The Complex Korteweg-de Vries Equation: A Deeper Theory of Shallow Water Waves
TThe Complex Korteweg-de Vries Equation: A Deeper Theory of Shallow Water Waves
M. Crabb and N. Akhmediev
Department of Theoretical Physics, Research School of Physics,The Australian National University, Canberra, ACT, 2600, Australia
Using Levi-Civit`a’s theory of ideal fluids, we derive the complex Korteweg-de Vries (KdV) equa-tion, describing the complex velocity of a shallow fluid up to first order. We use perturbation theory,and the long wave, slowly varying velocity approximations for shallow water. The complex KdVequation describes the nontrivial dynamics of all water particles from the surface to the bottom ofthe water layer. A crucial new step made in our work is the proof that a natural consequence of thecomplex KdV theory is that the wave elevation is described by the real KdV equation. The complexKdV approach in the theory of shallow fluids is thus more fundamental than the one based on thereal KdV equation. We demonstrate how it allows direct calculation of the particle trajectories atany point of the fluid, and that these results agree well with numerical simulations of other authors.
PACS numbers: 05.45.Yv, 42.65.Tg, 42.81.qb 47.35.-i
I. INTRODUCTION
Water waves can be classified into many types [1]. Oneshared feature is, as Feynman put it, that they have allthe possible complications that a wave can have. Forinstance, when treated with full generality, water wavesare commonly considered to be nonlinear phenomena [2].Consequently, to precisely describe water waves is infa-mously difficult. One convenient approximate methodof describing water waves is to give an evolution equa-tion for the elevation of the water surface [3, 4]; in thelowest order nonlinear approximation, this leads to thenonlinear Schr¨odinger equation for the complex envelopeof waves in deep water [5], and the real Korteweg-de Vries(KdV) equation for the elevation of a nonlinear shallowwater wave [6–8]. Higher-order physical effects can be in-corporated in each of these equations for more descriptivepower [4, 9–11]. However, these approaches still restrictus to only the motion of the surface.When a fluid is incompressible, we can also describeits state of motion with the potential. As is often de-tailed in many standard texts on hydrodynamics [12],a well-behaved potential can be treated as the indepen-dent variable in the description of the fluid, rather thana spatial coordinate. This approach was pioneered byLevi-Civit`a in 1907, who derived an equation satisfied byall fluid motions without singularities in a channel of ar-bitrary depth [13]. The theory was developed in the earlytwentieth century, but due to the difficult nature of theresulting equation it has fallen by the wayside. Moderninterest in this approach has been partly revived by Levi[14].In this work, we give a thorough derivation of a com-plex KdV equation describing first order perturbations inthe complex velocity around steady flow. Importantly,not only the dependent function in the KdV equationbut the spatial variable is considered to be complex aswell. This approach provides a complete description ofthe flow in the fluid up to first order, and is not limited todescribing only the water elevation. Thus, the main ad-vantage of the complex KdV equation in hydrodynamics is that it describes the dynamics of water particles notonly at the surface but also throughout the entire bodyof the fluid.A new step forward in our work is that if the complexKdV equation describes a first-order perturbation of thecomplex velocity for a shallow fluid, then the fluid’s eleva-tion is naturally determined by a solution of the standardreal KdV equation.As an example application of this theory, we give asimple demonstration of how the motion of the entirefluid may be described with a basic periodic solution tothe complex KdV equation. We illustrate how the parti-cle displacement from a given point may be determined,along with their trajectories, and show that in the limitas the period becomes infinite, the familiar soliton solu-tion may be correctly described.
