct Method in Soliton Theory. Cambridge University Press; 2004. First published in Japanese 1992 [6] Remoissenet M. Waves Called Solitons: Concepts and Experiments. Berlin: Springer; 1999. [7] Infeld E, Rowlands G. Nonlinear Waves, Solitons and Chaos. Cambridge University Press; 20 0 0. [8] Osborne AR. Nonlinear Ocean Waves And The Inverse Scattering Transform. Amsterdam: Academic Press; 2010. [9] Mei CC, Le Méhauté B. Note on the equations of long waves over an uneven bottom. J Geophys Res. 1966;71:393–400. [10] Grimshaw R. The solitary wave in water of variable depth. J Fluid Mech 1970;42:639–56. [11] Djordjevic´ VD, Redekopp LG. On the development of packets of surface gravity waves moving over an uneven bottom. J Appl Math Phys (ZAMP) 1978;29:950–62. [12] Benilov ES, Howlin CP. Evolution of packets of surface gravity waves over strong smooth topography. Stud Appl Math 2006;116:289–301. [13] Nakoulima O, Zahibo N, Pelinovsky E, Talipova T, Kurkin A. Solitary wave dynamics in shallow water over periodic topography. Chaos 2005;15:037107. [14] Grimshaw R, Pelinovsky E, Talipova T. Fission of a weakly nonlinear interfacial solitary wave at a step. Geophys Astrophys Fluid Dyn 2008;102:179–94.

