Notes on Multi-parameters Perturbation Method. for Dispersive and Nonlinear Partial Differential. Equations

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1 International Journal of Mathematical Analysis Vol. 9, 5, no. 4, - HIKARI Ltd, Notes on Multi-parameters Perturbation Method for Dispersive and Nonlinear Partial Differential Equations Edi Cahyono Department of Mathematics FMIPA Universitas Halu Oleo Kampus Bumi Tridharma Anduonohu Kendari 9 Indonesia La Ode Ngkoimani Department of Physics FMIPA Universitas Halu Oleo Kampus Bumi Tridharma Anduonohu Kendari 9 Indonesia Marwan Ramli Department of Mathematics Syiah Kuala University Banda Aceh Indonesia Copyright 5 Edi Cahyono et al. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Waves develop amplitude change during the propagation. For the simplest case, this has been shown by bichromatic waves, waves that contain two frequencies. The waves of larger amplitudes and smaller frequency differences showed more significant amplitude deformation. The amplitude deformation can be explained from the so-called side band interactions. Perturbation method for diffusive and

2 Edi Cahyono et al. nonlinear equation is applied to show the amplitude deformation. The solution is in the form of a series expansion of the amplitude as Stokes approach for single frequency waves. Stokes approach shows that the dispersion relation involves the amplitude. However, this does not explain the amplitude deformation. Therefore, this paper discusses bichromatic wave solution of diffusive and nonlinear equation. Moreover, the focus is on a hidden parameter appears in the series expansion, namely the wave number difference. The parameter of wave number difference may affect the convergence of the series expansion. The relation among the parameters of amplitude, wave number and wave number difference for the convergence of the series is presented. Keywords: Amplitude deformation, dispersive and nonlinear equation, perturbation method, side band interactions Introduction Waves show amplitude deformation during the propagation. In the oceans, this implies the phenomenon of freak waves (rogue waves or sometimes extreme waves), waves which reveal large amplitudes at some positions for some period of time. Although the presence of freak wavers is rare, they are responsible for some ship accidents. For example the freak wave collision with the ship in the Agulhas current analyzed in []. Waseda et al. analyzed that freak waves observed during several ship accidents reported in []. Therefore, information of the presence of freak waves is important for offshore activities. The presence has been often reported in media. Nikolkina and Didenkulova [] collected and analyzed freak waves reported in media in 6. Takuji Waseda et al. [4] conducted deep water observation of freak waves in the North West Pacific Ocean. Zakharov et al. [5] conducted a study on the statistics of arising of freak waves on a surface of an ideal heavy fluid is studied. To understand better the large amplitude deformation of waves, experiments have been conducted in hydrodynamics laboratories. Waves were generated and measured at several points. The signals of the waves at all points were studied. In this case, the relation among the signals was analyzed. For the simplest case, one might expect that a signal at one point was a translation of the other. Some simple waves evolve as expected, and can be considered as a linear relation of signals discussed in [6]. However, some bichromatic waves show that the relation of the signals is much more complicated. An experiment of amplitude deformation for bichromatic waves was conducted by Stansberg in [7]. More detailed experiments and numerical simulations were reported in [8, 9,, ]. It has been observed that the amplitudes and the frequency differences of the waves have been responsible for the amplitude change. Mathematically, freak wave phenomena is often studied based on NLS equation. Slunyaev [] applied NLS to show that there exists strong correlation between