II. THE LEVI-CIVIT `A THEORY OFPOTENTIAL FLOW IN INVISCID FLUIDS
Water of depth h occupies a channel with a flat bottom,and waves of a height η = η ( x, t ) above the mean levelpropagate along the surface, the axis of x being takenalong the bottom, and in the direction of propagation,while y represents a vertical coordinate. A diagram isshown in Fig.1. At any point x at a time t the watersurface is described by the equation y = h + η ( x, t ) . Thus, ∂η∂t + u ∂η∂x = v, where u and v are the horizontal and vertical componentsof the fluid velocity, respectively.When the ratio of wave height to wavelength is verysmall, we have the linearised theory ∂η∂t = ∂ψ∂x where ψ is the stream function, with the convention dψ = vdx − udy . If we additionally suppose that the motion a r X i v : . [ phy s i c s . f l u - dyn ] F e b xy hη ( x, t ) FIG. 1:
Schematic of water layer with average depth h.The function η ( x, t ) describes the water elevation abovethe average level. is irrotational, and that the wave is long enough thatsurface tension can be neglected, then the pressure justinside the fluid’s surface must be very nearly equal to thepressure just outside the fluid’s surface, and this impliesthat, approximately, gη (cid:39) ∂φ∂t , where φ is the velocity potential, with dφ = − udx − vdy .Since there is no flow through the bottom of the chan-nel, the stream function is constant on the bottom of thechannel, and we can choose ψ = 0 when y = 0. The com-plex potential w = φ + iψ is then real when y = 0 , and w can be analytically continued into the region − h (cid:54) y < . Doing so leads to Cisotti’s equation, ∂ ∂t { w ( z + ih, t ) + w ( z − ih, t ) } ++ ig ∂∂z { w ( z + ih, t ) − w ( z − ih, t ) } = 0 , (1)More details can be found in the well-known book byMilne-Thomson [12]. This equation is complex but lin-ear. However, we are also interested in nonlinear waves.Beginning from just Bernoulli’s principle, if we let q = √ u + v be the total speed of the fluid at a givenposition and time, then − ∂φ∂t + q + gy + pρ = f ( t )where ρ is the density of the fluid, p is the pressure, andthe time-derivative of the velocity potential is the energydue to acceleration within the fluid.Now we suppose that the fluid surface is free to movealong a variable curve given by y = h + η ( x, t ) . Along thefree surface, the pressure is constant, so we have − ∂φ∂t + q + gη = 0 . (2)We can define the velocity potential φ such that any con-stant or function of time on the right hand side can be setto zero. Note also that for surface waves, 0 (cid:54) | ψ | (cid:54) | ψ | , where ψ = ψ on the free surface y = h + η ( x, t ) . Let us denote w = φ + iψ, z = x + iy, and Υ = u − iv .We reiterate Υ = − ∂w∂z . (3)Since w and Υ are both real along the bottom of the chan-nel, and certainly assumed to be holomorphic throughoutthe body of fluid, we can analytically continue both func-tions to the region where − h (cid:54) y <
0. Also, since w isholomorphic at every point of the flow, we can invertthe relationship to write z, and the complex velocity Υ , as functions of the potential w rather than position. Thevelocity potential and stream function satisfy, by (3), thedifferential equation dx + idy = − dφ + idψu − iv . Streamlines are defined by dψ = 0 , so, with ds = dx + dy defining the line element on the free surface, and after col-lecting real and imaginary parts, which are respectively dx = − udφ − vdψq ,dy = − vdφ + udψq , we have ∂y∂s = − vq ∂φ∂s on a streamline. We also have, by definition, − dφ = udx + vdy, so it is easy to see that ∂φ∂s = − q, and consequently ∂η∂s = vq on a streamline. Bernoulli’s equation (2) can now bedifferentiated with respect to an arc length s along thefree surface y = h + η ( x, t ) to give ∂q∂t + q ∂q∂s + gvq = 0 , or ∂q ∂t + q ∂q ∂s + 2 gv = 0 . (4)As stated only shortly before, we can recast Bernoulli’sprinciple, now in the form (4), in terms of the complex ve-locity Υ and the complex potential w. We obtain througha simple change of variable dφ = − qds the equation ∂∂t | Υ | − | Υ | ∂∂φ | Υ | + ig (Υ − Υ) = 0 . (5)Here, instead of the condition that the free surface be de-fined by y = h + η ( x, t ), we have instead the free surfacedefined by the streamline ψ = ψ , and the complex veloc-ity should be understood as a function Υ = Υ( φ + iψ , t ) , with the conjugate velocity Υ given by Υ = Υ( φ − iψ , t ) . Lastly, since ψ = 0 and therefore w is real along the bot-tom of the channel, y = 0 , we can extend the differentialequation for Υ to all w , rather than just for w = φ. Thedifferential equation (5) can therefore be presented in theform − ∂∂t log { Υ( w + iψ , t )Υ( w − iψ , t ) } ++ ∂∂w { Υ( w + iψ , t )Υ( w − iψ , t ) } ++ ig (cid:26) w + iψ , t ) − w − iψ , t ) (cid:27) = 0 . (6)This equation was first derived by Levi-Civit`a [13] over acentury ago, albeit in a more limited form, dealing onlywith steady flow. However, due to the fact that a differ-ential equation of this type is difficult to solve, there hasbeen only relatively minor attention given to this particu-lar equation. Having said that, even though this equationis challenging to solve in the general case, it is nonethelesspossible to gain insight into possible fluid dynamics byapplying certain techniques, such as perturbation series. III. LINEAR PERTURBATIONS AROUND ASTEADY FLOW
Suppose that the complex velocity of the wave has theform of a small perturbation around an otherwise con-stant flow parallel to the bottom of the channel, soΥ( w, t ) = c { εα ( w, t ) } , where ε is a small parameter, α is a complex functionof the potential and c is a real constant. Since we arerestricting ourselves to considering only small perturba-tions around a steady horizontal flow, most of the flowacross the surface at any given point will be due to thehorizontal movement, and therefore we can take the fluxover the free surface to be approximately ψ = − ch. Thiswill clearly be true for periodic functions, but also for thegeneral problem of the solitary wave [12], since the fluidmust return to steady motion infinitely far away from thetravelling pulse.By hypothesis, ε (cid:39) , so we can disregard those terms of all but first order. To first order, the perturbation sat-isfies − ∂∂t log[1 + ε { α ( w + ich, t ) + α ( w − ich, t ) } ]++ cε ∂∂w { α ( w + ich, t ) + α ( w − ich, t ) } ++ igεc { α ( w + ich, t ) − α ( w − ich, t ) } = 0 . (7)This is the equation which will determine the stability ofa small perturbation in the velocity.For simplicity, first, we will consider motion which canbe reduced to steady flow, such as, for example, a solitarywave or periodic motion. When the motion is steady, allthe time dependence will be implicit, and contained inthe potential, so that (7) becomes (cid:26) ddw cos (cid:18) ch ddw (cid:19) − gc sin (cid:18) ch ddw (cid:19)(cid:27) α ( w ) = 0 . (8)Here, the sine and cosine terms should be understood,like the exponential, in the sense of formal power seriesof an operator. That is,cos (cid:18) ddw (cid:19) = ∞ (cid:88) n =0 ( − n (2 n )! d n dw n , (9)and similar for the sine operator.The ease of manipulation which results from the form(8) immediately suggests a simple solution for steady flowin the form α ( w ) = α e iµw , (10)where µ is a solution of the transcendental equationtanh chµ = c µg , (11)and α a constant. If α is small, integration gives theequation z = − c (cid:26) w + iα µ e iµw + O ( α ) (cid:27) and thus η = α µ e chµ cos µcx + O ( α ) (12)for the stationary form of the free surface, up to firstorder. We see that µ = k/c, approximately, where k isthe wavenumber of the flow, and that (11) is just thedispersion relation for waves in an inviscid fluid of depth h. Now suppose the motion is steady. We can write thepotential purely as a function of z ; w = w ( z ) , and tolowest order in ε , we have w = − cz + O ( ε ) . To lowest order in the z -plane: − ddz { β ( z + ih ) + β ( z − ih ) } = igc { β ( z + ih ) − β ( z − ih ) } . (13)This is effectively equivalent to Cisotti’s equation (1).For the case in which the channel is shallow, with aflat bottom, we can obtain an approximate equation ofmotion by expanding in powers of h , which will be aconvergent series when h is sufficiently small. IV. THE NONLINEAR THEORY
We can more generally consider a perturbation seriesin powers of the small parameter ε of the formΥ = c (1 + εβ + ε β + · · · + ε n β n + · · · ) , (14)where β n = β n ( z, t ) . We can expand (6) to each orderof ε, and solve the equations obtained at each order se-quentially. The constant flow velocity c will have distinctvalues for the shallow and deep regimes, and these mustbe treated as two separate limiting cases to make theproblem manageable. A. The Complex KdV Equation
