Energy invariant for shallow water waves and the Korteweg -- de Vries equation. Is energy always an invariant?
aa r X i v : . [ phy s i c s . f l u - dyn ] S e p Energy invariant for shallow water waves and the Korteweg – de Vries equation.Is energy always an invariant?
Anna Karczewska ∗ Faculty of Mathematics, Computer Science and EconometricsUniversity of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra, Poland
Piotr Rozmej † Institute of Physics, Faculty of Physics and AstronomyUniversity of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra, Poland
Eryk Infeld ‡ National Centre for Nuclear Research, Hoża 69, 00-681 Warszawa, Poland (Dated: September 14, 2015)It is well known that the KdV equation has an infinite set of conserved quantities. The first threeare often considered to represent mass, momentum and energy. Here we try to answer the questionof how this comes about, and also how these KdV quantities relate to those of the Euler shallowwater equations. Here Luke’s Lagrangian is helpful. We also consider higher order extensions ofKdV. Though in general not integrable, in some sense they are almost so, these with the accuracyof the expansion.
PACS numbers: 02.30.Jr, 05.45.-a, 47.35.Bb, 47.35.FgKeywords: Soliton, shallow water waves, nonlinear equations, invariants of KdV
I. INTRODUCTION
There exists a vast number of papers dealing with theshallow water problem. Aspects of the propagation ofweakly nonlinear, dispersive waves are still beeing stud-ied. Last year we published two articles [1, 2] in whichKorteveg–de Vries type equations were derived in weaklynonlinear, dispersive and long wavelength limit. The sec-ond order KdV type equation was derived. The secondorder KdV equation [3, 4], sometimes called "extendedKdV equation", was obtained for the case with a flatbottom. In derivation of the new equation we adaptedthe method described in [4]. In [2], an analytic solutionof this equation in the form of a particular soliton wasfound, as well.It is well known, see, e.g. [5–8], that for the KdV equa-tion there exists an infinite number of invariants, that is,integrals over space of functions of the wave profile andits derivatives, which are constants in time. Looking foranalogous invariants for the second order KdV equationwe met with some problems even for the standard KdVequation (which is first order in small parameters). Thisproblem appears when energy conservation is considered.In this paper we reconsider invariants of the KdV equa-tion and formulas for the total energy in several differentapproaches and different frames of reference (fixed andmoving ones). We find that the invariant I (3) , some-times called the energy invariant, does not always have ∗ [email protected] † [email protected] ‡ [email protected] that interpretation. We also give a proof that for the sec-ond order KdV equation, obtained in [1–4], R ∞−∞ η dx isnot an invariant of motion.There are many papers considering higher-order KdVtype equations. Among them we would like to point outworks of Byatt-Smith [9], Kichenassamy and Olver [10],Marchant [3, 11–14], Zou and Su [15], Tzirtzilakis et.al. [16] and Burde [17]. It was shown that if some coef-ficients of the second order equation for shallow waterproblem (1) are diferent or zero then there exists a hier-archy of solition solutions. Kichenassamy and Olver [10]even claimed that for second order KdV equation solitarysolutions of appropriate form can not exist. This claimwas falsified in our paper [2] where the analytic solutionof the second order KdV equation (1) was found. Con-cerning the energy conservation there are indications thatcollisions of solitons [18, 19] which are solutions of higherorder equations of KdV type can be inelastic [15, 16].The paper is organized as follows. In Section II severalfrequently used forms of KdV equations are recalled withparticular attention to transformations between fixed andmoving reference frames. In Section III the form of thethree lowest invariants of KdV equations is derived fordifferent forms of the equations. In Section IV we showthat the energy calculated from the definition H = T + V has no invariant form. Section V describes the varia-tional approach in a potential formulation which gives aproper KdV equation but fails in obtaining second orderKdV equations. In the next section the proper invari-ants are obtained from Luke’s Lagrangian density. Sec-tion VII summarizes conclusions on the energy for KdVequation. In section VIII we apply the same formalismto calculate energy for waves governed by the extendedKdV equation (second order). We found that energy isnot conserved neither in fixed coordinate system nor inthe moving frame. II. THE EXTENDED KDV EQUATION
The geometry of shallow water waves is presented inFig. 1.In [1, 2] we derived an equation, second order in smallparameters, in the fixed reference system and with scalednondimensional variables containing terms for bottomfluctuations. They will not be considered here. h a η (x,t) α =a/h β =(h/l) η (x,t)undisturbed surfacebottom FIG. 1. Schematic view of the geometry.
For a flat bottom that equation reduces to the secondorder KdV type equation, identical with [4, Eq. (21)] for β = α , that is, η t + η x + α ηη x + β η x + α (cid:18) − η η x (cid:19) (1) + αβ (cid:18) η x η x + 512 ηη x (cid:19) + β η x = 0 . Subscripts denote partial differentiation. Small parame-ters α, β are defined by ratios of the wave amplitude a ,the average water depth h and mean wavelength lα = ah , β = (cid:18) hl (cid:19) . Equation (1) was earlier derived in [3] and called "theextended KdV equation".Limitation to the first order in small parameters yieldsthe KdV equation in a fixed coordinate system η t + η x + α ηη x + β η x = 0 . (2)Transformation to a moving frame in the form ¯ x = ( x − t ) , ¯ t = t, ¯ η = η, (3)allows us to remove the term η x in the KdV equation ina frame moving withthe velocity of sound √ gh ¯ η ¯ t + α
32 ¯ η ¯ η ¯ x + β
16 ¯ η x = 0 . (4) The explicit form of the scaling leading to equations (1)– (4) is given by (29).Problems with mass, momentum and energy conser-vation in the KdV equation were discussed in [20] re-cently. In this paper the authors considered the KdVequations in the original dimensional variables. Thenthe KdV equatios are η t + cη x + 32 ch ηη x + ch η xxx = 0 , (5)in a fixed frame of reference and η t + 32 ch ηη x + ch η xxx = 0 , (6)in a moving frame. In both, c = √ gh , and (6) is obtainedfrom (5) by setting x ′ = x − ct and dropping the primesign.In our present paper we discuss the energy formulas ob-tained both in fixed and moving frames of reference forKdV (2), (4), (5), (6) . There seem to be some contradic-tions in the literature because the form of some invariantsand the energy formulas differ in different sources becauseof using different reference frames and/or different scal-ings. In this paper we address this problems.The second goal is to present some invariants for aKdV type equation of the second order (1). III. INVARIANTS OF KDV TYPE EQUATIONS
What invariants can be attributed to equations (1) –(2) and (5) – (6) ?It is well known, see, e.g. [7, Ch. 5], that an equationof the form ∂T∂t + ∂X∂x = 0 , (7)where neither T (an analog to density) nor X (an analogto flux) contain partial derivatives with respect to t ,corresponds to some conservation law . It can be applied,in particular, to KdV equations (where there exist aninfinite number of such conservation laws) and to theequations of KdV type like (1). Functions T and X may depend on x, t, η, η x , η x , . . . , h, h x , . . . , but not η t .If both functions T and X x are integrable on ( −∞ , ∞ ) and lim x →±∞ X = const (soliton solutions), then integrationof equation (7) yields dd t (cid:18)Z ∞−∞ T dx (cid:19) = 0 or Z ∞−∞ T dx = const. , (8)since Z ∞−∞ X x dx = X ( ∞ , t ) − X ( −∞ , t ) = 0 . (9)The same conclusion applies for periodic solutions(cnoidal waves), when in the integrals (8), (9) limits of in-tegration ( −∞ , ∞ ) are replaced by ( a, b ) , where b − a = Λ is the space period of the cnoidal wave (the wave length). A. Invariants of the KdV equation
For the KdV equation (2) the two first invariants canbe obtained easily. Writing (2) in the form ∂η∂t + ∂∂x (cid:18) η + 34 αη + 16 βη xx (cid:19) = 0 . (10)one immediately obtains the conservation of mass (vol-ume) law I (1) = Z ∞−∞ η dx = const . (11)Similarly, multiplication of (2) by η gives ∂∂t (cid:18) η (cid:19) + ∂∂x (cid:18) η + 12 αη − βη x + 16 βηη xx (cid:19) = 0 , (12)resulting in the invariant of the form I (2) = Z ∞−∞ η dx = const . (13)In the literature of the subject, see, e.g. [7, 20], I (2) isattributed to momentum conservation.Invariants I (1) , I (2) have the same form for all KdVequations (2), (4), (A2), (5), (6).Denote the left hand side of (2) by KDV ( x, t ) and take η × KDV ( x, t ) − βα η x × ∂∂x KDV ( x, t ) . (14)The result, after simplifications is ∂∂t (cid:18) η − βα η x (cid:19) + ∂∂x (cid:18) αη + 12 βη x η (15) − βη x η + η + 118 β α η x − β α η x η x − βα η x (cid:19) = 0 . Then the next invariant for KdV in the fixed referenceframe (2) is I (3)fixed frame = Z ∞−∞ (cid:18) η − βα η x (cid:19) dx = const . (16)The same invariant is obtained for the KdV in the mov-ing frame (4). The same construction like (14) but forequation (4) results in ∂∂t (cid:18) η − βα η x (cid:19) + ∂∂x (cid:18) αη + 12 βη x η (17) − βη x η + η + 118 β α η x − β α η x η x (cid:19) = 0 . Then the next invariant for KdV equation in moving ref-erence frame (2) is I (3)moving frame = Z ∞−∞ (cid:18) η − βα η x (cid:19) dx = const . (18) The procedure similar to those described above leadsto the same invariants for both equations (5) and (6)where KdV equations are written in dimensional vari-ables. To see this, it is enough to take η × kdv ( x, t ) − h ∂∂x kdv ( x, t ) = 0 , where kdv ( x, t ) is the lhs either of(5) or (6). For equation (5) the conservation law is ∂∂t (cid:18) η − h η x (cid:19) + ∂∂x (cid:18) cη − c h η − ch η x (19) − ch ηη x + 12 ch η η xx + 118 ch η xx − ch η x η xxx (cid:19) = 0 , whereas for equation (6) the flux term is different ∂∂t (cid:18) η − h η x (cid:19) + ∂∂x (cid:18) c h η − ch ηη x (20) + 12 ch η η xx + 118 ch η xx − ch η x η xxx (cid:19) = 0 . But in both cases the same I (3) invariant is obtained as I (3)dimensional = Z ∞−∞ (cid:18) η − h η x (cid:19) dx = const . (21) Conclusion
Invariants I (3) have the same form for fixedand moving frames of reference when the transformationfrom fixed to moving frame scales x and t in the sameway (e.g. x ′ = x − t and t ′ = t ). When scaling factorsare different, like in (A1), then the form of I (3) in themoving frame differs from the form in the fixed frame,see Appendix A. For those solutions of KdV which preserve their shapesduring the motion, that is, for cnoidal solutions and singlesoliton solutions, integrals of any power of η ( x, t ) and anypower of arbitrary derivative of the solution with respectto x are invariants. That is, I ( a,n ) = Z ∞−∞ ( η nx ) a dx = const , (22)where n = 0 , , , . . . , and a ∈ R is an arbitrary realnumber. Then an arbitrary linear combination of I ( a,n ) is an invariant, as well. B. Invariants of the second order equations
Can we construct invariants for KdV type equations ofthe second order? Let us try to take T = η for equation(1). Then we find that all terms, except η t , can bewritten as X x , as Z (cid:20) η x + α ηη x + β η x + α (cid:18) − η η x (cid:19) + αβ (cid:18) η x η x + 512 ηη x (cid:19) + β η x (cid:21) dx = η + 34 αη + 16 βη x − α η (23) + αβ (cid:18) η x + 512 ηη x (cid:19) + 19360 β η x . As (23) depends on η and space derivatives and alsosince all those functions vanish when x → ±∞ , theconservation law for mass (volume) Z ∞−∞ η ( x, t ) dx = const., (24)holds for the second order equation.(Conservation law (24) holds for the equation with anuneven bottom, as well.)Until now we did not find any other invariants for thesecod order equations. Moreover, we can show that theintegral I (2) (13) is no longer an invariant of the secondorder KdV equation (1).Upon multiplication of equation (1) by η one obtains ∂∂t (cid:18) η (cid:19) + ∂∂x (cid:20) η + 12 αη + 16 β (cid:18) − η x + ηη x (cid:19) − α η + 19360 β (cid:18) η xx − η x η x + ηη x (cid:19) (25) + 512 αβ η η x (cid:21) + 18 αβ ηη x η x . The last term in (25) can not be expressed as ∂∂x X ( η, η x , . . . ) . Therefore R + ∞−∞ η dx is not a conservedquantity. IV. ENERGY
The invariant I (3) is usually referred to as the energyinvariant. Is this really the case? A. Energy in a fixed frame as calculated from thedefinition
The hydrodynamic equations for an incompressible, in-viscid fluid, in irrotational motion and under gravity ina fixed frame of reference, lead to a KdV equation of theform ˜ η ˜ t + ˜ η ˜ x + α
32 ˜ η ˜ η ˜ x + β
16 ˜ η x = 0 . (26)We will find the function ˜ f ˜ x = ˜ η − α ˜ η + 13 β ˜ η ˜ x ˜ x , (27)obtained as a byproduct in derivation of KdV, useful inwhat follows. For more details see Appendix B or [24,Chapter 5]. Tildas denote scaled dimensionless quanti-ties.Now construct the total energy of the fluid from thedefinition. The total energy is the sum of potential and kinetic en-ergy. In our two-dimensional system the energy in origi-nal (dimensional coordinates) is E = T + V = Z + ∞−∞ Z h + η ρv dy ! dx (28) + Z + ∞−∞ Z h + η ρgy dy ! dx . Equations (26) and (27) are obtained after scaling [1,2, 4]. We now have dimesionless variables, according to ˜ φ = hla √ gh φ, ˜ x = xl , ˜ η = ηa , ˜ y = yh , ˜ t = tl/ √ gh , (29)and V = ρgh l Z + ∞−∞ Z α ˜ η ρ ˜ y d ˜ y d ˜ x, (30) T = 12 ρgh l Z + ∞−∞ Z α ˜ η (cid:18) α ˜ φ x + α β ˜ φ y (cid:19) d ˜ y d ˜ x. (31)Note, that the factor in front of the integrals has thedimension of energy.In the following, we omit signs ∼ , having in mind thatwe are working in dimensionless variables. Integration in(30) with respect to y yields V = 12 gh lρ Z ∞−∞ (cid:0) α η + 2 αη + 1 (cid:1) dx (32) = 12 gh lρ (cid:20)Z ∞−∞ (cid:0) α η + 2 αη (cid:1) dx + Z ∞−∞ dx (cid:21) . After renormalization (substraction of constant term R ∞−∞ dx ) one obtains V = 12 gh lρ Z ∞−∞ (cid:0) α η + 2 αη (cid:1) dx. (33)In kinetic energy we use the velocity potential ex-pressed in the lowest (first) order φ x = f x − βy f xxx and φ y = − βyf xx , (34)where f x was defined in (27). Now the bracket in theintegral (31) is (cid:18) α φ x + α β φ y (cid:19) = α (cid:0) f x + βy ( − f x f xxx + f xx ) (cid:1) . (35)Inegration with respect to the vertical corrdinate y gives,up to the same order, T = 12 ρgh l Z + ∞−∞ α (cid:2) f x (1 + αη )+ β ( − f x f xxx + f xx ) 13 (1 + αη ) (cid:21) dx (36) = 12 ρgh l Z + ∞−∞ α (cid:20) f x + αf x η + 13 β (cid:0) f xx − f x f xxx (cid:1)(cid:21) dx. In order to express energy through the elevation funcion η we use (27). We then substitute f x = η in terms ofthe third order and f x = η − αη + βηη xx in termsof the second order T = 12 ρgh l Z + ∞−∞ α (cid:20)(cid:18) η − αη + 23 