3 Notes on multi-parameters perturbation method spectrum and spectral peak of the wave groups. Onorato et al. [] applied NLS to show an angle of two wave trains which was responsible for establishing a freak wave sea. Specifically, angles between to are the most probable for establishing a freak wave. Cheng et al. [4] studied controllable rogue waves in coupled nonlinear Schrödinger equations with varying potentials and nonlinearities. Shrira and Geogjaev [5] consider the Peregrine soliton as a prototype of freak waves. However, the large amplitude deformation has also been discussed from side band interaction point of view based on dispersive and nonlinear model of KdV equation [6, 7, 8]. Although some exact solutions of KdV type are available and have been discussed in several papers such as [9,, ], these kinds of methods cannot be applied to explain the amplitude change. Especially in [6], the amplitude deformation has been explained analytically by applying perturbation method the solution expressed in a series expansion of the amplitude. Perturbation method is often used to solve partial differential equation, especially if the equation cannot be solved exactly. One may refer to [] and the references therein for perturbation method for nonlinear partial differential equations and diffusion equations. In the simplest case, the solution in a series expansion of amplitude is just Stokes approach for KdV equation, [, page 47-47]. In Stokes wave, the convergence of the series depends on the amplitude parameter and another parameter related to wave number. Because Stokes wave has only one frequency, it does not exhibit side band interaction. The side band interactions of bichromatic waves (or waves contain multi frequencies) firstly appear in the third order. These interactions contain an additional parameter that is frequency difference (or wave number difference). The parameter of wave number difference may imply the convergence of the series expansion. Stokes Approach for Solution of KdV Equation Let x, t R and u be real function of x and t. In this paper, the model is KdV equation in the form u t uxxx 6uu x () The focus is a solution of () in the form of series expansion of small parameter ε. In other word, the series expansion u f f f () is sought to satisfy (). In [, page 47-47] such series expansion has been applied to discuss Stokes approach on KdV equation. In () f is function of θ, kx t. The angular frequency is also expanded in a power series. () j

4 4 Edi Cahyono et al. This leads to the following equations that depend on the hierarchical order of ' ''' f k f, (4) ' ''' ' f k f 6kf f f ', (5) ''' ' f f f k f 6k f. (6) Equation (4) leads to the (linear) dispersion relation k, (7) and the solution f cos. (8) It is more convenient to write it in exponential notation f cos exp( i) cc.. (9) Notation cc. stands for complex conjugate. Higher order frequency for j is obtained from taking the resonant term j of the corresponding equation to vanish. Solving the corresponding equation without the resonant term(s) gives the solution of f j. For example, is ' obtained from equation (5). In this case, the only resonant term of (5) is f. Taking this resonant term to be zero implies. Solving (5) without the resonant term gives the solution f exp( i) cc. cos, (9). () k Equation (6) consists of resonant terms f and partly in 6k f f '. Observe that ' 6k f f ik exp( i ) exp( i) cc. Hence, the resonant term of (6) is ik exp( i ) i exp( i) cc. Taking these to vanish, results in k. () 4k '

5 Notes on multi-parameters perturbation method 5 The third order solution is sought from ' ''' 9 f k f ik exp i cc. which gives the solution f exp( i) cc. cos 9 9 k. () 4 7k 5k Hence, the solution of u is 9 u cos cos cos, () 4 k 5k 9 k. (4) 4k Notes: Second order term of () will dominate the first order for k which breaks the convergence of the series expansion. Similarly, if, then the third k order dominate the previous orders. In general, one may show that for n n the nth order will dominate the first order. Compared to cosine k wave, the solution () has sharper crests and flatter troughs. The exact solution of such wave is known as cnoidal wave expressed in Jacobi elliptic function. There are many discussions on Jacobi elliptic function, among others [4, 5]. The reason of the discussion on Stokes approach is to introduce the side band interaction. Series Expansion for Bichromatic Solution Although the idea is almost similar to Stokes approach, the solution of bichromatic wave cannot be expanded in a function of single variable θ. It is because the solution contains two frequencies. However, the solution should be in the form u u u u (5) The solution of u is determined in advance in the form u, (6) cos cos

6 6 Edi Cahyono et al. k x t for j,. Again, it is more convenient to write (6) j j in the form of u exp exp cc. (7) For j,, the frequencies are also expanded in power series. (8) The equations (up to third orders) that depend on the hierarchical order of are u u, (9) t xxx u u, () t xxx and u u, () t xxx () () u exp i exp i 6u x cc., () () u exp i exp i 6 u x cc.. The solution (6) or (7) satisfies (9), provided that k j j,. for, () The term 6 uu x in does not consist resonance. Hence, taking the resonant terms on to vanish results in for j,, (). Observe that exp i k exp i k k exp i k k exp i k 6u u i cc., 4 Substituting this into () via and applying (), the solution of the form i, exp i i i, exp u, exp cc., exp u is in (4)

