Generalized KdV-type equations versus Boussinesq's equations for uneven bottom -- numerical study
GGeneralized KdV-type equations versus Boussinesq’s equations for uneven bottom -numerical study
Anna Karczewska ∗ Faculty of Mathematics, Computer Science and EconometricsUniversity of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra, Poland
Piotr Rozmej † Faculty of Physics and Astronomy, University of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra, Poland (Dated: July 7, 2020)The paper’s main goal is to compare the motion of solitary surface waves resulting from twosimilar but slightly different approaches. In the first approach, the numerical evolution of solitonsurface waves moving over the uneven bottom is obtained using single wave equations. In thesecond approach, the numerical evolution of the same initial conditions is obtained by the solutionof a coupled set of the Boussinesq equations for the same Euler equations system. We discuss fourphysically relevant cases of relationships between small parameters α, β, δ . For the flat bottom,these cases imply the Korteweg-de Vries equation (KdV), the extended KdV (KdV2), fifth-orderKdV (KdV5), and the Gardner equation (GE). In all studied cases, the influence of the bottomvariations on the amplitude and velocity of a surface wave calculated from the Boussinesq equationsis substantially more significant than that obtained from single wave equations.
PACS numbers: 02.30.Jr, 05.45.-a, 47.35.Bb, 47.35.FgKeywords: KdV-type equations, Gardner equation, uneven bottom, numerical evolution
I. INTRODUCTION - THE CONCEPT OF THESTUDY
Nonlinear waves are the subject of a vast number ofstudies in many fields of science. They appear in hy-drodynamics, propagation of optical and acoustic waves,plasma physics, electrical circuits, biology, and many oth-ers. These equations usually appear as approximations ofmore basic laws describing the behavior of relevant sys-tems, usually too complicated for non-numerical analy-sis. These approximations assume that some parameterscharacterizing the system are small, and then a pertur-bative approach can be used. In this way, one can derivevarious nonlinear wave equations, e.g., the
Korteweg-deVries equation (KdV), the extended Korteweg-de Vriesequation (KdV2), or the
Gardner equa-tion. All these equations can be derived from the Eulerequations describing the model of the irrotational motionof an inviscid and incompressible fluid in a container witha flat, impenetrable bottom.The real world, however, is not that simple. In partic-ular, bottoms of oceans, seas, rivers are non-flat. There-fore, it would be desirable to find a relatively simplemathematical description that would take into accountbottom variations. In the past, there were many at-tempts to attack this problem. In this article, we onlybriefly remind some of these works. Some first resultswere obtained by Mei and Le Méhauté [1], and Grimshaw[2]. Several authors [3, 4] studied these problems us-ing variable coefficient nonlinear Schrödinger equation ∗ [email protected] † [email protected] (NLS). Some research groups developed approaches com-bining linear and nonlinear theories [5–7]. The Gardnerequation was also extensively investigated in this context[8–11]. The Hamiltonian approach was utilized by VanGroeasen and Pudjaprasetya [12, 13]. Another widely ap-plied method consists in taking an appropriate averageof vertical variables, which results in the Green-Naghdiequations [14–16]. Several authors derived variable co-efficient KdV equation (vcKdV) [24–28] in attempts todescribe the evolution of a solitary wave moving onto ashelf. Article [29] is the only one known to us (apart fromour approach) in which the authors introduce besides twosmall standard parameters, the third one associated withan uneven bottom. We presented a broader discussion ofsome of the current attempts and methods to account foruneven bottoms in [30].In the paper [30], we derived equations of the KdVtype for an uneven bottom for various relationships be-tween small parameters α, β, δ . For a flat bottom, onecan always eliminate the w function from the Boussinesqequations and get a single wave equation for the η func-tion (surface distortion from the equilibrium state). Foran uneven bottom, this can only be done for the lowestpossible order of the perturbation approach, and onlyif the bottom is a piecewise linear function. In othercases, there is no w function that makes the Boussi-nesq equations compatible. Therefore, for testing surfacewaves in the case of an uneven bottom studying the setof Boussinesq’s equation seems to be more appropriate.The present work supplements [30] with a comparisonof these two methods, including the study of the Gard-ner equation and calculations for much longer evolutiontimes.In [30], we derived four new wave equations, which a r X i v : . [ n li n . PS ] J u l generalize for the case of uneven bottom the Korteweg-de Vries equation (KdV), the extended KdV (KdV2),the fifth-order KdV, and the Gardner equation (com-bined KdV - mKdV). The first is obtained for α = O ( β ) , δ = O ( β ) , the second for α = O ( β ) , δ = O ( β ) , thethird for α = O ( β ) , δ = O ( β ) and the fourth for β = O ( α ) , δ = O ( β ) . In all cases, the generalized waveequations could be derived only for a particular class ofbottom functions, namely the piecewise linear ones. Onthe way to these results, we derived corresponding setsof the Boussinesq equations, which are valid for bottomsof arbitrary shapes.However, it seems that in numerical simulations ofwave evolution according to these generalized equations,all of them can be used for arbitrary bottom functions.The reason consists in the discretization of numericalcodes. The knowledge of the bottom function is neededonly in the mesh points, like when the bottom functionis a piecewise linear one.In the paper, we numerically test the results of the evo-lution of the nonlinear waves obtained from the Boussi-nesq equations with those obtained from the correspond-ing single KdV-type equations generalized for the unevenbottom in [30]. We assume that initial conditions corre-spond to solitons appropriate to the particular case. Suchsoliton can be formed in a region of flat bottom, and nextenter the region where the bottom is varying.The paper is organized as follows. In section II webriefly remind the reader of the Euler equations for theirrotational motion of the inviscid, incompressible fluid,which arises for the shallow water problem. This set ofequations can serve as a starting point for the derivationof both Boussinesq’s equations and the single wave equa-tion for each particular case of ordering of small parame-ters. In section III the case of generalized KdV equationis analyzed. In section IV we discuss the generalizedextended KdV (KdV2). Next, in section V the gener-alized fifth-order KdV is studied. Section VI is devotedto the generalized Gardner equation. In section VII, westudied some examples in which the initial conditions aresubstantially different from the solitons appropriate forparticular equations. The conclusions are contained inSection VIII. II. EULER EQUATIONS FOR AN UNEVENBOTTOM