βηη xx (cid:19) + αη + 13 β (cid:0) η x − ηη xx (cid:1)(cid:21) dx = 12 ρgh l α (cid:20)Z + ∞−∞ (cid:18) η + 12 αη (cid:19) dx (37) + Z + ∞−∞ β (cid:0) η x + ηη xx (cid:1) dx (cid:21) . The last term vanishes as Z + ∞−∞ (cid:0) η x + ηη xx (cid:1) dx = Z + ∞−∞ η x dx + ηη x | + ∞−∞ − Z + ∞−∞ η x dx = 0 . (38)Therefore the total energy in the fixed frame is given by E tot = T + V = ρgh l Z ∞−∞ (cid:18) αη + ( αη ) + 14 ( αη ) (cid:19) dx (39)= ρgh l (cid:18) αI (1) + α I (2) + 14 α I (3) + 112 α β Z ∞−∞ η x dx (cid:19) The energy (39) in a fixed frame of reference has non in-variant form . The last term in (39) results in small de-viations from energy conservation only when η x changesin time in soliton’s reference frame, what occurs only dur-ing soliton collision. This deviations are discussed andillustrated in Section VI E.The result (39) gives the energy in powers of η only.The same structure of powers in η was obtained by theauthors of [20], who work in dimensional KdV equations(5) and (6). On page 122 they present a non-dimensionalenergy density E in a frame moving with the velocity U .Then, if U = 0 is set, the energy density in a fixed frameis proportional to αη + α η as the formula is obtainedup to second order in α . However, the third order term is α η , so the formula up to the third order in α becomes E ∼ αη + α η + 14 α η . (39)This energy density contains the same terms like (39) anddoes not contain the term η x , as well.Energy calculated from the definition does not containa proper invariant of motion. B. Energy in a moving frame
Now consider the total energy according to (28) calcu-lated in a frame moving with the velocity of sound c = √ gh . Using the same scaling (29) to dimensionless vari-ables we note that in these variables c = 1 . As pointed by Ali and Kalisch [8,Sect. 3] working in such system onehas to replace φ x by the horizontal velocity in a movingframe, that is by ˜ φ ˜ x − α = α ˜ η − α ˜ η + β (cid:16) − y (cid:17) ˜ η ˜ x ˜ x − α .Then repeating the same steps as in the previous subsec-tion yields the energy expressed by invariants E tot = ρgh l Z ∞−∞ (cid:20) − α ˜ η + 14 ( α ˜ η ) + 12 α (cid:18) ˜ η − βα ˜ η x (cid:19)(cid:21) d ˜ x = ρgh l (cid:18) − αI (1) + 14 α I (2) + 12 α I (3) (cid:19) . (40)The crucial term − α β ˜ η x in (40) appears due to inte-gration over vertical variable of the term βα ˜ η ˜ x ˜ x suppliedby ( ˜ φ ˜ x − α ) . V. VARIATIONAL APPROACHA. Lagrangian approach, potential formulation
Some attempts at the variational approach to shallowwater problems are summarized in G.B. Whitham’s book[21, Sect 16.14].For KdV as it stands, we can not write a variationalprinciple directly. It is necessary to introduce a velocitypotential. The simplest choice is to take η = ϕ x . Thenequation (2) in the fixed frame takes the form ϕ xt + ϕ xx + 32 αϕ x ϕ xx + 16 βϕ xxxx = 0 . (41)The appropriate Lagrangian density is L fixed frame := − ϕ t ϕ x − ϕ x − α ϕ x + β ϕ xx . (42)Indeed, the Euler–Lagrange equation obtained from La-grangian (42) is just (41).For our moving reference frame the substitution η = ϕ x into (4) gives ϕ xt + 32 αϕ x ϕ xx + 16 βϕ xxxx = 0 . (43)So, the appropriate Lagrangian density is L moving frame := − ϕ t ϕ x − α ϕ x + β ϕ xx . (44)Again, the Euler–Lagrange equation obtained from La-grangian (44) is just (43). B. Hamiltonians for KdV equations in thepotential formulation
The Hamiltonian for the KdV equation in a fixed frame(2) can be obtained in the following way. Defining gen-eralized momentum π = ∂ L ∂ϕ t , where L is given by (42),one obtains H = Z ∞−∞ [ π ˙ ϕ − L ] dx = Z ∞−∞ (cid:20) ϕ x + α ϕ x − β ϕ xx (cid:21) dx = Z ∞−∞ (cid:20) η + 14 α (cid:18) η − β α η x (cid:19)(cid:21) dx . (45)The energy is expressed by invariants I (2) , I (3) so it is aconstant of motion.The same procedure for KdV in a moving frame (4)gives H = Z ∞−∞ [ π ˙ ϕ − L ] dx = Z ∞−∞ (cid:20) α ϕ x − β ϕ xx (cid:21) dx = 14 α Z ∞−∞ (cid:18) η − β α η x (cid:19) dx . (46)The Hamiltonian (46) only consists I (3) .The constant of motion in a moving frame is E = 14 I (3) = const . (47)The potential formulation of the Lagrangian, describedabove, is succesful for deriving KdV equations both forfixed and moving reference frames. It fails, however, forthe second order KdV equation (1). We proved that thereexists a nonlinear expression of L ( ϕ t , ϕ x , ϕ xx , . . . ) , suchthat the resulting Euler–Lagrange equation differs verylittle from equation (1). The difference lies only in thevalue of one of the coefficients in the second order term αβ (cid:0) η x η x + ηη x (cid:1) . Particular values of coefficientsin this term also cause the lack of the I (2) invariant forsecond order KdV equation, (see (25)). VI. LUKE’S LAGRANGIAN AND KDVENERGY
The full set of Euler equations for the shallow waterproblem, as well as KdV equations (2), (A2), and sec-ond order KdV equation (1) can be derived from Luke’sLagrangian [23], see, e.g. [3]. Luke points out, how-ever, that his Lagrangian is not equal to the difference ofkinetic and potential energy. Euler–Lagrange equationsobtained from L = T − V do not have the proper form atthe boundary. Instead, Luke’s Lagrangian is the sum ofkinetic and potential energy suplemented by the φ t termwhich is necessary in dynamical boundary condition. A. Derivation of KdV energy from the originalEuler equations according to [24]
In Chapter 5.2 of the Infeld and Rowlands book theauthors present a derivation of the KdV equation fromthe Euler (hydrodynamic) equations using a single smallparameter ε . Moreover, they show that the same methodallows us to derive the Kadomtsev–Petviashvili (KP) equation [29] for water waves [30, 31, 33, 34] and alsononlinear equations for ion acoustic waves in a plasma[32, 35]. The authors first derive equations of motion,then construct a Lagrangian and look for constants ofmotion. For the purpose of this paper and for comparisonto results obtained in the next subsections it is convenientto present their results starting from Luke’s Lagrangiandensity. That density can be written as (here g = 1 ) L = Z η (cid:20) φ t + 12 ( φ x + φ z ) + z (cid:21) dz . (48)In Chapter 5.2.1 of [24] the authors introduce scaledvariables in a movimg frame ( ε plays a role of small pa-rameter and if ε = α = β , then KdV equation is ob-tained). Then (for details, see Appendix B or [24, Chap-ter 5.2]) φ z = − ε zf ξξ , φ x = εf ξ − ε z f ξξξ ,φ t = − εf ξ + ε (cid:18) f τ + z f ξξξ (cid:19) − ε z f ξξτ . (49)Substitution of the above formulas into the expression [ ]under the integral in (48) gives [ ] = z − εf ξ + ε (cid:18) f τ + 12 f ξ + z f ξξξ (cid:19) (50) + ε z (cid:2) − f ξξτ + ( f ξξ − f ξ f ξξξ ) (cid:3) + O ( ε ) . Remark
The full Lagrangian is obtained by integrationof the Lagrangian density (48) with respect to x . In-tegration limits are ( −∞ , ∞ ) for a soliton solutions, or [ a, b ] , where b − a = X –wave length (space period) forcnoidal solutions. Integration by parts and properties ofthe solutions at the limits, see (9), allow us to use theequivalence R ∞−∞ ( f ξξ − f ξ f ξξξ ) dξ = R ∞−∞ f ξξ dξ .Therefore [ ] = z − εf ξ + ε (cid:18) f τ + 12 f ξ + z f ξξξ (cid:19) (51) + ε z (cid:2) − f ξξτ + 2 f ξξ (cid:3) + O ( ε ) . Integration over y gives (note that η = ⇒ εη ) L = 12 (1 + εη ) + (1 + εη ) (cid:20) − εf ξ + ε (cid:18) f τ + 12 f ξ (cid:19)(cid:21) + 13 (1 + εη ) (cid:20) ε f ξξξ − ε f ξξτ + ε f ξξ (cid:21) . (52)Write (52) up to third order in εL = L (0) + εL (1) + ε L (2) + ε L (3) + O ( ε ) . It is easy to show, that L (0) = 12 , L (1) = η − f ξ ,L (2) = f τ + 12 η − ηf ξ + 12 f ξ + 16 f ξξξ , (53) L (3) = ηf τ + 12 ηf ξ + 12 ηf ξξξ − f ξξτ + 13 f ξξ . The Hamiltonian density reads as H = f τ ∂L∂f τ + f ξξτ ∂L∂f ξξτ − L (54) = − (cid:20)