16

A. Karczewska and P. Rozmej / Commun Nonlinear Sci Numer Simulat 82 (2020) 105073

[15] Pelinovsky E, Choi BH, Talipova T, Woo SB, Kim DC. Solitary wave transformation on the underwater step: theory and numerical experiments. Appl Math Comput 2010;217:1704–18. [16] RHJ G, Smyth NF. Resonant flow of a stratified fluid over topography. J Fluid Mech 1986;169:429–64. [17] Smyth NF. Modulation theory solution for resonant flow over topography. Proc R Soc Lond A 1987;409:79–97. [18] Pelinovskii EN, Slyunayev AV. Generation and interaction of large-amplitude solitons. JETP Lett 1998;67:655–61. [19] Kamchatnov AM, Y-H K, T-C L, Horng T-L, Gou S-C, Clift R, El GA, RHJ G. Undular bore theory for the gardner equation. Phys Rev E 2012;86:036605. [20] van Greoesen E, Pudjaprasetya SR. Uni-directional waves over slowly varying bottom. Part I: derivation of a KdV-type of equation. Wave Motion 1993;18:345–70. [21] Pudjaprasetya SR, van Greoesen E. Uni-directional waves over slowly varying bottom. Part II: quasi-homogeneous approximation of distorting waves. Wave Motion 1996;23:23–38. [22] Green AE, Naghdi PM. A derivation of equations for wave propagation in water of variable depth. J Fluid Mech 1976;78:237–46. [23] Kim JW, Bai KJ, Ertekin RC, Webster WC. A derivation of the green-naghdi equations for irrotational flows. J Eng Math 2001;40:17–42. [24] Nadiga BT, Margolin LG, Smolarkiewicz PK. Different approximations of shallow fluid flow over an obstacle. Phys Fluids 1996;8:2066–77. [25] Selezov IT. Propagation of unsteady nonlinear surface gravity waves above an irregular bottom. Int J Fluid Mech 20 0 0;27:146–57. [26] Niu X, Yu X. Analytic solution of long wave propagation over a submerged hump. Coastal Eng 2011;58:143–50. 2011 Liu H-W.Xie, J-J. Discussion of “Analytic solution of long wave propagation over a submerged hump” by Niu and Yu (2011). Coastal Eng 2011 58 948–952 [27] Israwi S. Variable depth KdV equations and generalizetions to more nonlinear regimes. ESAIM 2012;44:347–70. [28] Duruflé M, Israwi S. A numerical study of variable depth KdV quations and generalizetions of camassa-holm-like equations. J Comp Appl Math 2010;236:4149–265. [29] Yuan C, Grimshaw R, Johnson E, Chen X. The propagation of internal solitary waves over variable topography in a horizontally two-dimensional framework. J Phys Ocean 2018;48:283–300. [30] Fan L, Yan W. On the weak solutions and persistence properties for the variable depth KDV general equations. Nonlinear Anal 2018;44:223–45. [31] Stepanyants Y. The effects of interplay between the rotation and shoaling for a solitary wave on variable topography. Stud Appl Math 2019;142:465–86. [32] Madsen O, Mei C. The transformation of a solitary wave over an uneven bottom. J Fluid Mech 1969;39:781–91. [33] Kakutani T. Effect of an uneven bottom on gravity waves. J Phys Soc Japan 1971;30:272–6. [34] Johnson RS. Some numerical solutions of a variable-coefficient Korteweg-de Vries equation (with applications to solitary wave development on a shelf). J Fluid Mech 1972;54:81–91. [35] Johnson R. On the development of a solitary wave moving over an uneven bottom. Math Proc Cambridge Phil Soc 1973;73:183–203. [36] Benilov ES. On the surface waves in a shallow channel with an uneven bottom. Stud Appl Math 1992;87:1–14. [37] Rosales RR, Papanicolau GC. Gravity waves in a channel with a rough bottom. Stud Appl Math 1983;68(2):89–102. [38] Nachbin A, Papanicolau GC. Water waves in shallow channels of rapidly varying depth. J Fluid Mech 1992;241:311–32. [39] Craig W, Guyenne P, Nichols DP, Sulem C. Hamiltonian long-wave expansions for water waves over a rough bottom. Proc Roy Soc A 2005;461:839–73. [40] Peregrine DH. Long waves in a uniform channel of arbitrary cross-section. J Fluid Mech 1968;32(2):353–65. [41] Chassagne R, Filippini AG, Ricchiuto M, Bonneton P. Dispersive and dispersive-like bores in channels with sloping banks. J Fluid Mech 2019;870:595–616. [42] Craig W, Lannes D, Sulem C. Water waves over a rough bottom in the shallow water regime. Ann I H Poincaré-AN 2012;29:233–59. [43] Karczewska A, Rozmej P, Infeld E. Shallow water soliton dynamics beyond KdV. Phys Rev E 2014;90:012907. [44] Burde GI, Sergyeyev A. Ordering of two small parameters in the shallow water wave problem. J Phys A 2013;46:075501. [45] Rozmej P, Karczewska A. Comment on the paper “the third-order perturbed Korteweg-de Vries equation for shallow water waves with a non-flat bottom”. Eur Phys J Plus, 132. Fokou M, Kofané TC, Mohamadou A, Yomba E, editors; 2017. arXiv: 1804.01940. The review by the anonymous referee of the paper. [46] Marchant TR, Smyth NF. The extended Korteweg-de Vries equation and the resonant flow of a fluid over topography. J Fluid Mech 1990;221:263–88. [47] Infeld E, Karczewska A, Rowlands G, Rozmej P. Exact cnoidal solutions of the extended KdV equation. Acta Phys Pol A 2018;133:1191–9. [48] Rozmej P, Karczewska A, Infeld E. Superposition solutions to the extended KdV equation for water surface waves. Nonlinear Dyn 2018;91:1085–93. [49] Rozmej P, Karczewska A. New exact superposition solutions to KdV2 equation. Adv Math Phys 2018;2018:5095482. [50] Hunter JK, Scheurle J. Existence of perturbed solitary wave solutions to a model equation for water waves. Phys D 1988;32:253–68. [51] Grimshaw R, Malomed B, Benilov E. Solitary waves with damped oscillatory tails – an analysis of the 5th-order Korteweg-de Vries equation. Phys D 1994;77:473–85. [52] Dey B, Khare A, Kumar CN. Stationary solitons of the fifth order KdV-type. Equations and their stabilization. Phys Lett A 1996;223:449–52. [53] Bridges TJ, Derks G, Gottwald G. Stability and instability of solitary waves of the fifth order KdV equation: a numerical framework. Phys D 2002;172:190–216. [54] Rosenau P, Hyman JM. Compactons: solitons with finite wavelength. Phys Rev Lett 1993;70:564–7.