To make the paper self-contained, we briefly remindthe approach to the shallow water problem in a moregeneral case when the bottom of the fluid is not even.The model applies to the waves on both the surface ofthe liquid and the interface between two immiscible flu-ids. A detailed description of the model and methods ofderiving relevant nonlinear wave equations is presentedin our work [30].The set of Euler equations, written in nondimensional variables has the following form βφ xx + φ zz = 0 , (1) η t + αφ x η x − β φ z = 0 , (2) φ t + 12 αφ x + 12 αβ φ z + η − τ β η x (1 + α βη x ) / = 0 , (3) φ z − βδ ( h x φ x ) = 0 . (4)Equation (1) is the Laplace equation for the velocity po-tential valid for the whole volume of the fluid. Equations(2) and (3) are so-called kinematic and dynamic bound-ary conditions at the surface, that is for z = 1 + αη ,respectively. The equation (4) represents the boundarycondition at the non-flat unpenetrable bottom, i.e. for z = δh ( x ) . In (3), the Bond number τ = T(cid:37)gh , where T isthe surface tension coefficient. For surface gravity waves,this term can be safely neglected, since τ < − (whenthe fluid depth is of the order of meters), but it can be im-portant for waves in thin fluid layers. For abbreviation allsubscripts in (1)-(4) denote the partial derivatives withrespect to particular variables, i.e. φ t ≡ ∂φ∂t , η x ≡ ∂ η∂x ,and so on.The parameters α, β, δ in the set (1)-(4) have thefollowing meaning. Besides standard small parameters α = aH and β = (cid:0) Hl (cid:1) we introduced the third one, de-fined as δ = a h H . Here a represents the wave amplitude, H - average depth of the basin, l - average wavelengthand a h - amplitude of bottom variations. For the pertur-bation approach, all of them should be small, howevernot necessarily of the same order. Therefore for differentordering of these parameters one can derive different setsof the Boussinesq equations and in consequence differentwave equations. The cases with flat bottom ( δ = 0) arepresented in [31]. We already introduced the third smallparameter δ = a h H in [34] in order to generalize the ex-tended KdV equation (KdV2) for the case of the unevenbottom. Unfortunately, the derivation presented in [34]is not fully consistent, and the final equation contains animproper term additionally.As usual, the velocity potential is seeking in the formof power series in the vertical coordinate φ ( x, z, t ) = ∞ (cid:88) m =0 z m φ ( m ) ( x, t ) , (5)where φ ( m ) ( x, t ) are yet unknown functions. The Laplaceequation (1) determines φ in the form, which involvesonly 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 ) = ∞ (cid:88) m =0 ( − m β m (2 m )! ∂ m f ( x, t ) ∂x m z m (6) + ∞ (cid:88) m =0 ( − m β m +1 (2 m + 1)! ∂ m +1 F ( x, t ) ∂x m +1 z m +1 . The explicit form of this velocity potential reads as φ = f − βz f x + 124 β z f x − β z f x + · · · + βzF x − β z F x + 1120 β z F x + · · · (7)In the next step, one uses the boundary condition atthe bottom (4). For a standard flat bottom case it fol-lows that F x = 0 and only f and its even x -derivativesremain in (7). For an uneven bottom, the situation ismore complicated, and one can express F x explicitly by f only in some low order. Precisely this order dependson the relation between β and δ parameters. Below weshow this step explicitly for the case δ = O ( β ) . For othercases, in which the procedure is analogous, śwe refer to[30]. Insertion of the velocity potential (7) into (4) gives(with z = δh ( x ) ) the following complicated relation be-tween the functions F x and fF x − δ ( hf x ) x − βδ ( h F x ) x + 16 βδ ( h f x ) x + 124 β δ ( h F x ) x + · · · = 0 . (8)Keeping only terms lower than third order leaves F x = δ ( hf x ) x , (9)which allows us to express the x -dependence of the ve-locity potential through f, h , and their x -derivatives upto second order. This fact limits the velocity potential tothe form φ = f − βz f x + 124 β z f x + βδz ( hf x ) x (10)valid only up to second order in small parameters. At-tempts to go to higher orders would require solving theequation (8) for F with arbitrary h , which is impossibleto do. III. CASE α = O ( β ) , δ = O ( β ) -GENERALIZATION OF KDV This case corresponds to shallow water waves. Sincethe coefficient of surface tension is very small, one cansafely neglect the appropriate term in the Euler equa-tions.Due to the presence of the term − β φ z in (2), theBoussinesq equations resulting from the substitution of(10) into (2) and (3) are correct only up to first order in α, β and δ . They take the following form (see, [30], eqs.(17)-(18)) η t + w x + α ( ηw ) x − βw x − δ ( hw ) x = 0 , (11) w t + η x + αww x − βw xt = 0 . (12) α =0.2424, β =0.2, δ =0.0 η ( x ,t ) , w ( x ,t ) x KdVBus- η Bus-w