12 + ε ( η − f ξ ) + ε (cid:18) η − ηf ξ + 12 f ξ + 16 f ξξξ (cid:19) + ε (cid:18) ηf ξ + 12 ηf ξξξ + 13 f ξξ (cid:19)(cid:21) . Dropping the constant term one obtains the total energyas E = Z ∞−∞ (cid:20) ε ( η − f ξ ) + ε (cid:18) η − ηf ξ + 12 f ξ + 16 f ξξξ (cid:19) + ε (cid:18) ηf ξ + 12 ηf ξξξ + 13 f ξξ (cid:19)(cid:21) dξ. (55)Now, we need to express f ξ and its derivatives by η and its derivatives. We use (27) replacing α and β by ε ,that is, f ξ = η − εη + 13 εη ξξ . (56)Then the total energy in a moving frame is expressedin terms of the second and the third invariants E = − (cid:20) ε Z ∞−∞ η dx + ε Z ∞−∞ (cid:18) η − η ξ (cid:19) dx (cid:21) . (57)Note that the term η ξ occuring in the third orderinvariant originates from three terms appearing in φ z , φ x and φ t (see terms f ξξ and f ξξξ in (49)). B. Luke’s Lagrangian
The original Lagrangian density in Luke’s paper [23] is L = Z h ( x )0 ρ (cid:20) φ t + 12 ( φ x + φ y ) + gy (cid:21) dy . (58)After scaling as in [1, 2, 4] ˜ φ = hla √ gh φ, ˜ x = xl , ˜ η = ηa , ˜ y = yh , ˜ t = tl/ √ gh , (59)we obtain φ t = ghα ˜ φ ˜ t , φ x = ghα ˜ φ x , φ y = gh α β ˜ φ y . (60)The Lagrangian density in scaled variables becomes( dy = hd ˜ y ) L = ρgha Z αη (cid:20) ˜ φ ˜ t + 12 (cid:18) ˜ φ x + α β ˜ φ y (cid:19)(cid:21) d ˜ y + 12 ρgh (1 + αη ) . (61) So, in dimensionless quatities Lρgha = Z αη (cid:20) ˜ φ ˜ t + 12 (cid:18) α ˜ φ x + αβ ˜ φ y (cid:19)(cid:21) d ˜ y + 12 αη , (62)where the constant term and the term proportional to η in the expansion of (1 + αη ) are omitted. The form (62)is identical with Eq. (2.9) in Marchant & Smyth [3].The full Lagrangian is obtained by integration over x .In dimensionless variables ( dx = l d ˜ x ) it gives L = E Z ∞−∞ (cid:20)Z αη (cid:20) ˜ φ ˜ t + 12 (cid:18) α ˜ φ x + αβ ˜ φ y (cid:19)(cid:21) d ˜ y + 12 αη (cid:21) d ˜ x. (63)The factor in front of the integral, E = ρghal = ρgh l α ,has the dimension of energy.Next, the signs ( ∼ ) will be omitted, but we have toremember that we are working in scaled dimensionlessvariables in a fixed reference frame. C. Energy in the fixed reference frame
Express the Lagrangian density by η and f = φ (0) .Now, up to first order in small parameters φ = f − βy f xx , φ t = f t − βy f xxt ,φ x = f x − βy f xxx , φ y = − βyf xx . (64)Then the expression under the integral in (62) becomes [ ] = f t − βy f xxt + 12 αf x + 12 αβy (cid:0) − f x f xxx + f xx (cid:1) . (65)From properties of solutions at the limits (cid:0) − f x f xxx + f xx (cid:1) ⇒ f xx . Integration of (65) over y yields Lρgha = (cid:18) f t + 12 αf x (cid:19) (1 + αη ) − βf xxt
13 (1 + αη ) + αβf xx
13 (1 + αη ) + 12 αη . (66)The dimensionless Hamiltonian density is( f t ∂L∂f t + f xxt ∂L∂f xxt − L ) Hρgh l = − α (cid:20) αf x (1 + αη ) + αβf xx
13 (1 + αη ) + 12 αη (cid:21) . (67)Again, we need to express the Hamiltonian by η and itsderivatives, only. Inserting f x = η − αη + 13 βη xx (68)into (67) and leaving terms up to third order one obtains Hρgh l = − α (cid:20) αη + 14 α η + 13 αβ ( η x + ηη xx ) (cid:21) . (69)The energy is Eρgh l = − α Z ∞−∞ (cid:20) αη + 14 α η + 13 αβ ( η x + ηη xx ) (cid:21) dx = − (cid:20) α Z ∞−∞ η dx + 14 α Z ∞−∞ η dx (cid:21) (70)since the integral of the αβ term vanishes. Here, in thesame way as in calculations of energy directly from thedefinition (39), the energy is expressed by integrals of η and η . The term proportional to αη is not present in(70), because it was dropped earlier [3]. D. Energy in a moving frame
Transforming into the moving frame ¯ x = x − t, ¯ t = αt, ∂ x = ∂ ¯ x , ∂ t = − ∂ ¯ x + α∂ ¯ t . (71) φ = f − βy f ¯ x ¯ x , φ x = f ¯ x − βy f ¯ x ¯ x ¯ x , φ y = − βyf ¯ x ¯ x , (72) φ t = − f ¯ x + 12 βy f ¯ x ¯ x ¯ x + α ( f ¯ t − βy f ¯ x ¯ x ¯ t ) . (73)Up to second order (cid:18) αφ x + αβ φ y (cid:19) = 12 (cid:2) αf x + αβy ( − f ¯ x f ¯ x ¯ x ¯ x + f x ¯ x ) (cid:3) = 12 αf x + αβy f x ¯ x . (74)Therefore the expression under the integral in (62) is [ ] = − f ¯ x + 12 βy f ¯ x ¯ x ¯ x + α ( f ¯ t − βy f ¯ x ¯ x ¯ t )+ 12 αf x + αβy f x ¯ x . (75)Integration yields Lρgha = (cid:18) − f ¯ x + αf ¯ t + 12 αf x (cid:19) (1 + αη ) (76) + 13 (1 + αη ) (cid:18) β ( f ¯ x ¯ x ¯ x − f ¯ x ¯ x ¯ t ) + αβf x ¯ x (cid:19) + 12 αη . Like in (67) above, the Hamiltonian density is
Hρgh l = − α (cid:20)(cid:18) − f ¯ x + 12 αf x (cid:19) (1 + αη ) (77) + 13 (1 + αη ) (cid:18) βf ¯ x ¯ x ¯ x + αβf x ¯ x (cid:19) + 12 αη (cid:21) . Expressing f ¯ x by (68) one obtains Hρgh l = − α (cid:20) − αη + 13 βη xx − α η (78) + αβ (cid:18) − η x − ηη xx (cid:19) − β η xxxx (cid:21) . Finally the energy is given by
Eρgh l = α Z ∞−∞ η dx + α Z ∞−∞ (cid:18) η − βα η x (cid:19) dx (79)since integrals from terms with β, β vanish at inte-gration limits, and − ηη xx ⇒ η x . The invariantterm proportional to αη is not present in (79), becauseit was dropped in (62). If we include that term, thetotal energy is a linear combination of all three lowestinvariants, I (1) , I (3) , I (3) . Comment
An almost identical formula for the energyin a moving frame, for KdV expressed in dimensionalvariables (6), was obtained in [20]. That energy is ex-pressed by all three lowest order invariants E = − c Z ∞−∞ η dx + 14 c h Z ∞−∞ η dx (80) + 12 c h Z ∞−∞ (cid:18) η − h η x (cid:19) dx, as well. Translation of (80) to nondimensional variablesyields E ̺ = ̺gh l (cid:18) − αI (1) + 14 α I (2) + 12 α I (3) (cid:19) . E. How strongly is energy conservation violated? E FIG. 2. Precision of energy conservation for 3-soliton solution.Energies are plotted as open circles ( E ) and open squares( E ) for 40 time instants. The total energy in the fixed frame is given by equa-tion (39). Taking into account its non-dimensional partwe may write E ( t ) = T + V̺gh l = Z ∞−∞ (cid:20) αη + ( αη ) + 14 ( αη ) (cid:21) dx = αI (1) + α I (2) + 14 Z ∞−∞ ( αη ) dx (81)In order to see how much the changes of E violate energyconservation we will compare it to the same formula butexpressed by invariants E ( t ) = αI (1) + α I (2) + 14 α I (3) . (82)The time dependence of E and E is presented inFig. 2 for a 3-soliton solution of KdV (2). Presented istime evolution in the interval t ∈ [ − , . The shapeof the 3-soliton solution is presented only for three times t = − , − , in order to show shapes changing duringthe collision.For presentation the example of a 3-soliton solutionwith amplitudes equal 1,5, 1 and 0.5 was chosen. In Fig. 3the positions of solutions at given times were artificiallyshifted to set them closer to each other. The plots inFigs. 2 and 3 for t > are symmetric to those which areshown in the figures.For this example the relative discrepancy of the enregy E from the constant value, is very small δE = E ( t = − − E ( t = 0) E ( t = − ≈ . . (83)However, the E energy is conserved with numerical pre-cision of thirteen decimal digits in this example. In asimilar example with a 2-soliton solution (with apmli-tudes 1 and 0.5) the relative error (83) was even smaller,with the value δE ≈ . . This suggests that the de-gree of nonconservation of energy increases with n , where n is the number of solitons in the solution. η ( x ,t ) x t=-12 t = -6 t = 0 FIG. 3. Shape evolutiom of 3-soliton solution during collision.
