A Split-Step Fourier Scheme for the Dissipative Kundu-Eckhaus Equation and its Rogue Wave Dynamics
aa r X i v : . [ n li n . C D ] A p r A Split-Step Fourier Scheme for the Dissipative Kundu-EckhausEquation and Its Rogue Wave Dynamics
Cihan Bayındır ∗ and Hazal Yurtbak Associate Professor, Engineering Faculty,˙Istanbul Technical University, 34469 Maslak, ˙Istanbul, Turkey.Adjunct Professor, Engineering Faculty, Bo˘gazi¸ci University, 34342 Bebek, ˙Istanbul, Turkey.International Collaboration Board Member,CERN, CH-1211 Geneva 23, Switzerland. Department of Civil Engineering, I¸sık University, 34485 Maslak, ˙Istanbul, Turkey
We investigate the rogue wave dynamics of the dissipative Kundu-Eckhaus equation. With thismotivation, we propose a split-step Fourier scheme for its numerical solution. After testing theaccuracy and stability of the scheme using an analytical solution as a benchmark problem, weanalyze the chaotic wave fields generated by the modulation instability within the frame of thedissipative Kundu-Eckhaus equation. We discuss the effects of various parameters on rogue waveformation probability and we also discuss the role of dissipation on occurrences of such waves.
PACS numbers: 03.65.w, 05.45.-a, 03.75.b
I. INTRODUCTION
The Eckhaus equation is a nonlinear partial differential equation which is an extended version ofthe well-known nonlinear Schr¨odinger equation (NLSE). This equation was introduced by Kundu[1] and Eckhaus [2] independently, therefore it is commonly known as Kundu-Eckhaus equation(KEE). The KEE admits many different types of analytical solutions including but not limitedto the single, dual and N-solitary waves, seed solutions and rogue wave solutions [3–6]. KEE isused to model various phenomena such as fiber optical waveforms, water waves, fluids, ion-acousticwaves just to name a few [3–6].One of the most striking features of the nonlinear systems such as the KEE is their abilityto sufficiently describe unexpectedly large waves. These waves, which are unexpected and haveheights on the order of at least two times the significant wave height in a chaotic wave field, areknown as rogue waves. Rogue waves appear in optics, hydrodynamics, plasmas and in finance[5–8].The effect of losses or gain are taken into consideration in some nonlinear models i.e. thedissipative nonlinear Schrdinger equation [9]. However, to our best knowledge, such effects arenot studied within the frame of the KEE before. With this motivation, we study the dissipativeKundu-Eckhaus equation (dKEE) in this paper. We first derive a simple analytical solution andthen use that solution as a benchmark problem to analyze the stability and accuracy of a split-stepscheme we propose for the numerical solution of the dKEE. We show that modulation instabilityleads to rogue wave formation within the frame of the dKEE. We discuss the effect of the dissipationparameter on the probability of occurrences of rogue waves. ∗ Electronic address: [email protected]
II. METHODOLOGY
The dissipative Kundu-Eckhaus equation (dKEE) can be written as i ∂U∂t + µ ∂ U∂ξ + µ | U | U + iµ U + µ | U | U − µ i (cid:16) | U | (cid:17) ξ U = 0 , (1)where t is the time and ξ is the space parameter. In this equation, the parameter µ is thedispersion constant, the parameter µ is the cubic nonlinearity constant and the parameter µ isthe quintic nonlinearity and Raman scattering constant. The parameter µ controls the dissipationor gain, depending on its sign [9]. Seeking a solution to the dKEE in the form of U ( ξ, t ) = a ( t ) e i [ kξ − Ω( t )] (2)one can obtain a simple solution as U = Ae − µ t e i (cid:20) kξ − µ k t − µ µ A e − µ t − µ A µ e − µ t + c (cid:21) (3)where A and c are constants. We use this simple exponential solution as a benchmark problem totest the stability and accuracy of the split-step Fourier scheme we implement in the next section. A. A Split-Step Fourier Method for the Numerical Solution of the dKEE
