Energy invariant for shallow water waves and the Korteweg – de Vries equation.
Is energy always an invariant?
It is well known that the KdV equation has an infinite set of conserved quantities. The first three are often considered to represent mass, momentum and energy. Here we try to answer the question of how this comes about, and also how these KdV quantities relate to those of the Euler shallow water equations. Here Luke’s Lagrangian is helpful. We also consider higher order extensions of KdV. Though in general not integrable, in some sense they are almost so, these with the accuracy of the expansion.
pacs:02.30.Jr, 05.45.-a, 47.35.Bb, 47.35.Fg
There exists a vast number of papers dealing with the shallow water problem. Aspects of the propagation of weakly nonlinear, dispersive waves are still beeing studied. Last year we published two articles KRR ; KRI in which Korteveg–de Vries type equations were derived in weakly nonlinear, dispersive and long wavelength limit. The second order KdV type equation was derived. The second order KdV equation MS ; BS , sometimes called "extended KdV equation", was obtained for the case with a flat bottom. In derivation of the new equation we adapted the method described in BS . In KRI , an analytic solution of this equation in the form of a particular soliton was found, as well.
It is well known, see, e.g. Miura ; MGK ; DrJ ; Newell85 , that for the KdV equation there exists an infinite number of invariants, that is, integrals over space of functions of the wave profile and its derivatives, which are constants in time. Looking for analogous invariants for the second order KdV equation we met with some problems even for the standard KdV equation (which is first order in small parameters). This problem appears when energy conservation is considered.
In this paper we reconsider invariants of the KdV equation and formulas for the total energy in several different approaches and different frames of reference (fixed and moving ones). We find that the invariant , sometimes called the energy invariant, does not always have that interpretation. We also give a proof that for the second order KdV equation, obtained in KRR ; KRI ; MS ; BS , is not an invariant of motion.
There are many papers considering higher-order KdV type equations. Among them we would like to point out works of Byatt-Smith B-S87 , Kichenassamy and Olver SK_PO , Marchant MS ; MS96 ; Mar99 ; Mar02 ; Mar02a , Zou and Su ZouSu , Tzirtzilakis et.al. TMAB and Burde Burde . It was shown that if some coefficients of the second order equation for shallow water problem (1) are diferent or zero then there exists a hierarchy of solition solutions. Kichenassamy and Olver SK_PO even claimed that for second order KdV equation solitary solutions of appropriate form can not exist. This claim was falsified in our paper KRI where the analytic solution of the second order KdV equation (1) was found. Concerning the energy conservation there are indications that collisions of solitons Hirota72 ; Hir which are solutions of higher order equations of KdV type can be inelastic ZouSu ; TMAB .
The paper is organized as follows. In Section II several frequently used forms of KdV equations are recalled with particular attention to transformations between fixed and moving reference frames. In Section III the form of the three lowest invariants of KdV equations is derived for different forms of the equations. In Section IV we show that the energy calculated from the definition has no invariant form. Section V describes the variational approach in a potential formulation which gives a proper KdV equation but fails in obtaining second order KdV equations. In the next section the proper invariants are obtained from Luke’s Lagrangian density. Section VII summarizes conclusions on the energy for KdV equation. In section VIII we apply the same formalism to calculate energy for waves governed by the extended KdV equation (second order). We found that energy is not conserved neither in fixed coordinate system nor in the moving frame.
Ii The extended KdV equation
The geometry of shallow water waves is presented in Fig. 1.
In KRR ; KRI we derived an equation, second order in small parameters, in the fixed reference system and with scaled nondimensional variables containing terms for bottom fluctuations. They will not be considered here.
For a flat bottom that equation reduces to the second order KdV type equation, identical with (BS, , Eq. (21)) for , that is,
Subscripts denote partial differentiation. Small parameters are defined by ratios of the wave amplitude , the average water depth and mean wavelength
Limitation to the first order in small parameters yields the KdV equation in a fixed coordinate system
Transformation to a moving frame in the form
allows us to remove the term in the KdV equation in a frame moving withthe velocity of sound
Problems with mass, momentum and energy conservation in the KdV equation were discussed in Kalisch recently. In this paper the authors considered the KdV equations in the original dimensional variables. Then the KdV equatios are
in a fixed frame of reference and
In our present paper we discuss the energy formulas obtained both in fixed and moving frames of reference for KdV (2), (4), (5), (6) . There seem to be some contradictions in the literature because the form of some invariants and the energy formulas differ in different sources because of using different reference frames and/or different scalings. In this paper we address this problems.