VII. CONCLUSIONS FOR KDV EQUATION
The main conclusions can be formulated as follows • The invariants of KdV in fixed and moving frameshave the same form. (Of course when we have thesame scaling factor for x and t in the transforma-tion between frames). • We confirmed some known facts. Firstly, that theusual form of the energy H = T + V is not alwaysexpressed by invariants only. The reason lies in thefact, as pointed out by Luke in [23], that the Euler–Lagrange equations obtained from the Lagrangian L = T − V do not supply the right boundary condi-tions. Secondly, the variational approach based onLuke’s Lagrangian density provides the right Eulerequations at the boundary and allows for a deriva-tion higher order KdV equations. • In the frame moving with the velocity of soundall energy components are expressed by invariants.Energy is conserved. • Numerical calculations confirm that invariants I (1) , I (2) , I (3) in the forms (11), (13), (16), (18)are exact constants of motion for two- and three-soliton solutions, both for fixed and moving coordi-nate systems. In all performed tests the invariantswere exact up to fourteen digits in double precisioncalculations. • For the extended KdV equation (1) we have onlyfound one invariant of motion I (1) (24). • The total energy in the fixed coordinate systemas calculated in (39) is not exactly conserved butonly altered during collisions, even then by minutequantities (an order of magnitude smaller than ex-pected). Details in figure caption of figure 2.A summary of these conclusions can be found in Ta-ble I.
VIII. EXTENDED KDV EQUATION
In this section we calculate energy formula corespond-ing to a wave motion governed by second order equationsin scaled variables, that is the equation (1) for the fixedcoordinate system and the correponding equation for amoving coordinate system. As previosly we compare en-ergies calculated from the definition with those Luke’sLagrangian.
A. Energy in a fixed frame calculated fromdefinition
Now , instead of (2) we consider the second order KdVequation, that is (1) called by Marchant & Smyth [3]"extended KdV".In section IV A, total energy of the wave governed byKdV equation, that is the equation (2) with terms onlyup to first order in small parameters was obtained in (39).In calculation according to eq. (1) the potential energyis expressed by the same formula (32) as previously forKdV equation. In the expression for kinetic energy the0
TABLE I. Comparison of different energy formulas. Here η (3) = Z ∞−∞ η dx . † Formulas in this column are written in
E̺gh l .Euler Luke’s Integrals Potential KdVequations Lagrangian T + V Lagrangian dimensional † Fixed frame αI (1) + α I (2) + α η (3) α I (2) + α η (3) αI (1) + α I (2) + α η (3) 12 I (2) + αI (3) αI (1) + α I (2) + α η (3) (39) (70) (39) (45) (39)Moving frame α I (2) + α I (3) 14 α I (2) + α I (3) − αI (1) + α I (2) + α I (3) 14 αI (3) − αI (1) + αI (2) + α I (3) (57) (79) (40) (46) (80) velocity potential has to be expanded to second order insmall parameters φ = f − βy f xx + 124 β y f xxxx , (84)with derivatives φ x = f x − βy f xxx + β y f xxxxx ,φ y = − βyf xx + β y f xxxx . (85)Integrating over y and retaining terms up to fourth orderyields T = 12 ρgh l Z + ∞−∞ α (cid:20) f x + αηf x + 13 β (cid:0) f xx − f x f xxx (cid:1) + αβ ( ηf xx − ηf x f xxx ) (86) + β (cid:18) f xxx − f xx f xxxx + 160 f x f xxxxx (cid:19)(cid:21) dx. Expression (86) limited to first line gives kinetic energyfor KdV equation, see (36).Now, we use the expression for f x (and its derivatives)up to second order, see e.g. [3, Eq. (2.7)], [2, Eq. (17)] f x = η − αη + 13 βη xx + 18 α η (87) + αβ (cid:18) η x + 12 ηη xx (cid:19) + 110 β η xxxx . Insertion (87) and its derivatives into (86) gives T = 12 ρgh l Z + ∞−∞ α (cid:20) η + 12 αη + 13 β (cid:0) η x + ηη xx (cid:1) − α η + αβ (cid:18) ηη x + 34 η η xx (cid:19) (88) + β (cid:18) η xx + 745 η x η xxx + 19180 ηη xxxx (cid:19)(cid:21) dx. From properties of solutions at x → ±∞ terms with β and β in square bracket vanish and the term with αβ can be written form. Finally one obtains T = 12 ρgh l Z + ∞−∞ α (cid:20) η + 12 αη − α η − αβηη x (cid:21) dx. (89) Then total energy is the sum of (33) and (89) E tot = ρgh l Z ∞−∞ (cid:20) αη + ( αη ) + 14 ( αη ) (90) −
332 ( αη ) − α βηη x (cid:21) dx. The first three terms are identical as in KdV energyformula (39), the last two terms are new for extendedKdV equation (1).