In this section we propose a split-step Fourier method (SSFM) for the numerical solution of thedKEE. As in the other spectral methods [10–12], the SSFM calculates the spatial derivatives usingFFT routines in periodic domains [13–19]. However, temporal derivatives are calculated using astepping procedure. In SSFM, the governing equation is splitted into two parts generally, namelythe linear and nonlinear part. Various order splittings are possible for the utilization of the SSFM.As a possible first order splitting, we split the nonlinear part of the dKEE as iU t = − ( µ | U | + µ | U | − iµ ( | U | ) ξ + iµ ) U (4)which can be integrated to give˜ U ( ξ, t + ∆ t ) = e i ( µ | U | + µ | U | − iµ ( | U | ) ξ + iµ )∆ t U (5)where ∆ t is the time step and U = U ( ξ, t ) is the initial condition. One can evaluate the spatialderivate in this equation using the Fourier transforms˜ U ( ξ, t + ∆ t ) = e i ( µ | U | + µ | U | − iµ F − { ikF [ | U | ] } + iµ ) ∆ t U (6)where k is the Fourier transform parameter. In here, F and F − denote the forward and inverseFourier transforms, respectively. All Fourier transforms are evaluated using efficient FFT routinesin this study. The remaining linear part of the dKEE can be written as iU t = − µ U ξξ (7)Using the Fourier series one can evaluate the linear part as U ( ξ, t + ∆ t ) = F − h e − iµ k ∆ t F [ ˜ U ( ξ, t + ∆ t )] i (8)where k is as before. Therefore, pluging Eq.(6) into Eq.(8), the complete form of the SSFM forthe numerical solution of the dKEE can be written as U ( ξ, t + ∆ t ) = F − h e − iµ k ∆ t F [ e i ( µ | U | + µ | U | − iµ F − [ ikF [ | U | ]]+ iµ )∆ t U ] i (9)Throughout this study, the number of spectral components are selected as N = 1024 and ∆ t = 10 − which does not cause any instability in the SSFM simulations. III. RESULTS AND DISCUSSIONA. Comparisons of the Analytical and Numerical Solutions of the DKEE
In this section, we provide a comparison of the analytical solution of the dKEE given by Eq.(2)and its numerical solutions obtained using the SSFM. With this purpose, in Fig. (1), we compare thereal part and absolute value of those complex valued solutions at t = 0 for A = 0 . , c = 0 , µ = 1, µ = 2, µ = 0 . , µ = 2 / -200 -150 -100 -50 0 50 100 150 200-101 U t=0 -200 -150 -100 -50 0 50 100 150 20000.51 |U| t=0 Split-Step SchemeExact Solution
FIG. 1: Comparison of the split-step vs exact solution of the dKEE at t = 0 . µ = 1 , µ = 2 , µ =0 . , µ = 2 / , A = 0 . -200 -150 -100 -50 0 50 100 150 200-101 U t=7.6 -200 -150 -100 -50 0 50 100 150 20000.51 |U| t=7.6 Split-Step SchemeExact Solution
FIG. 2: Comparison of the split-step vs exact solution of the dKEE at t = 7 . µ = 1 , µ = 2 , µ =0 . , µ = 2 / , A = 0 . -200 -150 -100 -50 0 50 100 150 200-101 U t=0 -200 -150 -100 -50 0 50 100 150 20000.51 |U| t=0 Split-Step SchemeExact Solution
FIG. 3: Comparison of the split-step vs exact solution of the dKEE at t = 0 . µ = 1 , µ = 2 , µ =1 , µ = 2 / , A = 0 . -200 -150 -100 -50 0 50 100 150 200-101 U t=7.6 -200 -150 -100 -50 0 50 100 150 20000.51 |U| t=7.6 Split-Step SchemeExact Solution
FIG. 4: Comparison of the split-step vs exact solution of the dKEE at t = 7 . µ = 1 , µ = 2 , µ =1 , µ = 2 / , A = 0 . As one can realize by checking the figure, the two solutions at the initial stage is in agreement.After time stepping is performed using the SSFM, the numerical and analytical solutions are stillin good agreement at t = 7 .