For shallow fluids, we consider perturbations around aconstant flow c = √ gh. We can expand Υ( w ± ich, t ) inpowers of h, since h is small. We then apply the changeof variable Υ( w, t ) (cid:55)→ Υ( z, t ), under which the derivativetransforms as ∂ Υ ∂w (cid:55)→ − ∂∂z log Υ( z, t ) . (15)Higher order derivatives transform as ∂ Υ ∂w (cid:55)→ ∂ ∂z log Υ ,∂ Υ ∂w (cid:55)→ − ∂ ∂z log Υ + 1Υ ∂ Υ ∂z ∂ ∂z log Υ , and so on. Truncating the series at third order gives theapproximate equation − Υ ∂ Υ ∂z + ( gh ) ∂∂z log Υ++ gh (cid:18) ∂ Υ ∂z − ∂ Υ ∂z ∂ Υ ∂z (cid:19) ++ g h (cid:18) ∂ Υ ∂z ∂ ∂z log Υ − ∂ ∂z log Υ (cid:19) (cid:39) ∂ Υ ∂t after a change of variable to the z -plane. We then assumea perturbation series solution of the formΥ( z, t ) √ gh = ∞ (cid:88) n =0 ε n β n ( z, t ) . The lowest order term is identically zero with this sub-stitution, as we have seen. The only surviving coefficientof ε gives the first order approximation2( gh ) (cid:18) h ∂ β ∂z − √ gh ∂β ∂t (cid:19) ε, while the second-order terms read (cid:26) − β ∂β ∂z − h ∂β ∂z ∂ β ∂z + h β ∂ β ∂z −− √ gh β ∂β ∂t + 2 h ∂ β ∂z − √ gh ∂β ∂t (cid:27) ( gh ) ε . Since the same stray terms that appear as the coefficientsof ε appear as coefficients of ε with β instead of β , itis reasonable to suspect that these terms actually belongto a higher order of smallness.If we let a typical wave in the fluid have a characteristiclength scale l, then to say that the fluid is shallow is onlythe statement that the fraction h/l is much less than 1.For a wave which is typically long, the wave will normallyappear to change only slowly, and over a long horizontaldistance of the same order as l. If we put (cid:18) hl (cid:19) = δ, where δ is small, then this suggests defining the dimen-sionless variable Z = X + iY = zl . The condition that the fluid is shallow is now that thevertical coordinate Y is much smaller than 1 everywherein the fluid; more specifically, Y = O ( δ ). We will alsoassume that δ = ε. Writing equations in terms of Z, we see that the varia-tion of the first order perturbation β must be infinitesi-mal with respect to t. This is physically intuitive since thecharacteristic length scale of the wave is large. However,upon the introduction of a longer time scale, t (cid:48) = εt, β can be seen to have a nontrivial slow variation; we have atfirst order a complex Korteweg-de Vries (KdV) equation: l √ gh ∂β ∂t (cid:48) = − β ∂β ∂Z + 13 ∂ β ∂Z . If we take the ratio of t (cid:48) to the characteristic time l/ √ gh ,we can define a dimensionless long time scale T by t (cid:48) T = l √ gh . In terms of Z and T, we have the fully dimensionless formof the complex KdV equation, ∂β ∂T = − β ∂β ∂Z + 13 ∂ β ∂Z . (16)The conditions of validity are that the wave is generallylong, and the perturbation varies slowly with time.The idea of reviving the theory of the complex KdVequation comes from the work of Levi [14]. However,some fundamental errors made in previous work [14, 15]did not allow this theory to be used in practice. Namely,in [14], the water elevation h was directly replaced by theexpression h = h ε with h being an unspecified pa-rameter. The smallness parameter introduced this wayis incompatible with approximations required for estab-lishing correspondence with the real KdV equation. Inthe work [15], an intended small parameter ε was notdimensionless, making the idea of smallness in a pertur-bation series poorly defined. Here, by instead scaling thecomplex variable by a characteristic length scale l , thetypical length of the wave enters through the derivativeterms, providing a physically clear interpretation of howsmall each of the terms are. The two scales of the flow, h and l , then naturally form the dimensionless parameter ε = ( h/l ) , which is also the same order of smallness asin the real KdV equation [16], up to a possible constantfactor.Unlike the better-known real KdV equation [7], thecomplex KdV equation describes a perturbation of thecomplex velocity around an otherwise steady, uniformflow. It thus describes the entire fluid motion, not justthe elevation of the free surface. Consequently, our ap-proach allows us to describe the trajectories of the fluidparticles across the whole layer of liquid. This has notactually been carried out at all up until now, althoughthis is one of the important advantages of the complexKdV equation. Below, we provide the first demonstra-tion of how this is done, as well as point out its excellentqualitative agreement with numerical results of other au-thors.But before we do, we will show, for the first time, howthe complex KdV theory leads naturally to the standardtheory of the real KdV equation for the surface elevationof shallow water. B. Relation to the Real KdV Equation