FIG. 1. Time evolution of the KdV soliton (17) obtained ac-cording to KdV equation (13) – black lines and that obtainedfrom the Boussinesq set (11)-(12) – blue lines. Additionally,the evolution of w ( x, t ) function is displayed with green lines.Flat bottom ( δ = 0 ) is assumed. Elimination of w from (11)-(12) in order to obtain a sin-gle wave equation for η appears to be possible only when h x = 0 , that is when the bottom function is the piece-wise linear one. In such case the system (11)-(12) can bemade compatible, and reduced to the single KdV-typeequation ([30], eq. (28)) η t + η x + 32 αηη x + 16 βη x − δ (2 hη x + h x η ) = 0 . (13)On the other hand, the Boussinesq equations do not re-quire the condition h x = 0 , the bottom function h canbe arbitrary. From this point of view the Boussinesqequations (11)-(12) are more general (more fundamen-tal) than the single wave equation (13).It is worth to emphasize that the above properties aregeneral. They are the same for all cases (all wave equa-tions) discussed in this paper. For more details on thederivation of nonlinear wave equations generalized for theuneven bottom, we refer to [30]. -0.2 0 0.2 0.4 0.6 0.8 1 1.2-50 0 50 100 150 200 250 300 350 400 α =0.1, β =0.1, δ =0.2 η ( x ,t ) x 2k*dt(2k+1)*dth(x) FIG. 2. Time evolution of the KdV soliton (17) obtainedaccording to KdV equation (13) for the bottom given by (18)with δ = 0 . . Subsequent profiles correspond to times t n = n ∗ , with n = 0 , , . . . , . The shape of the bottom functionis drawn in an arbitrary scale. In numerical simulations, we can apply the FDM (finitedifference method) with leap-frog, which stability is welldetermined for appropriate relation between time step ∆ t and mesh size ∆ x .For the equation (13) the appropriate algorithm is thefollowing η j +1 i = η j − i − t (cid:18) ( η x ) j − i + 32 αη j − i ( η x ) j − i (14) + 16 β ( η x ) j − i −
14 (2 h i ( η x ) j − i + ( h x ) i η j − i ) (cid:19) . For the Boussinesq set (11)-(12), we have to evolve twoequations simultaneously η j +1 i = η j − i − t (cid:104) ( w x ) j − i + α (cid:16) ( η x ) j − i w j − i + η j − i ( w x ) j − i (cid:17) − β ( w x ) j − i − δ (cid:16) ( h x ) i w j − i + h i ( w x ) j − i (cid:17) (cid:105) (15) w j +1 i = w j − i − t (cid:18) ( η x ) j − i + αw j − i ( w x ) j − i − β ( w xt ) j − i (cid:19) . (16)In (14)-(16) , i = 0 , . . . , N − is the index of the meshpoint x i and j enumerates time step. Periodic boundaryconditions in x are used. Time increment ∆ t = (∆ x ) assures stability of the time integration. Setting δ = 0 one obtains the set of equations corresponding to theKorteweg-de Vries equation.In first tests of the code we use initial condition in theform of the KdV soliton, that is, η ( x, , where η ( x, t ) = A sech (cid:20)(cid:114) α β A (cid:16) x − t (cid:16) A α (cid:17)(cid:17)(cid:21) = A sech [ B ( x − vt )] . (17)Then the initial condition for w is given by w = η − αη + 13 βη x at t = 0 . In Fig. 1, numerical results of the KdV soliton (17)evolution for α = 0 . , β = 0 . and δ = 0 , that isfor the flat bottom, are shown. The KdV soliton am-plitude is chosen to be A = 1 for comparison with theKdV2 case shown in Fig. 5. In both Figs. 1 and 3, timeseparation between displayed wave profiles is dt = 16 .Results, shown in Fig. 1, can be considered as a check ofthe numerical code. In the KdV case, the soliton moveswith the constant velocity ( v = 1 + α A ) and a fixed pro-file. Since initial conditions are chosen as KdV soliton,the η and w functions evolving according to Boussinesq’sequations develop very small tails and move with slightlydifferent velocity, but profiles of their main parts exhibita soliton motion.Next, we calculate the case in which the KdV soliton,formed on a flat bottom area enters the region over an -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-50 0 50 100 150 200 250 300 350 400 α =0.1, β =0.1, δ =0.2 η ( x ,t ) , w ( x ,t ) x Bus- η Bus-wh(x)
FIG. 3. Time evolution of the KdV soliton (17) and corre-sponding w function obtained according to the Boussinesqset (11)-(12) for the bottom given by (18) with δ = 0 . and α = β = 0 . . Subsequent profiles correspond to times t n = n ∗ , with n = 0 , , . . . , . To avoid overlaps, profilesof w function are shifted by 0.3 up and by 8 left. extended bump of the shape given by the function h ( x ) = 12 ( tanh [0 . x − tanh [0 . − x )]) . (18)The results of numerical evolution of the KdV solitonaccording to the equation (13) (precisely, according to itsdiscretized version (14)) for the case α = β = 0 . , δ = 0 . are presented in Fig. 2. Time separation between consec-utive wave profiles is dt = 16 . These results show thataccording to the generalized KdV equation (13), the un-even bottom implies only minimal variations of solitonsamplitude and velocity and creates a kind of small tail.In Fig. 3, we present the sequence of profiles obtainedin numerics for the set of Boussinesq’s equations (11)-(12). Contrary to results from the KdV generalized forpiecewise linear bottom function (13), in this case, wehave almost ideal soliton shapes, without secondary soli-ton trains. Moreover, both η and w evolve similarly, withrelative changes of w bigger than those of η . These rel-ative changes are magnified in Fig. 4. One has to stressthat the changes in the surface wave amplitude and ve-locity obtained from the set of Boussinesq equations (11)-(12) are substantially greater than those obtained fromKdV equation (13), presented in Fig. 2. These propertiesof results remain similar for a wide range of parameters α, β when the bottom is the same. IV. CASE α = O ( β ) , δ = O ( β ) -GENERALIZATION OF KDV2 In this case (see details in [30]), from the boundarycondition at the bottom we obtain F x = βδ ( hf x ) x , (19) α =0.1, β =0.1, δ =0.2 η ( x ,t ) , w ( x ,t ) x Bus- η Bus-w
FIG. 4. Details of three profiles of η ( x, t ) and w ( x, t ) displayedin Fig. 3 coresponding to time instants t = 0 , and 352.The second and third profile was shifted near the initial onefor comparison. valid up to fourth order in β which inserted into (7) givesthe velocity potential valid up to fourth order φ = f − βz f x + 124 β z f x − β z f x + β δz ( hf x ) x + 140320 β z f x + O ( β ) . (20)In principle, the Boussinesq equations can be consistentlyderived up to third order (remember term − β φ x in (2)).However, we will proceed to second order, only.Keeping only terms up to second order (for consistencywith the order of approximation used in bottom bound-ary condition) one arrives at the second order Boussinesqset (see, [30], eqs. (37)-(38)) η t + w x + α ( ηw ) x − βw x − αβ ( ηw x ) x + 1120 β w x − δ ( hw ) x = 0 , (21) w t + η x + αww x − β w xt + 124 β w xt (22) + 12 αβ ( − ηw xt ) x + w x w x − ww x ) = 0 . In the case of the flat bottom, that is when δ = 0 , anappropriate form of w , precisely w = η − α η + β η x + α η (23) + αβ (cid:18) η x + 12 ηη x (cid:19) + β η x makes the equations (21)-(22) identical. The resultedequation is known as the extended KdV [33] or KdV2 [35] η t + η x + α ηη x + β η x − α η η x (24) + αβ (cid:18) η x η x + 512 ηη x (cid:19) + β η x = 0 . We proved recently that the extended KdV equations(24), despite its nonintegrability, possesses three kinds ofanalytic solutions of the same form as the correspondingKdV solutions, with slightly different coefficients. In [34],we found single soliton solution of the form η ( x, t ) = A sech [ B ( x − vt )] . This form is the same as the formof the KdV soliton (17), but the coefficients are slightlydifferent. In [36], we found cnoidal solutions of the form η ( x, t ) = A cn [ B ( x − vt )] + D whereas in [37, 38] we foundso called ’superposition’ periodic solutions of the form η ( x, t ) = A ( dn [ B ( x − vt )] ± √ m cn [ B ( x − vt )] dn [ B ( x − vt )]) , where cn , dn are Jacobi elliptic functions. It isworth to emphasize that contrary to the KdV case, exactmulti-soliton solutions to the KdV2 do not exist [39]. α =0.2424, β =0.2 η ( x ,t ) , w ( x ,t ) x KdV2Bus- η Bus-w