B. Energy in a fixed frame calculated from Luke’sLagrangian
Calculate energy in the same way as in Section VI, C,but in one order higher. In scaled coordinates Lagrangiandensity is expressed by (62) (here we keep infinite con-stant term) L = ρgh l (cid:26)Z αη α (cid:20) φ t + 12 (cid:18) αφ x + αβ φ y (cid:19)(cid:21) dy + 12 (1 + αη ) (cid:27) . (91)From (84) we have φ t = f t − βy f xxt + 124 β y f xxxxt . (92)Inserting (92) and (85) into (91), integrating over y andretaining terms up to third order one obtains (constantterm id dropped) Lρgh l = α (cid:26) ( η + f t ) + α (cid:18) η + ηf t + 12 f x (cid:19) − βf xxt + 12 α ηf x + αβ (cid:18) f xx − ηf xxt − f x f xxx (cid:19) + 1120 β f xxxxt + 12 α β (cid:0) ηf xx − η f xxt − ηf x f xxx (cid:1) + αβ (cid:18) f xxx − f xx f xxxx (93) + 124 ηf xxxxt + 1120 f x f xxxxx (cid:19) − β f xxxxxxt (cid:27) . H = f t ∂L∂f t + f xxt ∂L∂f xxt + f (4 x ) t ∂L∂f (4 x ) t + f (6 x ) t ∂L∂f (6 x ) t − L is Hρgh l = − αη − α (cid:0) η + f x (cid:1) − α ηf x + α β (cid:18) − f xx + 16 f x f xxx (cid:19) + α β (cid:18) − ηf xx + 12 ηf x f xxx (cid:19) (94) + α β (cid:18) − f xxx + 130 f xx f xxxx − f x f xxxxx (cid:19) . Now, we use f x in the second order (87) and its deriva-tives. Insertion these expressions into (94) nd retentionterms up to thired order yields Hρgh l = − αη − α η − α η + 332 α η + α β (cid:18) − η x − ηη xx (cid:19) (95) + α β (cid:18) − ηη x − η η xx (cid:19) + α β (cid:18) − η xx − η x η xxx − ηη xxxx (cid:19) . The energy is obtained by integration of (95) over x (us-ing integration by parts and properties of η and its deriva-tives at x → ±∞ ). Then terms with αβ and αβ vanish.The final result is E = − ρgh l Z + ∞−∞ (cid:20) αη + ( αη ) + 14 ( αη ) (96) −
332 ( αη ) − α βηη x (cid:21) dx, the same as (90) but with the opposite sign. C. Energy in a moving frame from definition
Let us follow arguments given by Ali and Kalisch [20,Sec. 3] and used already in Section IV B. Working ina moving frame one has to replace φ x by the horizontalvelocity in a moving frame, that is, φ x − α . Then in aframe moving with the sound velocity we have φ x = f x − βy f xxx + β y f xxxxx − α ,φ y = − βyf xx + β y f xxxx . (97) Then the expression under integral over y in (31) becomes(in the following terms up to fourth order are kept) (cid:0) α φ x + α β φ y (cid:19) = 1 − αf x + α f x + y α βf xx (98) + y αβf xxx − y α βf x f xxx − y αβ f xxxxx + y α β (cid:18) f xxx − f xx f xxxx + 112 f x f xxxxx (cid:19) . After integration over y one obtains T = 12 ρgh l Z + ∞−∞ (cid:2) α ( η − f x ) + α (cid:0) − ηf x + f x (cid:1) + 13 αβf xxx + α ηf x − αβ f xxxxx + α β (cid:18) f xx + ηf xxx − f x f xxx (cid:19) + α β (cid:0) ηf xx + η f xxx − ηf x f xxx (cid:1) + α β (cid:18) f xxx − f xx f xxxx (99) − ηf xxxxx + 160 f x f xxxxx (cid:19)(cid:21) dx. Then insertion f x (87) and its derivatives yields T = 12 ρgh l Z + ∞−∞ (cid:20) − αη − α η − αβη xx + 34 α η − α β (cid:18) η x + 12 ηη xx (cid:19) − αβ η xxxx − α η + α β (cid:18) ηη x + 38 η η xx (cid:19) (100) + α β (cid:18) η xx + 233360 η x η xxx + 119360 ηη xxxx (cid:19) + 136 αβ η xxxxxx (cid:21) dx, where constant term is dropped. Using properties of so-lutions at x → ±∞ this expression can be simplified to T = 12 ρgh l Z + ∞−∞ (cid:20) − αη − α η + 34 α η − α η + 724 α β η x + 112 α β ηη x + 120 α β η xx (cid:21) dx. (101)Then total energy is E tot = ρgh l Z + ∞−∞ (cid:20) αη + 14 α η + 38 α η − α η + 748 α β η x + 124 α β ηη x + 140 α β η xx (cid:21) dx. (102)In special case α = β this formula simplifies to E tot = ρgh l Z + ∞−∞ (cid:20) αη + 14 α η + α (cid:18) η + 748 η x (cid:19) + α (cid:18) − η + 124 ηη x + 140 η xx (cid:19)(cid:21) dx. (103)2 D. Energy in a moving frame from Luke’sLagrangian
Follow considerations in Section VI, but with KdV2equation (1). Transforming into the moving framethrough (71) we have now φ = f − βy f ¯ x ¯ x + 124 β y f ¯ x ¯ x ¯ x ¯ x , (104) φ x = f ¯ x − βy f ¯ x ¯ x ¯ x + 124 β y f ¯ x ¯ x ¯ x ¯ x ¯ x , (105) φ y = − βyf ¯ x ¯ x + 16 β y f ¯ x ¯ x ¯ x ¯ x , (106) φ t = − f ¯ x + 12 βy f ¯ x ¯ x ¯ x − β y f ¯ x ¯ x ¯ x ¯ x ¯ x (107) + α ( f ¯ t − βy f ¯ x ¯ x ¯ t + 124 β y f ¯ x ¯ x ¯ x ¯ x ¯ t ) . Inserting (104)–(107) into (91) one obtains Lagrangiandensity in moving frame as (constant term is droppedas previously) Lρgh l = α ( η − f ¯ x ) + α (cid:18) η + f ¯ t − ηf ¯ x + 12 f x (cid:19) + 16 αβf ¯ x ¯ x ¯ x + α (cid:18) ηf ¯ t + 12 ηf x (cid:19) − αβ f ¯ x ¯ x ¯ x ¯ x ¯ x + α β (cid:18) f x ¯ x − f ¯ x ¯ x ¯ t + 12 ηf ¯ x ¯ x ¯ x − f ¯ x f ¯ x ¯ x ¯ x (cid:19) + α β (cid:18) ηf x ¯ x − ηf ¯ x ¯ x ¯ t + 12 η f ¯ x ¯ x ¯ x − ηf ¯ x f ¯ x ¯ x ¯ x (cid:19) + α β (cid:18) f x ¯ x ¯ x − f ¯ x ¯ x f ¯ x ¯ x ¯ x ¯ x + 1120 f ¯ x ¯ x ¯ x ¯ x ¯ t − ηf ¯ x ¯ x ¯ x ¯ x ¯ x + 1120 f ¯ x f ¯ x ¯ x ¯ x ¯ x ¯ x (cid:19) . (108)Then Hamiltonian density H = f ¯ t ∂L∂f ¯ t + f ¯ x ¯ x ¯ t ∂L∂f ¯ x ¯ x ¯ t + f ¯ x ¯ x ¯ x ¯ x ¯ t ∂L∂f ¯ x ¯ x ¯ x ¯ x ¯ t − L (109)after insertion of (108) into (109) yields Hρgh l = α ( − η + f ¯ x ) + α (cid:18) − η + ηf ¯ x − f x (cid:19) − αβf ¯ x ¯ x ¯ x − α ηf x + 1120 αβ f ¯ x ¯ x ¯ x ¯ x ¯ x + α β (cid:18) − f x ¯ x − ηf ¯ x ¯ x ¯ x + 16 f ¯ x f ¯ x ¯ x ¯ x (cid:19) (110) + α β (cid:18) − ηf x ¯ x − η f ¯ x ¯ x ¯ x + 12 ηf ¯ x f ¯ x ¯ x ¯ x (cid:19) + α β (cid:18) − f x ¯ x ¯ x + 130 f ¯ x ¯ x f ¯ x ¯ x ¯ x ¯ x + 124 ηf ¯ x ¯ x ¯ x ¯ x ¯ x − f ¯ x f ¯ x ¯ x ¯ x ¯ x ¯ x (cid:19) . In order to express (110) by η only we use f ¯ x in the form(87) and its derivatives. It gives Hρgh l = − α η + 16 αβη ¯ x ¯ x − α η (111) + α β (cid:18) η x + 14 ηη xx (cid:19) + 19360 αβ η ¯ x ¯ x ¯ x ¯ x + 732 α η − α β (cid:18) ηη x + 316 η η ¯ x ¯ x (cid:19) − α β (cid:18) η x ¯ x + 233720 η ¯ x η ¯ x ¯ x ¯ x + 119720 ηη ¯ x ¯ x ¯ x ¯ x (cid:19) − αβ η ¯ x ¯ x ¯ x ¯ x ¯ x ¯ x Then energy is given by the integral E = ̺gh l Z + ∞−∞ (cid:20) − α η + 16 αβη ¯ x ¯ x − α η (112) + α β (cid:18) η x + 14 ηη xx (cid:19) + 19360 αβ η ¯ x ¯ x ¯ x ¯ x + 732 α η − α β (cid:18) ηη x + 316 η η ¯ x ¯ x (cid:19) − αβ η ¯ x ¯ x ¯ x ¯ x ¯ x ¯ x − α β (cid:18) η x ¯ x + 233720 η ¯ x η ¯ x ¯ x ¯ x + 119720 ηη ¯ x ¯ x ¯ x ¯ x (cid:19)(cid:21) dx. From properties of solution integrals of terms with αβ, αβ , αβ vanish and terms with α β, α β, α β canbe simplified. Finally, energy is given by the follwingexpression E = ̺gh l Z + ∞−∞ (cid:20) − α η − α η + 732 α η (113) − α βη x − α βηη x − α β η x ¯ x (cid:21) dx. In special case when β = α the result is E = ̺gh l Z + ∞−∞ (cid:20) − α η − α (cid:18) η + 748 η x (cid:19) + α (cid:18) η − ηη x − η x ¯ x (cid:19)(cid:21) dx. (114)If the invariant term I (1) ≡ R αη dx is dropped in (90)or (96) then the energy calculated in the moving frame(113) have the same value but with oposite sign. E. Numerical tests