We can construct a similarly dimensionless potential˜ w by the scaling˜ w = w √ ghl = − Z + εγ + ε γ + · · · (17)The corresponding first order perturbation of the poten-tial in dimensionless form will give β through − ∂γ ∂Z = β ( Z, T ) . (18)Substituting this into (16) gives the potential KdV equa-tion for the function γ : ∂γ ∂T = 32 (cid:18) ∂γ ∂Z (cid:19) + 13 ∂ γ ∂Z . (19) Equation (19) also gives one of the simplest methodsfor obtaining the elevation η . We have assumed that,whatever the form of the free surface, it corresponds tothe constant value of the stream function ψ = − ch. Theimaginary part of the dimensionless potential ˜ w = ˜ φ + i ˜ ψ on the surface is then just˜ ψ = − hl . Suppose that a holomorphic solution for the first orderperturbation of the potential γ = γ ( Z, T ) is obtained.If γ has imaginary part ˜ ψ , then we must have on thefree surface y = h + η the equation ηl (cid:39) ε ˜ ψ | Y =( h + η ) /l (20)in terms of the dimensionless variables, accurate to firstorder in ε . In general, this will give an implicit equationfor η .In fact, since Y has already been required to be small,of order √ ε, we can avoid the complications of implicitequations for the free surface by approximating the firstorder perturbation of the potential as γ ( Z, T ) (cid:39) γ ( X, T ) − iY β ( X, T ) . Recalling that the stream function vanishes on the bot-tom of the channel, it follows that γ ( X, T ) is real, so thefirst order perturbation in the stream function must beapproximately ˜ ψ (cid:39) − Y β ( X, T ) . From (20), the surface elevation η can be given in termsof β ( X, T ) as η ( X, T ) (cid:39) − ε { h + η ( X, T ) } β ( X, T ) , or η ( X, T ) (cid:39) − hεβ ( X, T ) (21)to first order in ε .The vanishing of the stream function along the bottomis equivalent to the vanishing of the vertical componentof the velocity. We see then that β ( X, T ) must be real.Given that β ( X, T ) is a real solution to (16) with thereal variable X instead of the complex variable Z, wehave obtained in (21) the result that the surface eleva-tion η of a shallow fluid is described by a real solutionof the KdV equation in X and T , up to a constant ofproportionality, and accurate to first order in ε .We therefore make the claim that the complex KdVequation is more fundamental than the real KdV equa-tion, since we have shown that the description of thesurface elevation by the real KdV equation naturallyemerges as a consequence of a perturbation of the com-plex velocity being described by the complex KdV equa-tion. The complex KdV equation, however, also allowsfor a full picture of the motion of the entire fluid up tofirst order, not just the surface elevation to which we arelimited by considering only its real solutions. Not onlymathematically, then, but also physically, we have thusjustified the claim that it is the complex KdV equationwhich takes the most fundamental place in the hydrody-namic theory of shallow fluids.It is also worth mentioning that the τ -function for thecomplex KdV equation, which provides maybe the sim-plest mathematical lens through which to view the KdVequation [17], can be simply related to the first orderperturbation of the potential, sincelog τ ( Z, T ) ∝ (cid:90) Z γ ( Z (cid:48) , T ) dZ (cid:48) . (22)In the complex KdV equation, the τ -function is thus alogarithmic measure of flux in the fluid due to perturba-tions around steady flow. To develop any more sophisti-cated physical connection between the complex potentialand the τ -function further would be beyond the scope ofthis work, so we leave this as what we believe to be aninteresting comment.Terms of the third order of smallness give an equationfor β . Naturally, the calculations are more complicatedfor higher orders of the perturbation series, and are lessimmediately enlightening, so we relegate these to the ap-pendix. However, we stress that higher-order correctionsto the wave motion can be naturally calculated in anextended version of this theory.