FIG. 5. The same as in Fig. 1, but for the extended KdV(KdV2) equation (24) and second order Boussinesq’s set (21)-(22) with δ = 0 . Equations (21) and (22) can be made compatible onlyfor when h x = 0 . In such case, the generalization ofthe KdV2 (24) contains additional terms originating fromthe bottom variations (the bottom term is the same asin (13)) η t + η x + α ηη x + β η x − α η η x (25) + αβ (cid:18) η x η x + 512 ηη x (cid:19) + β η x − δ (2 hη x + h x η ) = 0 . In numerical calculations, we use the same FDMmethod as that described by equations (14)-(16), ex-tended by including appropriate terms, second order insmall parameters. As initial condition for η ( t = 0) theKdV2 solitons are used, whereas the initial conditionfor w is given by (23) with substitution η = η ( t = 0) .So, for the evolution shown in Fig. 5 the initial condi-tion has the the same form (17) but with coefficients: A ≈ . α , B ≈ (cid:113) . αβ A and v ≈ . . The param-eter α = 0 . assures the amplitude equal one.Now, we will compare the time evolution of the KdV2soliton, obtained according to second order equations(KdV2 or extended KdV ). In Fig. 6, we display profiles ofKdV2 soliton, which enters the region of the uneven bot-tom. The time evolution is obtained from the generalizedKdV2 equation (25). The behavior of solutions, despitedifferent values of small parameters, remains very similarto that presented in Fig. 2 for the first order equation. -0.2 0 0.2 0.4 0.6 0.8 1 1.2-50 0 50 100 150 200 250 300 350 400 α =0.24, β =0.3, δ =0.15 η ( x ,t ) x 2k*dt(2k+1)*dth(x) FIG. 6. The same as in Fig. 2, but for the extended KdV(KdV2) equation (25) and δ = 0 . . In Fig. 7, the initial KdV2 soliton evolves accordingto second order Boussinesq’s equations (21)-(22). In thiscase, similarly as in Fig. 3, one observes the much greaterinfluence of the bottom variation on changes of soliton’samplitude and velocity. -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4-50 0 50 100 150 200 250 300 350 400 α =0.2424, β =0.3, δ =0.15 η ( x ,t ) , w ( x ,t ) x Bus- η Bus-wh(x)
FIG. 7. The same as in Fig. 3, but for the second orderBoussinesq’s set (21)-(22).
V. CASE α = O ( β ) , δ = O ( β ) -GENERALIZATION OF FIFTH-ORDER KDVEQUATION In this case, since δ = O ( β ) , the forms of the func-tion F x and the velocity potential are given by (19)-(20).Keeping only terms up to second order one arrives at the second order Boussinesq system (see, [30], eqs. (61)-(62)) η t + w x − β w x + α ( wη ) x + 1120 β w x − δ ( hw ) x = 0 , (26) w t + η x − β (cid:18) w xt + τ η x (cid:19) + α ww x + 124 β w xt = 0 . (27)Here, one has to keep terms from surface tension τ (cid:54) = 0 .These terms are important because for the flat bottom( δ = 0 ), the equations (26)-(27) can be made compati-ble leading to so-called fifth-order KdV equation derivedby Hunter and Sheurle in [40] as a model equation forgravity-capillary shallow water waves of small amplitude.Similarly like in the previous sections for uneven bot-tom, the equations (26)-(27) can be made compatibleonly when the bottom function is piecewise linear. Theresulting wave equation, a generalization of the fifth-order KdV equation has the following form (see, eq. (68)in [30]) η t + η x + 32 αηη x + β − τ η x + β − τ − τ η x − δ (2 hη x + h x η ) = 0 . (28)The equation (28) differs from the fifth-order KdV equa-tion by the last term only.In numerical simulations, we again want to comparethe time evolution of surface waves obtained from thesingle wave equation (28) with time evolution obtainedfrom the Boussinesq set (26)-(27).It is well known, see, e.g. [41, 42], that the fifth orderKdV equation has a soliton solution in the form η ( x, t ) = A sech [ B ( x − vt )] . (29)For the fifth order KdV equation in the form (28) oneobtains the following values of the coefficients: A = 700(1 − τ ) −
19 + 30 τ + 45 τ ) α , (30) B = (cid:115) − τ )13( −
19 + 30 τ + 45 τ ) β and v = − τ + 10845 τ −
19 + 30 τ + 45 τ ) . (31)Real solutions require τ > . Using τ = 0 . weobtain A ≈ − . /α , B ≈ . /β and v ≈ . . To begin evolution according to theBoussinesq equations one needs the initial condition for w function which has the following form w ( x, t ) = η + β − τ η x − αη + β − τ − τ η x . (32)The numerical results of the time evolution of 5th-orderKdV soliton according to equation (28) are presented inFig. 8. The evolution of the same initial 5th-order KdVsoliton according to Boussinesq’s equations (26)-(27) isdisplayed in Fig. 9. Similarly, as in the previous section,the impact of the bottom variation on the surface wavemanifests more evident in the case of Boussinesq’s equa-tions. -0.015-0.01-0.005 0-50 0 50 100 150 200 250 300 350 400 α =0.24, β =0.3, δ =0.15 η ( x ,t ) x 2k*dt(2k+1)*dth(x) FIG. 8. The same as in Fig. 2 but for the 5th-order KdVequation (28). -0.015-0.01-0.005 0 0.005-50 0 50 100 150 200 250 300 350 400 α =0.24, β =0.3, δ =0.15 η ( x ,t ) , w ( x ,t ) x Bus- η Bus-wh(x)
