Signatures of chaotic dynamics in wave motion according to the extended KdV equation
aa r X i v : . [ n li n . C D ] J a n Signatures of chaotic dynamics in wave motionaccording to the extended KdV equation
Anna Karczewska and Piotr Rozmej Faculty of Mathematics, Computer Science and Econometrics,University of Zielona G´ora, 65-246 Zielona G´ora, Poland,
[email protected] ,WWW home page: http://staff.uz.zgora.pl/akarczew Faculty of Physics and Astronomy, University of Zielona G´ora,65-246 Zielona G´ora, Poland,
Abstract.
In this communication we test the hypothesis that for some initial con-ditions the time evolution of surface waves according to the extended KdV equa-tion (KdV2) exhibits signatures of the deterministic chaos.
Keywords:
Extended KdV equation, numerical evolution, deterministic chaos
Korteweg-de Vries equation (KdV) is the most famous nonlinear partial differ-ential equation modelling long-wave, weekly dispersive gravity waves of smallamplitude on a surface of shallow water. In scaled variables and a fixed referenceframe KdV takes the following form η t + η x + 32 αηη x + 16 βη xxx = 0 . (1)In (1), η ( x, t ) represents the wave profile, α = A/H , β = ( H/L ) , where A iswave’s amplitude, L - it’s average wavelength and H is water depth. Indexesindicate partial derivatives. KdV equation (1) is derived from Euler equations inperturbation approach, under assumption that parameters α ≈ β are small. It hasseveral analytic solutions: single soliton, multi-soliton, and periodic (cnoidal)ones. KdV is integrable. It also has the unique property of the infinite numberof integral invariants, see, e.g., [1].In 1990, Marchant and Smyth, extending perturbation approach to secondorder in small parameters α, β , derived the extended KdV equation (KdV2) [2] η t + η x + 32 αηη x + 16 βη xxx − α η η x (2) + αβ (cid:18) η x η xx + 512 ηη xxx (cid:19) + 19360 β η xxxxx = 0 . Anna Karczewska and Piotr Rozmej
Studying this equation, we showed that KdV2 has only one exact invariant, rep-resenting mass (volume) of displaced fluid. The other integral invariants are onlyapproximate, with deviations of the order of O ( α ) [3]. Next, we showed thatKdV2 equation, despite being non-integrable, has exact single soliton and pe-riodic solutions in the same form as KdV equation, but with slightly differentcoefficients [4,5,6].An exact single soliton solution of KdV2 has the same form as the KdVsoliton, that is, η ( x, t ) = A sech [ B ( x − vt )] . (3)However, coefficients A, B, v are uniquely determined by the coefficients of theKdV2 equation [4]. This property is entirely different from KdV solitons’ prop-erties, for which there is a one-parameter family of possible solutions. ThereforeKdV equation admits multi-soliton solutions, whereas the KdV2 equation doesnot. Since the equation (2) is second-order in small parameters α, β (assumingthat α ≈ β ), it should be a good approximation for much larger values of smallparameters than KdV. α =0.2424, β =0.3 η ( x ,t ) x 2k*dt(2k+1)*dt Fig. 1.
Snapshots of time evolution according to the KdV2 equation (2). Subsequent profilescorresponding to times t n = n ∗ , with n = 0 , , . . . , , are shifted up by one unit to avoidoverlaps. Initial condition in the form of the Gaussian of the same volume as the soliton, the sameamplitude and velocity. However, in the KdV2 case, for some initial conditions we obtained unex-pected results. In particular, when the initial condition was chosen in the formof depression (instead of ‘usual’ elevation), the time evolution calculated ac- ignatures of chaotic dynamics 3 -1 0 1 2 3 4 5 6 7-100 0 100 200 300 400 500 α =0.2424, β =0.3 η ( x ,t ) x 2k*dt(2k+1)*dt Fig. 2.
The same as in Fig. 1 but for initial condition in the form of the Gaussian which volumeis three times greater, with the same amplitude and velocity but the inverted displacement. cording to the KdV2 equation (2) appeared to be entirely different than thatwhen initial conditions are not much different from the exact soliton. We firstencountered these facts in [7]. We show an example of this behavior in Figs. 1and 2.At the first glance, time evolution presented in Fig. 2 (bottom) looks chaotic .In the next section, we try to verify this observation quantitatively.
It is well known [8] that the deterministic chaos occurs when trajectories ofthe system’s motion, starting from very close initial conditions, diverge expo-nentially with time. How can we define the distance between the trajectories?Let η ( x, t ) and η ( x, t ) denote two different trajectories. Define the followingmeasures of the distance between them: M ( t ) = Z ∞−∞ | η ( x, t ) − η ( x, t ) | dx (4) M ( t ) = Z ∞−∞ [ η ( x, t ) − η ( x, t )] dx. (5) Anna Karczewska and Piotr Rozmej
In numerical simulations, we utilize the finite difference method (FDM) de-scribed in detail in [4,7] with N of the order 5000-10000. Integrals are ap-proximated by sums, so M ( t ) ≈ P Ni =1 | η ( x i , t ) − η ( x i , t ) | dx and M ( t ) ≈ P Ni =1 [ η ( x i , t ) − η ( x i , t )] dx . In calculations, we use the periodic boundaryconditions. Therefore, the interval x ∈ [ x , x ] has to be much wider thanthe region where the surface wave is localized. In numerical calculations, pre-sented in [7], when the initial conditions were exact KdV2 solitons, the invariant I = R ∞−∞ η ( x, t ) dx was conserved with the precision − − − . In cal-culations shown in this note, since initial conditions are much different, I isconserved up to − . (t) SM (t) SM (t) GM (t) G Fig. 3.