The second goal is to present some invariants for a KdV type equation of the second order (1).
Iii Invariants of KdV type equations
It is well known, see, e.g. (DrJ, , Ch. 5), that an equation of the form
where neither (an analog to density) nor (an analog to flux) contain partial derivatives with respect to , corresponds to some conservation law. It can be applied, in particular, to KdV equations (where there exist an infinite number of such conservation laws) and to the equations of KdV type like (1). Functions and may depend on but not . If both functions and are integrable on and const (soliton solutions), then integration of equation (7) yields
The same conclusion applies for periodic solutions (cnoidal waves), when in the integrals (8), (9) limits of integration are replaced by , where is the space period of the cnoidal wave (the wave length).
iii.1 Invariants of the KdV equation
one immediately obtains the conservation of mass (volume) law
Similarly, multiplication of (2) by gives
resulting in the invariant of the form
Denote the left hand side of (2) by and take
The result, after simplifications is
Then the next invariant for KdV in the fixed reference frame (2) is
Then the next invariant for KdV equation in moving reference frame (2) is
The procedure similar to those described above leads to the same invariants for both equations (5) and (6) where KdV equations are written in dimensional variables. To see this, it is enough to take , where is the lhs either of (5) or (6). For equation (5) the conservation law is
whereas for equation (6) the flux term is different
But in both cases the same invariant is obtained as
Conclusion Invariants have the same form for fixed and moving frames of reference when the transformation from fixed to moving frame scales and in the same way (e.g. and ). When scaling factors are different, like in (116), then the form of in the moving frame differs from the form in the fixed frame, see Appendix A.
For those solutions of KdV which preserve their shapes during the motion, that is, for cnoidal solutions and single soliton solutions, integrals of any power of and any power of arbitrary derivative of the solution with respect to are invariants. That is,
where , and is an arbitrary real number. Then an arbitrary linear combination of is an invariant, as well.
iii.2 Invariants of the second order equations
Can we construct invariants for KdV type equations of the second order? Let us try to take for equation (1). Then we find that all terms, except , can be written as , as
As (III.2) depends on and space derivatives and also since all those functions vanish when , the conservation law for mass (volume)
holds for the second order equation.
(Conservation law (24) holds for the equation with an uneven bottom, as well.)
The invariant is usually referred to as the energy invariant. Is this really the case?
iv.1 Energy in a fixed frame as calculated from the definition
The hydrodynamic equations for an incompressible, inviscid fluid, in irrotational motion and under gravity in a fixed frame of reference, lead to a KdV equation of the form
We will find the function
Now construct the total energy of the fluid from the definition.
The total energy is the sum of potential and kinetic energy. In our two-dimensional system the energy in original (dimensional coordinates) is
Note, that the factor in front of the integrals has the dimension of energy.
In the following, we omit signs , having in mind that we are working in dimensionless variables. Integration in (30) with respect to yields
After renormalization (substraction of constant term ) one obtains
In kinetic energy we use the velocity potential expressed in the lowest (first) order
Inegration with respect to the vertical corrdinate gives, up to the same order,
In order to express energy through the elevation funcion we use (27). We then substitute in terms of the third order and in terms of the second order
The last term vanishes as
Therefore the total energy in the fixed frame is given by
The energy (39) in a fixed frame of reference has non invariant form. The last term in (39) results in small deviations from energy conservation only when changes in time in soliton’s reference frame, what occurs only during soliton collision. This deviations are discussed and illustrated in Section VI E.
The result (IV.1) gives the energy in powers of only. The same structure of powers in was obtained by the authors of Kalisch , who work in dimensional KdV equations (5) and (6). On page 122 they present a non-dimensional energy density in a frame moving with the velocity . Then, if is set, the energy density in a fixed frame is proportional to as the formula is obtained up to second order in . However, the third order term is , so the formula up to the third order in becomes
This energy density contains the same terms like (IV.1) and does not contain the term , as well.