FIG. 9. Profiles of functions η and w obtained from the secondorder Boussinesq set (26)-(27). In order to avoid overlapsprofiles of w function are shifted by 0.005 up and by 8 left. VI. CASE β = O ( α ) , δ = O ( α ) -GENERALIZATION OF THE GARDNEREQUATION In this case, the leading parameter is parameter α . Theboundary condition at the bottom requires F x − δ ( hf x ) x + 12 βδ ( h F x ) x + O ( α ) = 0 . Neglecting higher order terms we can use F x = δ ( hf x ) x + O ( α ) , (33) which ensures the expression of φ through only one un-known function f and its derivatives. Now, the Boussi-nesq set (up to second order) is given by (see, eqs. (85)-(86) in [30]) η t + w x + α ( ηw ) x − β w x − δ ( hw ) x = 0 , (34) w t + η x + αww x − β (cid:18) τ η x + 12 w xt (cid:19) = 0 . (35)Formally, the equations (34)-(35) are identical to theequations (11)-(12) obtained for the case α ≈ β ≈ δ , thatis 1st order equations that lead to the KdV equation when δ = 0 . This suggests that the solutions η, w of the systemof equations (34)-(35) may have identical functional formto those from the equation KdV.Similarly, as in the previous sections for the unevenbottom, the equations (34)-(35) can be made compatibleonly when the bottom function is piecewise linear. Theresulting wave equation, a generalization of the Gardnerequation has the following form (see, eq. (91) in [30]) η t + η x + 32 αηη x + α (cid:18) − η η x (cid:19) + 1 − τ β η x − δ (2 hη x + h x η ) = 0 . (36)Setting δ = 0 gives the well known Gardner equation(combined KdV-mKdV equation) η t + η x + 32 αηη x + α (cid:18) − η η x (cid:19) + 1 − τ β η x = 0 . (37)In this case the w function, limited to second order termsis, (see, e.g. [31, Eq. (A.1)]) w = η − αη + 18 α η + 2 − τ βη x . (38)It is well known, e.g. [43, 44], that for the Gardnerequation (37) there exists one parameter family of ana-lytic solutions in the form η ( x, t ) = A B cosh [( x − v t ) / ∆] . (39)The equation (37) imposes three conditions on coeffi-cients A, B, v, ∆ of solutions. So, three of them can beexpressed as functions of the single one. Choosing ∆ as the independent parameter one obtains the followingrelations A = 2 β α , B = ± (cid:114) − β , V = 1 + β . (40)Soliton’s amplitude is then η = A B = 2 β α ∆ (cid:18) ± (cid:113) − β (cid:19) . For B ∈ R , ∆ ≥ β . Assuming B ≥ one has limitingvalues of B as B = 0 , when ∆ = β , and B = 1 , when ∆ → ∞ . So, the corresponding limiting values of theamplitude are η = α and η = 0 , respectively. Theequations (40) are obtained by setting τ = 0 in (37),which is a fair approximation for surface gravity waves. A. Gardner equations for shallow water waves
Let us recall, that the Gardner equation (36) and (37)have been derived under assumptions that parameter α is small and parameters β and δ are of one order smaller,that is β ≈ δ ≈ O ( α ) . Therefore, for numerical simu-lations we take α = 0 . , β = 0 . , δ = 0 . . Thesevalues of α, β imply A = 0 . / ∆ , B = (cid:113) − . , and V = 1 + . . In Fig. 10 we display profiles of Gard-ner’s soliton obtained during the motion according to theGardner equation (37). These results can be comparedwith the evolution of the same initial Gardner’s solitonaccording to the Boussinesq equations (34)-(35), shownin Fig. 11. In the last case the initial condition for the w function is taken in the form (38). -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12-50 0 50 100 150 200 250 300 350 400 α =0.3, β =0.09, δ =0.09 η ( x ,t ) , w ( x ,t ) x 2k*dt(2k+1)*dth(x) FIG. 10. The same as in Fig. 2 but for the Gardner equation(36). Parameters α = 0 . , β = δ = 0 . , τ = 0 of the equationwere used. The value ∆ = 1 was chosen for the initial soliton(39). B. Gardner equation for thin liquid layers
In this case we have to take into account that the Bondnumber τ can be greater than 1/3. Then the coefficient − τ β in eq. (37) can become negative and the parameter B can be greater that 1. The parameters of the solution(39) are now A = 2(1 − τ ) β α ∆ , B = ± (cid:114) − (1 − τ ) β ,V = 1 + (1 − τ ) β , (41) -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14-50 0 50 100 150 200 250 300 350 400 α =0.3, β =0.09, δ =0.09 η ( x ,t ) , w ( x ,t ) x Bus- η Bus-wh(x)
FIG. 11. Profiles of functions η and w obtained from thesecond order Boussinesq’s set (34)-(35), the precursors of theGardner equation. Parameters are the same as in Fig. 10. Inorder to avoid overlaps, profiles of w function are shifted by0.005 up and by 8 left. with soliton’s amplitude given by η = A B = 2(1 − τ ) β α ∆ (cid:18) ± (cid:113) − (1 − τ ) β (cid:19) . The examples of time evolution of Gardner’s soliton forthe uneven bottom are displayed in Figs. 12 and 13. InFig. 12 we present results obtained from the Gardnerequation (37), whereas in Fig. 13 those which result fromBoussinesq’s set (34)-(35). The time step between subse-quent profiles is 16. In both cases we used the same ini-tial condition in the form of Gardner’s soliton (39) withparameters
A, B, V given by (41). For the Boussinesqsystem (34)-(35) the initial condition for the w functionis taken in the form (38).Comparing Figs. 10-13 we recognize the same quali-tative properties as in previous sections. The impact ofbottom changes on surface waves is more prominent whenthe evolution proceeds according to the Boussinesq equa-tions than in the case of the single Gardner equation. VII. NON-SOLITON INITIAL CONDITIONS
In all examples presented in previous sections, the ini-tial conditions were chosen in the form of soliton solutionsto particular wave equations. Such initial conditions ap-pear to be extremely resistant to disturbances introducedby varying bottom. This means that a bottom with asmall amplitude introduces only small changes of soli-ton’s amplitude and velocity, leaving the shape almostunchanged. On the other hand, in all considered cases,the impact of the bottom variations on the changes of sur-face waves is distinctly more significant when calculatedfrom the Boussinesq equations than when calculated fromsingle wave equations.Now, we study some examples of the time evolution ofinitial waves (elevation or depression), which shapes are -0.2-0.15-0.1-0.05 0 0.05-50 0 50 100 150 200 250 300 350 400 α =0.3, β =0.09, δ =0.09 η ( x ,t ) x 2k*dt(2k+1)*dth(x) FIG. 12. The same as in Fig. 2 but for the Gardner equation(36). Parameters α = 0 . , β = δ = 0 . , τ = 1 of the equationwere used. The value ∆ = 1 was chosen for the initial soliton(39). -0.2-0.15-0.1-0.05 0 0.05-50 0 50 100 150 200 250 300 350 400 α =0.3, β =0.09, δ =0.09 η ( x ,t ) , w ( x ,t ) x Bus η Bus wh(x)
FIG. 13. Profiles of functions η and w obtained from thesecond order Boussinesq’s set (34)-(35), the precoursors ofthe Gardner equation. Parameters are the same as in Fig. 10.In order to avoid overlaps, profiles of w function are shiftedby 0.005 up and by 8 left. different from solitons of particular equations. We studythese evolutions taking the initial shape of the wave in theform of a Gaussian with the amplitude equal to soliton’samplitude but with the width providing the volume ofthe deformation being substantially greater than that ofa soliton. In particular, we focus on the case, which,for the flat bottom, leads to the extended KdV equation(KdV2). In all other cases, the behavior of the evolutionof wave profiles appears qualitatively to be very similar. A. KdV case