Distances between trajectories, which start from almost identical initial conditions, asfunctions of time. Lines marked by S correspond to initial conditions in the form of exact KdV2soliton (3), whereas those marked by G correspond to initical conditions in the form of Gaussians.Open symbols indicate M measures, whereas the filled ones indicate M measures. First, we check the divergence of trajectories when the initial conditions areclose to the exact KdV2 soliton. This is presented in Fig. 3 with lines markedby S. The lines marked by G in Fig. 3 represent the divergence of trajectorieswhen η ( x, t = 0) was the Gaussian distortion having the same amplitude A as the KdV2 soliton, with the width σ providing the same volume. Then, for η ( x, t = 0) we chose the similar Gaussian form but with parameters slightlychanged, namely with A = A (1 + ε ) and σ = σ / (1 + ε ) , which ensuresthe same initial volume. In both cases, initial velocity was assumed to be equalto soliton’s velocity. Note, that profiles shown in Fig. 2 represent the evolutionof η ( x, t = 0) . Results displayed in Fig. 3 were obtained for ε = 10 − . For ε = 10 − and ε = 10 − the relative distances M ( t ) /M (0) behave almost ignatures of chaotic dynamics 5 exactly the same. It is clear that in all cases presented in Fig. 3 the distancesbetween trajectories which start from neighbor initial conditions diverge linearlywith time. For dynamical systems, it means that such conditions belong to thisphase-space region in which motion is regular. -9-8-7-6-5-4-3-2 0 50 100 150 200 250 300 l og ( M ) t Vol=1Vol=2Vol=3Vol=4 fit Vol=1fit Vol=2fit Vol=3fit Vol=4 Fig. 4.
Time dependence of measures M for ε = 10 − and different volumes of initial condi-tions. In the below-presented calculations, we studied the motion according to theKdV2 equation when initial displacements have the sign opposite from the soli-ton. For η ( x, t = 0) we chose the Gaussian distortion having the amplitude − A of the KdV2 soliton, but with the width σ ensuring multiple soliton’s vol-umes. For η ( x, t = 0) we chose the similar Gaussian form but with parametersslightly changed, namely with A = − A (1 + ε ) and σ = σ / (1 + ε ) , whichensures the same initial volume. In both cases, the initial velocity is chosen thesame as the velocity of the KdV2 soliton.Since in all cases shown below, the distances between trajectories increasedmuch faster than linearly with time, the next figures are plotted on a semiloga-rithmic scale. In Fig. 4, we show the time dependence of the distance measures M for ε = 10 − . The notation Vol= n , with n = 1 , , , denotes the initialvolume of the η ( x, t = 0) in the units of the KdV2 soliton volume. In Fig. 5 wepresent the analogous results but for M measures. In both figures we observealmost perfect exponential divergence of trajectories, even for n = 1 . Fits to theplots displayed in Fig. 4 give the following exponents: 0.00475, 0.00785, 0.0086and 0.0089 for Vol=1,2,3,4, respectively. For M measures the fitted exponents Anna Karczewska and Piotr Rozmej are: 0.00477, 0.0051, 0.00539 and 0.00561, respectively. All these exponentsare obtained by fitting in the interval t ∈ [150 : 300] . -9.5-9-8.5-8-7.5-7-6.5-6-5.5-5-4.5 0 50 100 150 200 250 300 l og ( M ) tVol=1Vol=2Vol=3Vol=4 fit Vol=1fit Vol=2fit Vol=3fit Vol=4 Fig. 5.
The same as in Fig. 4 but for M measures. The results shown above allow us to conclude that for initial conditionssubstantially different from the exact KdV2 soliton, the dynamics of motiongoverned by the KdV2 equation exhibits exponential divergence of trajectories.
References
1. Drazin, P.G., Johnson, T.S.:
Solitons: An Introduction.
Cambridge University Press, Cam-bridge 1989.2. Marchant, T.R., Smyth, N.F.:
The extended Korteweg-de Vries equation and the resonantflow of a fluid over topography.
J Fluid Mech 1990: :263-288.3. Karczewska, A., Rozmej, P., Infeld E., Rowlands G.:
Adiabatic invariants of the extendedKdV equation . Phys Lett A 2017: :270-275.4. Karczewska, A., Rozmej, P., Infeld, E.:
Shallow water soliton dynamics beyond Korteweg-deVries equation . Phys Rev E 2014: :012907.5. Infeld, I., Karczewska, A., Rowlands, G., Rozmej, P.: Exact cnoidal solutions of the extendedKdV equation . Acta Phys Pol 2018: :1191-1199.6. Rozmej, P., Karczewska, A., Infeld E.:
Superposition solutions to the extended KdV equationfor water surface waves.
Nonlinear Dynamics 2018: :1085-1093.7. Karczewska, A., Rozmej, P.: Generalized KdV-type equations versus Boussinesq’s equationsfor uneven bottom – numerical study.