V. PARTICLE MOTION IN PERIODIC WAVES
Ignoring the constant background flow, from a fixedpoint Z a fluid particle will move to the point Z + dZ (cid:48) ina time dT under the influence of the perturbation β . Inthis section, we will drop the subscript without confusion,since we are only working to first order in ε, in the KdVregime.The rate of change of the particle’s position will begiven by the differential equation dZ (cid:48) dT = β, (23)where, again, the bar denotes complex conjugation.A periodic wave will have the form β = β ( mZ − nT ) . Integrating with respect to T gives Z (cid:48) = γ ( mZ − nT ) n . (24)As before, we can make the lowest order approximation γ ( mZ − nT ) (cid:39) γ ( mX − nT ) − imY β ( mX − nT ) , so that the new position of the particle is given by thehorizontal and vertical coordinates X (cid:48) = 1 n γ ( mX − nT ) + C ,Y (cid:48) = mn Y { β ( mX − nT ) + C } , where C and C are constants of integration (when see-ing that a term of the form C Y can appear, it must beremembered that the initial position ( X, Y ) is fixed).To be definite, we will consider a particular periodicsolution of the complex KdV equation. From (16), wesee that we have a periodic solution of the form β ( Z, T ) = − k m cn ( mZ + nT, k ) , (25)where k is the modulus while the frequency is n = − (2 k − m . We also have one of the form β ( Z, T ) = − m dn ( mZ + nT, k ) , (26)where n = − ( k − m , and another of the form cn + k cn dn [18]. As k → , thefirst two tend to the same sech -type solution, while thethird tends to 0. Also, because of the identitydn ( u, k ) = k (cid:48) + k cn ( u, k ) , (27)where k (cid:48) is the complementary modulus, the cn anddn -type solutions can be simply transformed into oneanother through Galilean symmetry. We will thereforeuse the term cnoidal wave to refer to both of these solu-tions.Corresponding to the dn -solution, the perturbation inthe potential is γ ( Z, T ) = − mE (am ζ, k ) , (28)where E ( u, k ) is the elliptic integral of the second kind,and ζ = mZ − nT. If we look at the cn -solution instead, γ just picks up an extra term which is just a multipleof ζ from the relation (27). Again, because Z refers toa constant position in the fluid, not a coordinate whichfollows the particle, this will not lead to qualitativelydifferent behaviour up to Galilean symmetry.The coordinates of the fluid particle are given by X (cid:48) = − m n E (am ξ, k ) + C , (29) Y (cid:48) = − m Y n { dn ( ξ, k ) + C } , (30)where ξ = mX − nT. By use of the identity E (am u, k ) = Z ( u, k ) + EuK , where K = K ( k ), E = E ( k ) are the complete ellipticintegrals of the first and second kind, respectively, and Z ( u, k ) is the Jacobi zeta function with modulus k, wecan write ξ (cid:48) − ξ (cid:48) = − m n Z ( ξ, k ) . (31)Here ξ (cid:48) represents horizontal position in the comovingframe and ξ (cid:48) is a fixed constant. The fluid motion is noweasily seen to be periodic, with identical motion at pointsseparated by a horizontal distance 2 K in the comovingframe, i.e. ξ ∼ ξ + 2 K. The trajectory of any particle in the fluid must be aclosed curve (up to the constant horizontal flow). How-ever, it cannot be the usual ellipse, which is characteristicof linear flow. It is easy to see this by physical princi-ples alone. Because cnoidal waves are characterised bylong, relatively flat regions between narrow peaks, thetrajectory of a particle cannot be symmetric in two axes.Instead, it will travel along a nearly flat line, and then itsvertical velocity will increase and decrease quickly. Thismeans that its trajectory must be a curve which has abroad, flat base. The width of this curve will also in-crease with the modulus k, and be constant with respectto Y, since ξ (cid:48) is independent of Y . The height of the tra-jectory will also decrease until it is completely flat at thebottom of the channel.We give an illustration of the periodic motion of acnoidal wave in the complex KdV equation in Fig.2, tak-ing k large enough that the flat regions between peaksbecomes clearly defined. The surface has the form of acnoidal wave, while the curves on and within the fluiddisplay the trajectories of fluid particles as we descendto the bottom. The trajectories in the cnoidal wave areFIG. 