In Figs. 14 and 15 we show the profiles of the timeevolution of waves calculated according to equations (13)(KdV generalized for an uneven bottom) and (11)-(12)(the corresponding Boussinesq equations), respectively.In both cases, the initial condition was taken as theGaussian profile moving with the KdV soliton’s veloc-ity, the same amplitude, but with the triple volume of the fluid distortion from equilibrium. The parameters ofwave equations are α = β = δ = 0 . .The results show that the time evolution is dominatedby splitting of the initial wave into (at least) three mainsolitons. It seems that in long time evolution, one canexpect more distinct emergence of the fourth one. InFig. 15, one can notice the increase of the amplitude ofthe highest soliton during its motion over the bottombump, which is almost unnoticeable in Fig. 14.In Figs. 16 and 17 we present the cases of the timeevolution with equation parameters as in Figs. 14 and 15but assuming that the initial distortion has an inverseform than the appropriate soliton (depression instead el-evation). In these cases, the waves behave in an entirelydifferent way.The cases with α = β = δ = 0 . with the inverse ini-tial wave profile (not shown here) suggest chaotic dy-namics . α = β = δ =0.25 η ( x ,t ) x2k*dt(2k+1)*dth(x) FIG. 14. Time evolution obtained according to the KdV equa-tion (13). Initial Gaussian profile with the triple volume of theKdV soliton, the same velocity and amplitude. Here, timestep between the consecutive profiles is dt = 32 . α = β = δ =0.25 η ( x ,t ) x2k*dt(2k+1)*dth(x) FIG. 15. Time evolution obtained according to Boussinesq’sequations (11)-(12).Initial Gaussian profile with the triple vol-ume of the KdV soliton, the same velocity and amplitude.Here, time step between the consecutive profiles is dt = 32 . -1-0.5 0 0.5 1 1.5 2 2.5 3 3.5-100 0 100 200 300 400 α = β = δ =0.15 η ( x ,t ) x 2k*dt(2k+1)*dth(x) FIG. 16. Time evolution obtained according to the KdV equa-tion (13). Initial Gaussian profile with the triple volume ofthe KdV soliton, the same velocity, but the inverse amplitude.Here, time step between the consecutive profiles is dt = 64 . -1-0.5 0 0.5 1 1.5 2 2.5 3 3.5-100 0 100 200 300 400 α = β = δ =0.15 η ( x ,t ) x 2k*dt(2k+1)*dth(x) FIG. 17. Time evolution obtained according to Boussinesq’sequations (11)-(12). Initial Gaussian profile with the triplevolume of the KdV soliton, the same velocity, but the inverseamplitude. Here, time step between the consecutive profilesis dt = 64 . B. KdV2 case
In Figs. 18 and 19 we show the profiles of the timeevolution of waves calculated according to equations (25)(KdV2 generalized for an uneven bottom) and (21)-(22)(the corresponding Boussinesq equations), respectively.In both cases, the initial condition was taken as theGaussian profile moving with the KdV soliton’s veloc-ity, the same amplitude, but with the triple volume ofthe fluid distortion from equilibrium. The parameters ofwave equations are α = β = δ = 0 . . Since the equa-tions describe the macroscopic shallow water case, theparameter τ is set equal to zero.The results displayed in Figs. 18 and 19 show that thetime evolution is dominated by splitting of the initialwave into (at least) four solitons. It seems that in longtime evolution, one can expect more distinct emergenceof the fifth one. In Fig. 18, this splitting is accompany-ing by forwarding radiation of fast oscillations with tiny amplitude (the effect which also appeared in our earlierpapers [34, 35, 45]). In Fig. 19, one can notice the in-crease of the amplitude of the highest soliton during itsmotion over the bottom bump, which is difficult to seein Fig. 18.In next Figs. 20 and 21 we present the time evolutionwith the same parameters as those in Figs. 18 and 19.The only difference is that now the initial condition istaken as inverse of that in Figs. 18 and 19. This meansthat the initial condition has the form of depression in-stead elevation (normal for KdV2 equation). Surpris-ingly, time evolution obtained directly from the gener-alized KdV2 equation (25) displayed in Fig. 20 differssubstantially from the time evolution obtained from theappropriate Boussinesq’s equations (21)-(22). The timeevolution of w function, presented additionally in Fig. 22is qualitatively very similar to the evolution of η function.In contrast to these results obtained from the Boussi-nesq’s equations, the time evolution resulting from thegeneralized KdV2 equation (25) look chaotic . This be-havior may have the following cause. The KdV2 equationis only one of those considered in this paper, whose an-alytical solution is the so-called embedded soliton . Thispoint deserves further study. α =0.2424, β =0.3, δ =0.15 η ( x ,t ) x 2k*dt(2k+1)*dth(x) FIG. 18. Time evolution obtained according to the KdV2equation (25). Initial Gaussian profile with the triple volumeof the KdV2 soliton, the same velocity and amplitude. Here,time step between the consecutive profiles is dt = 16 . C. 5th-order KdV case
Let us recall, that analytic soliton solutions to the 5th-order KdV equation in the form η ( x, t ) = A Sech [ B ( x − vt )] exist only when < τ < √ − ≈ . . Prop-erties of wave motion when τ is close to and when τ isclose to . differ substantially from each other. There-fore, we present examples of time evolution of waves de-scribe by 5th-order KdV equation for two cases of τ .1 α =0.2424, β =0.3, δ =0.15 η ( x ,t ) x 2k*dt(2k+1)*dth(x) FIG. 19. Time evolution obtained according to Boussinesq’sequations (21)-(22). Initial Gaussian profile with the triplevolume of the KdV2 soliton, the same velocity and amplitude.Here, time step between the consecutive profiles is dt = 16 . -1 0 1 2 3 4 5 6 7-100 0 100 200 300 400 α =0.2424, β =0.3, δ =0.15 η ( x ,t ) x 2k*dt(2k+1)*dth(x) FIG. 20. Time evolution obtained according to the KdV2equation (25). Initial Gaussian profile with the triple volumeof the KdV2 soliton, the same velocity, but the inverse am-plitude. Here, time step between the consecutive profiles is dt = 64 .