1. Fixed frame
In order to check energy conservation for the extendedKdV equation (1) we performed several numerical tests.First, discuss energy conservation in a fixes frame. Wecalculated time evolution governed by the equation (1)of waves which initial shape was given by 1-, 2- and 3-soliton solutions of the KdV (first order) equations. For3 η ( x ,t ) xt=0t=120t=240t=315 FIG. 4. Example of time evolution of 3-soliton solution. presentation the following initial conditions were chosen.3-soliton solution have amplitudes 1.5, 1 and 0.25, 2-soliton solution have amplitudes 1 and 0.5 and 1-solitonsolution the amplitude 1. The changes of energy pre-sented in Figs. 6 and 5 are qualitatively the same also fordifferent amplitudes. An example of such time evolutionfor 3-soliton solution is presented in Fig. 4.Time range in Fig. 4 contains initial shape of 3-solitonsolution with almost separated solitons at t = 0 , inter-mediate shapes and almost ideal overlap of solitons at t = 315 . In order to do not obscure details the sub-sequent shapes are shifted verticaly with respect to theprevious ones. Note additional slower waves after themain one which are generated by second order terms ofthe equation (1), that were already discussed in [2]. E / E tEn 1-solEn 2-solEn 3-solMass FIG. 5. Energy (non)conservation for the extended KdVequation in the fixed frame (1). Symbols represent valuesof the total energy given by formulas (90) or (96). Full squaresymbols represent the invariant I (1) . We see that the total energy for waves which move ac-cording to the extended KdV equation is not conserved.Although energy variations are generally small (in time range considered they do not extend 0.001%, 0.004%and 0.005% for 1-, 2-, 3-soliton waves, respectively) theyincrease with more complicated waves. For additionalcheck of numerics the invariant I (1) = R + ∞−∞ αη ( x, t ) dx for the eaquation (1) was plotted as Mass . In spite ofapproximate integration the value of I (1) was obtainedconstant up to 10 digits for all initial conditions. F. Moving frame
Here we present variations of the energy calculated ina moving frame. The time evolution of the wave is givenby the equation (1) transformed with (71), that is theequation η ¯ t + 32 ηη ¯ x + 16 βα η x (115) − α η η ¯ x + β (cid:18) η ¯ x η x + 512 ηη x (cid:19) + 19360 β α η x = 0 . E / E tEn-1solEn-2solEn-3solMass FIG. 6. Energy (non)conservation for the extended KdVequation in the moving frame (115). Symbols represent valuesof the total energy given by the formula (102). Full squaresymbols represent the invariant I (1) . The time range of the evolution was chosen for a conve-nient comparison with the numerical results obtained infixed reference frame, that is 2- and 3-soliton waves movefrom separate solitons to fully colliding time instant. Theconvention of symbols is the same as in Fig. 5. the energyis calculated according to the formula (102). In movingcoordinate system energy variations are even greater thanin the fixed reference frame, because in the time periodconsidered it approaches values of 0.02%, 0.12% and 0.2%for 1-, 2- and 3-soliton waves, respectively. This increaseof relative time variations of energy can not be atributedonly to two times smaller leading term ( αη ) in (102)with respect to (90). Again, in spite of approximate in-tegration the value of I (1) was obtained constant up to10 digits for all initial conditions.4
1. Conclusions for extended KdV equation
We calculated energy of the fluid governed by the ex-tended KdV equation (1) in two cases:1. In a fixed frame (sections VIII A and VIII B).2. In the frame moving with the sound velocity (sec-tions VIII C and VIII D).In both cases we calculated energy using two methods:from definition and from Luke’s Lagrangian. Both meth-ods give consistent results. For fixed frame energies (90)and (96) are the same. For moving frame the energycalculated from the definition contains one term morethen energy calculated from Luke’s Lagrangian, but thisterm ( R αη dx ) is the invariant I (1) . When this term isdropped both energies in moving coordinate system (102)and (113) are the same and energies in both coordinatesystems differ only by sign.The general conclusion concerning energy conservationfor shallow water wave problem can be formulated as fol-lows. Since there exists the Lagrangian of the system(Luke’s Lagrangian) then exact solutions of Euler equa-tions have to conserve energy. However, when approx-imate equations of different orders resulting from exactEuler equations are considered, energy conservation isnot a priori determined. The KdV equations obtained infirst order approximation has a miraculous property, aninfinite number of invariants with energy among them.However, this astonishing property is lost in second or-der approximation to Euler equations and energy in thisorder may be conserved only approximately. Appendix A
The simplest, mathematical form of the KdV equationis obtained from (2) by passing to the moving frame withadditional scaling ¯ x = r
32 ( x − t ) , ¯ t = 14 r α t, u = η, (A1)which gives a standard, mathematical form of the KdVequation u ¯ t + 6 u u ¯ x + βα u ¯ x ¯ x ¯ x = 0 , or u ¯ t + 6 u u ¯ x + u ¯ x ¯ x ¯ x = 0 for β = α. (A2)Equations (A2), particularly with β = α are favored bymathematicians, see, e.g. [25]. This form of KdV is themost convenient for ISM (the Inverse Scattering Method,see, e.g. [26–28]).For the moving reference frame, in which the KdVequation has a standard (mathematical) form (A2), theinvariant I (3) is slightly different. To see this differencedenote the lhs of (A2) by KDVm ( x, t ) and construct η × KDVm ( x, t ) − βα η x × ∂∂x KDVm ( x, t ) = 0 . Then after simplifications one obtains ∂∂t (cid:18) η − βα η x (cid:19) + ∂∂x (cid:20) η − βα ηη x (A3) +3 βα η η xx − (cid:18) βα η xx (cid:19) + (cid:18) βα (cid:19) η x η xxx = 0 , which implies the invariant I (3) in the following form I (3)moving frame = Z ∞−∞ (cid:18) η − βα η x (cid:19) dx = const or (A4) I (3)moving frame = Z ∞−∞ (cid:18) η − η x (cid:19) dx = const for β = α. We see, however, that the difference between (A4) and(18) is caused by additional scaling.In the Lagrangian approach as described in Sect. V,the substitution u = ϕ x into (A2) gives ϕ xt + 6 ϕ x ϕ xx + ϕ xxxx = 0 . (A5)Then the appropriate Lagrangian density for equation(A2) with ( α = β ) is L standard KdV := − ϕ t ϕ x − ϕ x + 12 ϕ xx . (A6)Indeed, the Euler–Lagrange equation obtained from theLagrangian (A6) is just (A5).The Hamiltonian for KdV (A2) can be found e.g. in[22]. Defining generalized momentum π = ∂ L ∂ϕ t , where L is given by (A6), one obtains H = Z ∞−∞ [ π ˙ ϕ − L ] dx = Z ∞−∞ (cid:20) ∂ L ∂ϕ t ϕ t − L (cid:21) dx (A7) = Z ∞−∞ (cid:20) ϕ x − ϕ xx (cid:21) dx = Z ∞−∞ (cid:20) η − η x (cid:21) dx . This is the same invariant as I (3)moving frame in (A4). Appendix B