6, as depicted in Fig. (2). The effect of non-zero dissipation coefficientbecomes significant after time stepping, the waves and the envelope of the wave field, which can beobtained by using the Hilbert transforming wavefield, tends to decrease as depicted in the Fig. (2).Next, we turn our attention to the case where the dissipative effects are stronger. Changingthe dissipation coefficient µ , and selecting the same parameters as before, that is by setting A = 0 . , c = 0 , µ = 1 , µ = 2 , µ = 1 , µ = 2 /
3, we perform the numerical simulation again andplot the comparative results for t = 0 in Fig. (3) and for t = 7 . µ . The value of µ = 1 imposes a very strong dissipation in the frame of the dKEE and thesolutions decay within few dimensionless time units. B. Statistics of Rogue Waves of the DKEE and the Effect of Dissipation
Rogue waves are considered as the unexpected and high amplitude waves. They are generallydesired in fiber optical media, however their results can be catastrophic in the marine environment.There are some studies for their early detection [20]. One of the triggering mechanisms thattransforms sinusoidal wave trains into chaotic wave trains having abnormally high waves is theBenjamin-Feir instability. This instability is known as the Benjamin-Feir instability, or morecommonly as the modulation instability (MI) [21–25]. In order to discuss the effects of dissipationon the rogue wave formation probability within the frame of the dKEE, we trigger MI in ournumerical simulations. In order to trigger MI, a sinusoidal solution with a white noise is generallyused as an initial condition. Therefore, in order to create random wave fields having rogue wavecomponents, we use an initial condition for SSFM in the form of U = e imk ξ + βa (10)In here, m is a constant, k is the fundamental wave number which is equal to 2 π/L , β is MIparameter and a is a set of uniformly distributed random numbers in the interval of [ − , m and β are considered in this study, which may lead to different probabilitiesof rogue wave occurrences. -200 -150 -100 -50 0 50 100 150 200-505 U t=1.182 -200 -150 -100 -50 0 50 100 150 200024 |U| t=1.182 FIG. 5: A typical chaotic wave field generated in the frame of dKEE for m = 16 , β = 0 . , µ = 1 , µ =2 , µ = 0 , µ = 2 / In Fig. (5), we depict a typical chaotic wave field exhibiting rogue wave components generatedwithin the frame of dKEE. The parameters of computation are selected as m = 16 , β = 0 . , µ =1 , µ = 2 , µ = 0 , µ = 2 / | U | = 0 − | U | > |U| p ( | U | ) =0.1 =0.2 =0.3 =0.4 FIG. 6: Amplitude probability distribution in a chaotic wave field for m = 4 , β = 0 . , µ = 1 , µ = 2 , µ =2 / µ . In Fig. (6), we plot the amplitude probability distribution in a chaotic wave field for variousvalues of µ using m = 4 , β = 0 . , µ = 1 , µ = 2 , µ = 2 /
3. Each of the probability distributionsdepicted in Figs. (6)-(10) include approximately 10 wave components and are recorded after adimensionless adjustment time of t = 5 until to the dimensionless time of t = 10 at various timesteps. Checking this figure, one can realize that even the dissipation constant of µ = 0 . |U| p ( | U | ) =0.1 =0.2 =0.3 =0.4 FIG. 7: Amplitude probability distribution in a chaotic wave field for m = 4 , β = 0 . , µ = 1 , µ = 2 , µ =2 / µ . the parameter β on rogue wave formation probability, we depict Fig. (7) using the parameters as m = 4 , β = 0 . , µ = 1 , µ = 2 , µ = 2 /
3. It is known that a higher value of β leads to an increasein the rogue wave formation probability [5]. Comparing Fig. (6) and Fig. (7), one can realize thatthe same amount of increase in the dissipation parameter, µ , has a more dominant effect than anincrease in MI parameter β . |U| p ( | U | ) =0.1 =0.2 =0.3 =0.4 FIG. 8: Amplitude probability distribution in a chaotic wave field for m = 16 , β = 0 . , µ = 1 , µ = 2 , µ =2 / µ . |U| p ( | U | ) =0.1 =0.2 =0.3 =0.4 FIG. 9: Amplitude probability distribution in a chaotic wave field for m = 16 , β = 0 . , µ = 1 , µ = 2 , µ =2 / µ . Additionally, it is also known that an increase in m leads to an increase in the probabilityof rogue wave formation [5]. However, checking Fig. (8), it is possible to argue that the effectof dissipation constant is again more significant compared to the MI parameter m . The resultsdepicted in Fig. (8) are computed using m = 16 , β = 0 . , µ = 1 , µ = 2 , µ = 2 / β and m , we depictFig. (9) for which the parameters of computations are selected as m = 16 , β = 0 . , µ = 1 , µ =2 , µ = 2 /
3. Although an increase in both of the β and m lead to increases in the probability ofrogue wave formation, the effect of dissipation coefficient is still more significant than the combinedeffect of β and m . |U| p ( | U | ) =0.1 =0.0 FIG. 10: Amplitude probability distribution in a chaotic wave field for m = 4 , β = 0 . , µ = 1 , µ = 2 , µ =2 / µ = 0 . µ = 0. Lastly, we compare the effect of turning the dissipation parameter off. Setting µ = 0 turnsthe dKEE into KEE. As shown in Fig. (10), MI triggers generation of rogue waves in chaoticwave fields for both of the dKEE and KEE. The value of µ = 0 . IV. CONCLUSION