1. Small τ , close to lower limit Begin with τ = 0 . , as in [30, Sec. 8]. In Figs. 23 and24 we show the profiles of the time evolution of wavescalculated according to equations (28) (5th-order KdVgeneralized for an uneven bottom) and (26)-(27) (the cor-responding Boussinesq equations), respectively. In bothcases, the initial condition was taken as the Gaussianprofile moving with the KdV5 soliton’s velocity, the sameamplitude, but with the triple volume of the fluid distor-tion from equilibrium. The parameters of wave equationsare α = 0 . , β = 0 . , δ = 0 . .Surprisingly, in this case, the wave profiles remain al-most unchanged during the evolution, with only a slightincrease of the amplitude when the wave travels over thebottom bump. As in most other cases, the impact of thevarying bottom in the surface wave is more significant inBoussinesq’s equations.In Figs. 25 and 26, we present the cases of the time -1 0 1 2 3 4 5 6 7-100 0 100 200 300 400 α =0.2424, β =0.3, δ =0.15 η ( x ,t ) x 2k*dt(2k+1)*dth(x) FIG. 21. Time evolution of the surface wave η ( x, t ) obtainedaccording to Boussinesq’s equations (21)-(22). Initial Gaus-sian profile with the triple volume of the KdV2 soliton, thesame velocity, but the inverse amplitude. Here, time stepbetween the consecutive profiles is dt = 64 . -1 0 1 2 3 4 5 6 7-100 0 100 200 300 400 α =0.2424, β =0.3, δ =0.15 η ( x ,t ) x 2k*dt(2k+1)*dth(x) FIG. 22. Time evolution of the function w ( x, t ) obtained ac-cording to Boussinesq’s equations (21)-(22). Initial Gaussianprofile with the triple volume of the KdV2 soliton, the samevelocity, but the inverse amplitude. Here, time step betweenthe consecutive profiles is dt = 64 . evolution with equation parameters as in Figs. 23 and 24but assuming that the initial distortion has an inverseform than the appropriate soliton (elevation instead ofdepression). It is again surprising that in these cases,the profiles look like inverted profiles shown in Figs. 23and 24.
2. Large τ , close to upper limit Now, we use τ = 0 . . Figures 27 and 28 presentanalogous time evolution as Figs. 23 and 24. To avoidprofile overlaps we displayed profiles at larger time inter-vals dt = 64 . The profiles of w function are shifted by 32left and by 0.1 up. It is clear that in these cases the timeevolution is dominated by the process of splitting of theinitial wave into at least four solitons (during the time ofcalculation). Results obtained with single equation (28)2 -0.015-0.01-0.005 0-100 0 100 200 300 400 α =0.2424, β =0.3, δ =0.15, τ =0.35 η ( x ,t ) x 2k*dt(2k+1)*dt FIG. 23. Time evolution obtained according to the gener-alized 5th-order KdV equation (28). Initial Gaussian profilewith the triple volume of the KdV5 soliton, the same veloc-ity and amplitude. Here, time step between the consecutiveprofiles is dt = 16 . -0.015-0.01-0.005 0 0.005-100 0 100 200 300 400 α =0.2424, β =0.3, δ =0.15, τ =0.35 η ( x ,t ) x Bus- η Bus-wh(x)
FIG. 24. Time evolution obtained according to Boussinesq’sequations (26)-(27). Initial Gaussian profile with the triplevolume of the KdV5 soliton, the same velocity and amplitude.Here, time step between the consecutive profiles is dt = 16 . and Boussinesq’s equations (26)-(27) are very similar. Inthe former case the impact of the bottom bump is almostunnoticeable, in the latter is visible but also small.In Figs. 29 and 30 we present cases analogous to thoseshown in Figs. 25 and 26 but for τ = 0 . . The initialcondition is taken as the Gaussian with the volume threetimes greater than the volume of the soliton. However,the initial condition is inverse than the ’normal’ one. It isthe elevation instead of the depression. In Fig. 30 only η function is displayed. In these cases, the behavior of thewave evolution is qualitatively similar to correspondingcases (with inverse initial conditions) for different equa-tions. α =0.2424, β =0.3, δ =0.15, τ =0.35 η ( x ,t ) x 2k*dt(2k+1)*dt FIG. 25. Time evolution obtained according to the gener-alized 5th-order KdV equation (28). Initial Gaussian profilewith the triple volume of the KdV5 soliton, the same velocity,but the inverse initial distortion. Here, time step between theconsecutive profiles is dt = 16 . α =0.2424, β =0.3, δ =0.15, τ =0.35 η ( x ,t ) x Bus- η Bus-wh(x)
FIG. 26. Time evolution obtained according to Boussinesq’sequations (26)-(27). Initial Gaussian profile with the triplevolume of the KdV5 soliton, the same velocity, but the inverseinitial distortion. Here, time step between the consecutiveprofiles is dt = 16 . D. Gardner equation
1. Case corresponding to shallow water ( τ = 0) In Figs. 31 and 32, we show the profiles of the timeevolution of waves calculated according to equations (36)(the Gardner equation generalized for an uneven bot-tom) and (34)-(35) (the corresponding Boussinesq equa-tions), respectively. In both cases, the initial conditionwas taken as the Gaussian profile moving with the KdVsoliton’s velocity, the same amplitude, but with the triplevolume of the fluid distortion from equilibrium. The pa-rameters of wave equations are α = 0 . , β = δ = 0 . .Since the equations describe the macroscopic shallow wa-ter case, the parameter τ is set equal to zero. The pa-rameter ∆ = 1 is chosen for the Gardner soliton.The results displayed in Figs. 31 and 32 show that inthese cases, the time evolution is dominated by splittingof the initial wave into (at least) two solitons. The last3 -0.5-0.4-0.3-0.2-0.1 0 0.1-100 0 100 200 300 400 α =0.2424, β =0.3, δ =0.15, τ =0.38 η ( x ,t ) x FIG. 27. Time evolution obtained according to the gener-alized 5th-order KdV equation (28). Initial Gaussian profilewith the triple volume of the KdV5 soliton, the same veloc-ity and amplitude. Here, time step between the consecutiveprofiles is dt = 64 . -0.5-0.4-0.3-0.2-0.1 0 0.1 0.2-100 0 100 200 300 400 α =0.2424, β =0.3, δ =0.15, τ =0.38 η ( x ,t ) , w ( x ,t ) xBus- η Bus-wh(x)
FIG. 28. Time evolution obtained according to Boussinesq’sequations (26)-(27). Initial Gaussian profile with the triplevolume of the KdV5 soliton, the same velocity and amplitude.Here, time step between the consecutive profiles is dt = 64 . displayed profiles suggest that in long time evolution onecan expect more distinct emergence of the third one. Thisproperty is slightly better pronounced in Fig. 32, but thetime evolution of the surface wave is almost the same forboth figures.In next Figs. 33 and 34, we present the time evolutionwith the same parameters as those in Figs. 31 and 32.The only difference is that now the initial condition istaken as inverse of that in Figs. 31 and 32. This meansthat the initial condition has the form of depression in-stead of elevation (normal for shallow water case). Thetime evolution shown in these figures is entirely differ-ent from when initial displacement has a ’normal’ sign.On the other hand, results obtained from the generalizedGardner equation and the corresponding Boussinesq’ssystem are almost identical. -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4-100 0 100 200 300 400 α =0.2424, β =0.3, δ =0.15, τ =0.38 η ( x ,t ) x FIG. 29. Time evolution obtained according to the gener-alized 5th-order KdV equation (28). Initial Gaussian profilewith the triple volume of the KdV5 soliton, the same velocity,but the inverse initial distortion. Here, time step between theconsecutive profiles is dt = 64 . -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4-100 0 100 200 300 400 α =0.2424, β =0.3, δ =0.15, τ =0.38 η ( x ,t ) x FIG. 30. Time evolution obtained according to Boussinesq’sequations (26)-(27). Initial Gaussian profile with the triplevolume of the KdV5 soliton, the same velocity, but the inverseinitial distortion. Here, time step between the consecutiveprofiles is dt = 64 .