2: An illustration of the paths of the particles in arightward-propagating cnoidal wave (not to scale). Thesurface elevation is described by the real KdV equation,but the motion of the particles within the fluid isdescribed by the complex KdV equation (16). Themotion of the particle at a given point is determined by(30, 31). The trajectory depends on the wave modulus,frequency, and depth in the fluid. most similar to bean curves, rather than ellipses or cir- cles, which are familiar from the linear theory. However,when k is small enough, the trajectories may be well-approximated by ellipses. The particle trajectories are inclose agreement with earlier numerical approximations[19], but in the framework of the complex KdV theory,they appear naturally with very little work.As k → , we have also K → ∞ , so the motion is nolonger periodic. Instead, we have the well-known sech -shaped soliton solution. In this case, the particle trajec-tories become parabolic arcs [20].Lastly, we will give an example calculation of the el-evation from the complex KdV equation. The simplestcase to work with is that of a first order perturbation inthe potential of the form γ = a tanh a ( Z + a T ) . (32)This leads to the implicit equation for the surface ηl (cid:39) εa sin 2 a ( h + η ) /l cos 2 a ( h + η ) /l + cosh 2 a ( X + a T ) . An initial measurement taken at T = 0 will have a max-imum of η located at X = 0 , roughly given by the tran-scendental equation η l (cid:39) εa tan (cid:18) a h + η l (cid:19) . When h and l are known and the dimensionless param-eter a is fixed, this allows an approximation of η . As-suming that h + η is much smaller than l, we can taketan θ ∼ θ, and estimate the maximal elevation as η (cid:39) a hε, (33)which agrees with the result (21). It follows that η canbe put in terms of η as η (cid:39) η sech a ( X + a T ) (34)up to first order. This characterises the form of the freesurface traced out by a solitary wave. This was firstderived from the tanh-potential (32) by McCowan [20],but this connection is more quickly and easily obtainedby the simple relation (21), given a solution to (16) or(19). VI. CONCLUSIONS
In this paper, we derived the complex KdV equationfor shallow water waves. We showed that it describes asmall perturbation of the complex velocity around steadyflow, for the slowly-varying propagation of long waves ina shallow, incompressible, inviscid fluid. An importantconclusion of our derivation is that if the complex KdVequation describes a complex velocity for a shallow fluid,then the elevation must be given by a real solution ofthe KdV equation. In other words, the theory of the realKdV equation is just a particular case of what we havederived here.We reproduced the periodic waves in terms of ellipticfunctions and the well known result for the soliton so-lution. However, the complex KdV equation allowed usto calculate not only the surface wave profile, but alsothe trajectories of particles within the fluid, and showthat particles in cnoidal waves move along bean curve-like trajectories. Our results were in close agreement withnumerical work based on the real KdV equation [19], buthad the bonus of being a natural and easy consequence ofthe theoretical framework. This gives even more evidencefor the validity of the link between the two approachesestablished in the present work.Having complete information about the dynamics of allfluid particles, not only at the surface but also through-out the whole body of fluid, may make it possible tosolve more complicated problems. For example, this mayinclude the class of problems related to internal waveswhen the fluid is stratified, or may allow the techniquesof complex analysis [21] to be applied, such as confor-mal transformations, which may help to solve problemsinvolving special bottom profiles or underwater barriersand obstacles. For simplicity, we restricted ourselves toonly one type of boundary condition at the bottom, withzero friction. We have no doubt this can be generalisedto other conditions that will result in more general solu-tions. Generalisation to higher-order corrections is alsostraightforward in principle. The choices here have nolimits.We should also mention that the complex KdV equa-tion has a greater variety of solutions than the real one.This includes regular [22–24], blow-up [25, 26] or com-plexiton [27] solutions. This expands the range of phe- nomena that the complex KdV equation may describe.On the other hand, we stress that exact solutions, even ifthey are highly involved, can be found using well knowntechniques such as the Darboux transformation.Applications of the complex KdV equation are not lim-ited to water waves. This equation can be also used inthe seemingly unrelated problem of unidirectional crystalgrowth [28]. Last but not least, the theory of shallow wa-ter waves is spreading into the optical domain [29] whichcould be another area of future application.