Energy calculated from the definition does not contain a proper invariant of motion.
iv.2 Energy in a moving frame
Now consider the total energy according to (28) calculated in a frame moving with the velocity of sound . Using the same scaling (29) to dimensionless variables we note that in these variables . As pointed by Ali and Kalisch [8,Sect. 3] working in such system one has to replace by the horizontal velocity in a moving frame, that is by . Then repeating the same steps as in the previous subsection yields the energy expressed by invariants
The crucial term in (40) appears due to integration over vertical variable of the term supplied by .
V Variational approach
v.1 Lagrangian approach, potential formulation
Some attempts at the variational approach to shallow water problems are summarized in G.B. Whitham’s book (Whit, , Sect 16.14).
For KdV as it stands, we can not write a variational principle directly. It is necessary to introduce a velocity potential. The simplest choice is to take . Then equation (2) in the fixed frame takes the form
The appropriate Lagrangian density is
v.2 Hamiltonians for KdV equations in the potential formulation
The energy is expressed by invariants so it is a constant of motion.
The constant of motion in a moving frame is
The potential formulation of the Lagrangian, described above, is succesful for deriving KdV equations both for fixed and moving reference frames. It fails, however, for the second order KdV equation (1). We proved that there exists a nonlinear expression of , such that the resulting Euler–Lagrange equation differs very little from equation (1). The difference lies only in the value of one of the coefficients in the second order term . Particular values of coefficients in this term also cause the lack of the invariant for second order KdV equation, (see (III.2)).
Vi Luke’s Lagrangian and KdV energy
The full set of Euler equations for the shallow water problem, as well as KdV equations (2), (A), and second order KdV equation (1) can be derived from Luke’s Lagrangian Luke , see, e.g. MS . Luke points out, however, that his Lagrangian is not equal to the difference of kinetic and potential energy. Euler–Lagrange equations obtained from do not have the proper form at the boundary. Instead, Luke’s Lagrangian is the sum of kinetic and potential energy suplemented by the term which is necessary in dynamical boundary condition.
vi.1 Derivation of KdV energy from the original Euler equations according to Eigr
In Chapter 5.2 of the Infeld and Rowlands book the authors present a derivation of the KdV equation from the Euler (hydrodynamic) equations using a single small parameter . Moreover, they show that the same method allows us to derive the Kadomtsev–Petviashvili (KP) equation KP for water waves IRH ; KP_woda ; L-H&F ; Benj and also nonlinear equations for ion acoustic waves in a plasma KP_plazma ; InRo . The authors first derive equations of motion, then construct a Lagrangian and look for constants of motion. For the purpose of this paper and for comparison to results obtained in the next subsections it is convenient to present their results starting from Luke’s Lagrangian density. That density can be written as (here )
In Chapter 5.2.1 of EIGR the authors introduce scaled variables in a movimg frame ( plays a role of small parameter and if , then KdV equation is obtained). Then (for details, see Appendix B or (EIGR, , Chapter 5.2))
Substitution of the above formulas into the expression [ ] under the integral in (48) gives
Remark The full Lagrangian is obtained by integration of the Lagrangian density (48) with respect to . Integration limits are for a soliton solutions, or , where –wave length (space period) for cnoidal solutions. Integration by parts and properties of the solutions at the limits, see (9), allow us to use the equivalence .
Integration over gives (note that )
Write (52) up to third order in
It is easy to show, that
The Hamiltonian density reads as
Dropping the constant term one obtains the total energy as
Now, we need to express and its derivatives by and its derivatives. We use (27) replacing and by , that is,
Then the total energy in a moving frame is expressed in terms of the second and the third invariants
Note that the term occuring in the third order invariant originates from three terms appearing in , and (see terms and in (VI.1)).
vi.2 Luke’s Lagrangian
The original Lagrangian density in Luke’s paper Luke is
The Lagrangian density in scaled variables becomes ()
So, in dimensionless quatities
The full Lagrangian is obtained by integration over . In dimensionless variables () it gives
The factor in front of the integral, , has the dimension of energy.
Next, the signs ( ) will be omitted, but we have to remember that we are working in scaled dimensionless variables in a fixed reference frame.
vi.3 Energy in the fixed reference frame
Express the Lagrangian density by and . Now, up to first order in small parameters