2. Case corresponding to thin fluid layers ( τ > ) When ( τ > ) surface tension plays an important role.Such a situation appears when the thickness of the fluidlayer is very small. In the following examples, we set τ = 1 .In Figs. 35 and 36, we show the profiles of the timeevolution of waves calculated according to equations (36)(the Gardner equation generalized for an uneven bottom)and (34)-(35) (the corresponding Boussinesq equations),respectively. In both cases, the initial condition wastaken as the Gaussian profile moving with the Gardnersoliton’s velocity, the same amplitude, but with the triplevolume of the fluid distortion from equilibrium. The pa-rameters of wave equations are α = 0 . , β = δ = 0 . .The parameter ∆ = 1 is chosen for the Gardner soliton.Similarly, as in Figs. 31 and 32, the time evolution isdominated by the splitting of the initial wave into severalsolitons, at least three. The fourth one seems to emergein the last calculated profiles, as well. Here, the low-est solitons move faster than the higher ones, contrary4to usual cases. Similarly, as with τ = 0 , the results ob-tained with the Gardner equation and the correspondingBoussinesq equations are almost the same. In the lat-ter case, the impact of the bottom bump is slightly morepronounced.In Figs. 37 and 38, we used the same parameters asin Figs. 35 and 36, reversing only the sign of the initialdisplacement. The initial condition is then the elevationinstead of depression. Again, the results obtained withthe Gardner equation and the corresponding Boussinesqequations are almost the same. They are, however, en-tirely different from those in Figs. 35 and 36. α =0.3, β =0.09, δ =0.09 η ( x ,t ) x2k*dt(2k+1)*dth(x) FIG. 31. Time evolution obtained according to the Gardnerequation generalized for the uneven bottom (36) with τ = 0 .Initial Gaussian profile with the triple volume of the Gardnersoliton, the same amplitude, and velocity. dt = 16 . α =0.3, β =0.09, δ =0.09 η ( x ,t ) x2k*dt(2k+1)*dth(x) FIG. 32. Time evolution obtained according to Boussinesq’sequations, generalized for the uneven bottom (34)-(35) with τ = 0 . Initial Gaussian profile with the the triple volume ofthe Gardner soliton, the same amplitude, and velocity. Onlysurface wave η ( x, t ) is displayed. dt = 16 . VIII. CONCLUSIONS
In all considered cases for the uneven bottom, thenonlinear wave equations (13), (25), (28), and (36) are -0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 -100 0 100 200 300 400 α =0.32, β =0.09, δ =0.09 η ( x ,t ) x 2k*dt(2k+1)*dth(x) FIG. 33. Time evolution obtained according to the Gardnerequation (36) with τ = 0 . Initial Gaussian profile, repre-senting an elevation, with the triple volume of the Gardnersoliton, the same velocity, but the inverse amplitude. dt = 32 . -0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 -100 0 100 200 300 400 α =0.32, β =0.09, δ =0.09 η ( x ,t ) x 2k*dt(2k+1)*dth(x) FIG. 34. Time evolution obtained according to Boussinesq’sequations (34)-(35) with τ = 1 . Initial Gaussian profile, rep-resenting an elevation, with the the triple volume of Gardnersoliton, the same velocity, but the inverse amplitude. dt = 32 . non-integrable. Therefore the influence of the bottomvariations on surface waves has to be analyzed numeri-cally. One must remember that the validity of the derivedequations is limited to parameters α, β, δ that are smallenough.The main property of the results is the fact that the in-fluence of the uneven bottom on the surface wave η ( x, t ) obtained from the Boussinesq equations is always sub-stantially greater than that obtained from single KdV-type wave equations. It is worth emphasizing that usingthe Boussinesq equations does not need any conditionsimposed on the form of the bottom function, whereasthe compatibility condition, necessary for the existenceof single KdV-type wave equations, requires d hdx = 0 .The results of all simulations, performed according tothe Boussinesq equations reveal the fact that the rela-tive changes of w ( x, t ) functions are substantially moreprominent than that of η ( x, t ) functions.In all cases discussed above, when the initial conditionswere chosen in the form of soliton solutions to particu-5 -0.2-0.1 0 0.1 0.2 -100 0 100 200 300 400 α =0.3, β =0.09, δ =0.09 η ( x ,t ) x 2k*dt(2k+1)*dth(x) FIG. 35. Time evolution obtained according to the Gardnerequation (36) with τ = 1 . Initial Gaussian profile with volumethree times greater than that of the Gardner soliton, the sameamplitude, and velocity. dt = 32 . -0.2-0.1 0 0.1 0.2 -100 0 100 200 300 400 α =0.3, β =0.09, δ =0.09 η ( x ,t ) x 2k*dt(2k+1)*dth(x) FIG. 36. Time evolution obtained according to Boussinesq’sequations (34)-(35) with τ = 1 . Initial Gaussian profile withvolume three times greater than that of the Gardner soliton,the same amplitude, and velocity. dt = 32 . lar wave equations, the wave profiles appear extremely resistant to disturbances introduced by varying bottom. α =0.3, β =0.09, δ =0.09 η ( x ,t ) x2k*dt(2k+1)*dth(x) FIG. 37. Time evolution obtained according to the Gardnerequation (36) with τ = 1 . Initial Gaussian profile, repre-senting an elevation, with the triple volume of the Gardnersoliton, but the same amplitude, and velocity. dt = 32 . α =0.3, β =0.09, δ =0.09 η ( x ,t ) x2k*dt(2k+1)*dth(x) FIG. 38. Time evolution obtained according to Boussinesq’sequations (34)-(35) with τ = 1 . Initial Gaussian profile, rep-resenting an elevation, with the the triple volume of Gardnersoliton, but the same amplitude, and velocity. dt = 32 .[1] Mei CC. Le Méhauté B. Note on the equations of longwaves over an uneven bottom. 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