The set of Euler equations for irrotational motion ofan incompresible and inviscid fluid can be written (ne-glecting surface tension) in dimensionless form: ∇ φ = 0 (B1) φ z = 0 on z = 0 (B2) η t + φ x η x − φ x = 0 on z = 1 + η (B3) φ t + 12 (cid:0) φ x + φ z (cid:1) + η = 0 on z = 1 + η. (B4)We look for solutions to the Laplace equation (B1) inthe form φ = ∞ X n =0 z n f ( n ) ( x, y, t ) (B5)5yielding ∞ X n =0 h n ( n − z n − f ( n ) + z n ∇ f ( n ) i = 0 . (B6)In two dimensions ( x, z ) we obtain f ( n +2) = − n + 1)( n + 2)) ∂ f ( n ) ∂x . (B7)The boundary condition at the bottom, φ z = 0 at z = 0 implies f (1) = 0 and then all odd f (2 k +1) = 0 . Now φ = ∞ X n =0 ( − m z m (2 m )! ∂ m f∂x m , (B8)where f := f (0) . In the stretched coordinates ∂ x = ε∂ ξ so φ = ε f + ∞ X n =0 ( − m z m (2 m )! (cid:0) ε∂ ξ (cid:1) m f ! . (B9)Now both (B1) and (B2) are satisfied. We must alsosatisfy the boundary conditions on z = 1 + η .In the derivation of KdV and Kadomtsev-Petiashvili[29] from the Euler equations (B1)–(B4) Infeld and Row-lands [24] applied scaling assuming the following relationsvawelength : depth : amplitude as ε − / : 1 : ε. They then applied a transformation to a frame movingwith velocity of sound. The coordinates scales as ξ = ε ( x − t ) , τ = ε t (B10) ∂ t = − ε ∂ ξ + ε ∂ τ , ∂ x = ε ∂ ξ . (B11)For the wave amplitude and velocity potential the appro-priate scaling was η = εη (1) + ε η (2) + . . . , (B12)and φ = ε φ (1) + ε φ (2) + . . . . (B13)Then the lowest order expression for φ is φ ≈ ε f − ε z f ξξ (B14)Next, Infeld and Rowlands show that in order to simul-taneously satisfy (B3) and (B4) the next order contribu-tions to η and φ cancel. It is enough to keep η = 1 + εη (1) and φ = ε φ (1) (B15)and drop upper index (1) in what follows. [1] A. Karczewska, P. Rozmej and Ł. Rutkowski, A new non-linear equation in the shallow water wave problem , Phys-ica Scripta , 054026 (2014).[2] A. Karczewska, P. Rozmej and E. Infeld, Shallow-watersoliton dynamics beyond the Korteweg–de Vries equation ,Phys. Rev. E , 012907 (2014).[3] T.R. Marchant and N.F. Smyth, The extended Korteweg–de Vries equation and the resonant flow of a fluid overtopography , J. Fluid Mech. , 263-288 (1990).[4] G.I. Burde, A. Sergyeyev,
Ordering of two small parame-ters in the shallow water wave problem , J. Phys. A: Math.Theor. , 075501 (2013).[5] R.M. Miura, KdV equation and generalizations I. Aremarkable explicit nonlinear transformation , J. Math.Phys. , 1202-1204 (1968).[6] R.M. Miura, C.S. Gardner and M.D. Kruskal, KdV equa-tion and generalizations II. Existence of conservationlaws and constants of motion , J. Math. Phys. , 1204-1209 (1968).[7] P.G. Drazin and R.S. Johnson, Solitons: An Introduc-tion, Cambridge University Press, Cambridge, 1989.[8] A.C. Newell, Solitons in Mathematics and Physics ,Philadelphia: Society for Industraial and Applied Math-ematics, 1985.[9] J.G.B. Byatt-Smith,
On the change of amplitude of in-teracting solitary waves , J. Fluid Mech. , 495-497(1987). [10] S. Kichenassamy and P. Olver,
Existence and nonexis-tence of solitary wave solutions to higher-order modelevolution equations , SIAM J. Math. Anal., , 1141-1166(1992).[11] T.R. Marchant and N.F. Smyth, Soliton Interaction forthe Korteweg-de Vries equation , IMA J. Appl. Math. ,157-176 (1996).[12] T.R. Marchant, Coupled Korteweg - de Vries equationsdescribing, to higher-order, resonant flow of a fluid overtopography , Phys. Fluids , No. 7, 1797-1804 (1999).[13] T.R. Marchant, High-order interaction of solitary waveson shallow water , Studies in Appl. Math. , 1-17(2002).[14] T.R. Marchant,
Asymptotic solitons for a higher-ordermodified Korteveg-de Vries Equations , Phys. Rev. E ,046623(1-8) (2002).[15] Q. Zou and CH-H. Su, Overtaking collision between twosolitary waves , Phys. Fluids , No. 7, 2113-2123 (1986).[16] E. Tzirtzilakis, V. Marinakis, C. Apokis and T. Bountis, Soliton-like solutions of higher order wave equations ofKorteweg-de Vries type , J. Math. Phys. , No. 12, 6151-6165 (2002).[17] G.I. Burde, Solitary wave solutions of the higher-orderKdV models for bi-directional water waves , Commun.Nonlinear Sci. Numerical Simulat. , 1314-1328 (2011).[18] R. Hirota, Exact solution of the Korteweg-de Vries equa-tion for multiple collisions of solitons , Phys. Rev. Lett. , 1192-1194 (1972).[19] R. Hirota, The Direct Method in Soliton Theory , Cam-bridge University Press, Cambridge, (2004), first pub-lished in Japanese (1992).[20] A. Ali and H. Kalisch,
On the formulation of mass, mo-mentum and energy conservation in the KdV equation ,Acta Appl. Math. , 113-131 (2014).[21] G.B. Whitham,
Linear and Nonlinear Waves , Wiley, NewYork, 1974.[22] F. Cooper, C. Lucheroni, H. Shepard and P. So-dano,
Variational Method for Studying Solitons in theKorteweg-deVries Equation , Phys. Lett. A
A variational principle for a fluid with a freesurface , J. Fluid Mech. (1967), , part 2, 395-397.[24] E. Infeld and G. Rowlands, Nonlinear Waves, Solitonsand Chaos , 2nd edition, Cambridge University Press,Cambridge, 2000.[25] P. Lax,
Integrals of nonlinear equations of evolutionand solitary waves , Comm. Pure Applied Math. (5),467–490 (1968).[26] C.S. Gardner, J.M. Greene, M.D. Kruskal, andR.M. Miura, Method for Solving the Korteweg-deVriesEquation , Phys. Rev. Lett. , 1095-1097 (1967).[27] M. Ablowitz, H. Segur, Solitons and the Inverse Scatter- ing Transform , SIAM, Philadelphia, 1981.[28] M. Ablowitz, P. Clarkson,
Solitons, Nonlinear EvolutionEquations and Inverse Scattering , Cambridge UniversityPress, Cambridge, 1991.[29] B.B. Kadomtsev and V.I. Petviashvili,
On the stabilityof solitary waves in weakly dispersive media , Dokl. Akad.Nauk SSSR , 753-756 (1970); Sov. Phys. Dok. ,539-541 (1970).[30] E. Infeld, G. Rowlands and M. Hen, Three dimenionalstability of KdV waves and solitons , Acta Phys. Polon. A , 123-143, (1978).[31] E. Infeld and G. Rowlands, Three-dimensional stabilityof Korteweg de Vries waves and solitons, II , Acta Phys.Polon. A , 329-332, (1979).[32] E. Infeld, Three-dimensional stability of Korteweg deVries waves and solitons. III. Lagrangian methods, KdVwith positive dispersion , Acta Phys. Polon. A The stability of solitary waves , Proc. R.Soc. Lond. A , 153-183 (1972).[34] M.S. Longuet-Higgins and J.D. Fenton,
On the mass, mo-mentum, energy and circulation of a solitary wave. II ,Proc. R. Soc. Lond. A , 471-493, (1974).[35] E. Infeld and G. Rowlands,
Stability of nonlinear ionsound waves and solitons in plasma , Proc. R. Soc. Lond.A366