Appendix
For completeness, we give the results of the calcula-tions for second order perturbations. The second orderterms β in a shallow fluid satisfy the equation ∂β ∂T = β ∂β ∂Z + β ∂ β ∂Z − ∂β ∂Z ∂ β ∂Z ++ 19 ∂ β ∂Z − ∂ ( β β ) ∂Z . (35)where β = β ( Z, T ) is a solution of (16). Having β in explicit form provides the possibility for higher-orderextensions of the present theory. That is, working tohigher order in perturbation theory will result in equa-tions which include nonlinear and dispersive effects forshallow fluids, accurate to higher powers of the small pa-rameter ε = ( h/l ) . We note here that taking termswhich are of higher order in h/l will never result in atheory which is valid for anything other than shallow flu-ids, which are defined by h/l not being small in the firstplace. [1] A. Toffoli, E. M. Bitner-Gregersen, Types of ocean sur-face waves, wave classification, In: Encyclopedia of Mar-itime and Offshore Engineering, Basic Hydrodynamicsand Wave Dynamics (John Wiley & Sons, 2017).[2] A. Constantin, Nonlinear water waves: introduction andoverview, Phil. Trans. R. Soc. A , (2017).[3] Yu. V. Sedletskii, The fourth-order nonlinear Schr¨odingerequation for the envelope of Stokes waves on the surfaceof a finite-depth fluid J. Exp. Theor. Phys., , 180 – 193(2003).[4] A. V. Slunyaev, A high-order nonlinear envelope equationfor gravity waves in finite-depth water, J. Exp. Theor.Phys., , 926 – 941 (2005).[5] A. Osborne, Nonlinear ocean waves and the inverse scat-tering transform, (Elsevier, Amsterdam, 2010).[6] J. V. Boussinesq, Essai sur la the´orie des eaux courantes,(Imprimerie Nationale, Paris 1877).[7] D. J. Korteweg & G. De Vries, On the change of formof long waves advancing in a rectangular canal, and on anew type of long stationary waves, Phil. Mag. , 422 –443 (1895).[8] R. Grimshaw, Zihua Liu, Nonlinear periodic and solitary water waves on currents in shallow water, Studies in Ap-plied Math., , 60 – 77 (2017).[9] K. B. Dysthe, Note on a modification to the nonlinearSchr¨odinger equation for application to deep water waves,Proc. Royal Soc. A: Math., Phys, Eng. Sciences, , 105– 114 (1979).[10] K. Trulsen, K. B. Dysthe, A modified nonlinearSchr¨odinger equation for broader bandwidth gravitywaves on deep water, Wave Motion , 281 – 289 (1996).[11] T. R. Marchant, High-order interaction of solitary waveson shallow water, Stud. Appl. Math., , 1 – 17 (2002).[12] L. M. Milne-Thomson, Theoretical Hydrodynamics,Fifth Edition, (Dover, New York, 1968).[13] T. Levi-Civit`a, Sulle onde progressive di tipo perma-nente, Rend. Acc. Lincei , 777 – 790 (1907).[14] D. Levi, Levi-Civit`a theory for irrotational waver wavesin a one-dimensional channel and the complex Korteweg-de Vries equation, Theor. Math. Phys. , 705 – 709(1994).[15] D. Levi & M. Sanielevici, Irrotational water waves andthe complex Korteweg-de Vries equation, Physica D ,510 – 514 (1996). [16] M. Toda, Nonlinear waves and solitons, (KTK ScientificPublishers, Tokyo, 1989).[17] R. Hirota, The direct method in soliton theory, (Cam-bridge University Press, Cambridge, 2004).[18] A. Khare, A. Saxena, Linear superposition for a classof nonlinear equations, Phys. Lett. A , 2761 – 2765(2013).[19] H. Borluk, H. Kalisch, Particle dynamics in the KdVapproximation, Wave Motion , 691 – 709 (2016).[20] J. M. McCowan, On the solitary wave, Phil. Mag. Series5, , Issue 194, 45 – 58 (1891)[21] N. E. Joukowsky, ¨Uber die Konturen der Tragfl¨achen derDrachenflieger, Zeitschrift f¨ur Flugtechnique und Motor-luftschiffahrt, , 281 – 284 (1910).[22] Yi Zhang, Yi-neng Lv, Ling-ya Ye, Hai-qiong Zhao, Theexact solutions to the complex KdV equation, Phys. Lett.A, , 465 – 472 (2007).[23] Juan-Ming Yuan, Jiahong Wu, The complex KdV equa-tion with or without dissipation, Discrete and ContinuousDynamical Systems, Ser. B, , 489 – 512 (2005).[24] J. R. Miller, Stability properties of solitary waves in acomplex modified KdV system, Mathematics and Com-puters in Simulation (MATCOM), , 557 – 565 (2001).[25] Y. Charles Li, Simple explicit formulae for finite timeblow up solutions to the complex KdV equation, Chaos,Solitons and Fractals, , 369 – 337 (2009).[26] B. Birnir, An example of blow-up, for the complex KdVequation and existence beyond the blow-up, SIAM J.Appl. Math., , 710 – 725 (1987).[27] Hong-Li An, Yong Chen, Numerical complexiton solu-tions for the complex KdV equation by the homotopyperturbation method, Appl. Math. and Computation, , 125 – 133 (2008).[28] M. Kerszberg, A simple model for unidirectional crystalgrowth, Phys. Lett. A , 241 – 244 (1984).[29] S. Wabnitz, C. Finot, J. Fatome, G. Millot, Shallow waterrogue wavetrains in nonlinear optical fibers, Phys. Lett.A,337