7 Notes on multi-parameters perturbation method 7,, (5) k,, (6) k k k,, (7) k k k k k k,. (8) k k k k Observe that,,, exp i,,, exp i, exp i, exp i uu,, exp i cc. 4.,, exp i,, exp i,, exp i And,,,, k exp i,,, k exp i, k exp i, k exp i 6 6uu x i,, k k exp i cc. 4,, k k exp i,, k k exp i,, k k exp i (9) Resonant terms of () are and () k i,,, exp () k i,,, exp. Taking these resonant terms to vanish, it gives k j, j,, j,. for, ()

8 8 Edi Cahyono et al. The third order solution is obtained from non resonant terms of (),k exp i, k exp i,, k k exp i,, k k exp i,, k k exp i k k exp i 6 u t u xxx i cc. () 4,, This yields a solution in the form, i, exp i i i i i, exp, exp u, exp cc., exp, exp 9 k () 7k 9, () 7k, 9 5k k, 4 (), () 9 5k 4, (4), () k () k k k, (5), () k () k k k, (6), () k () k k k, (7), () k () k k k, (8) Summerizing this, the solution of u up to third order is given in the form

9 Notes on multi-parameters perturbation method 9 and u cos cos, cos cos, cos,, cos,cos, cos, cos, cos 6k j k j, j,,,, cos cos, (9). (4) 4 Side Band Interactions The focus of side band interactions is on the coefficients, and Applying () and simple algebraic calculation k k, kk k, (4) and k k,. k k k (4) Observe that as k k tends to zero, then, and, tend to cos and infinity. Hence, the third order side band terms, cos,,. of the solution (9) also tend to infinity. This breaks the convergence of (5). Especially for, the third order side band terms of (9) will dominate the first order terms. One may continue to the higher orders, and he or she will obtain that the coefficients of the (n+)th order side band n terms are in the order of. Again, for, the (n+)th order side band terms of (9) will dominate the first order terms. 5 Conclusion Some waves, including the simplest bichromatic waves, develop amplitude change during the propagation. The waves of larger amplitudes and smaller frequency differences showed more significant amplitude deformation. This can be explained by applying perturbation method for KdV equation, the solution is in the form of a series expansion of the amplitude,.

10 Edi Cahyono et al. Applying the series expansion for a single frequency wave to obtain the solution is known as Stokes approach. Applying stokes to KdV equation, one needs to consider parameter of wave number k, the presence of k may break the convergence of the series, especially when k. For bichromatic waves, there exists another parameter namely wave number difference,. This parameter appears firstly in the third order of the series. In fact, is in the (n+)th order of the series, for n,,. This parameter may break the convergence of the series as k does for Stokes wave. Especially for, the third order dominate the first order and so do the (n+)th order side band terms. For the case the proportion of the wave number difference and the amplitude breaks the convergence of the series, another technique should be sought to explain the amplitude change. References [] I. V. Lavrenov, The wave energy concentration at the Agulhas current of South Africa, Natural Hazards, 7 (998), [] Takuji Waseda, Hitoshi Tamura, Takeshi Kinoshita, Freakish sea index and sea states during ship accidents, Journal of Marine Science and Technology, 7 (), [] I. Nikolkina and I. Didenkulova, Catalogue of rogue waves reported in media in 6, Natural Hazards, 6 (), [4] Takuji Waseda, Masato Sinchi, Keiji Kiyomatsu, Tomoya Nishida, Shunsuke Takahashi, Sho Asaumi, Yoshimi Kawai, Hitoshi Tamura and Yasumasa Miyazawa, Deep water observations of extreme waves with moored and free GPS buoys, Ocean Dynamics, 64 (4), [5] V.E. Zakharov, A.I. Dyachenko, and R.V. Shamin, How probability for freak wave formation can be found, The European Physical Journal Special Topics, 85 (), [6] E. Cahyono, On the linear relation of signals, WSEAS Transactions on Signal Processing, (4),

11 Notes on multi-parameters perturbation method [7] C. T. Stansberg. On the nonlinear behaviour of ocean wave groups In: B. L. Edge and J. M. Hemsley (eds), Ocean Wave Measurement and Analysis, Reston, VA, USA: American Society of Civil Engineers (ASCE) (997), 7-4. [8] J. Westhuis, E. van Groesen and R. H. M. Huijsmans, Long time evolution of unstable bichromatic waves. Proc. 5th IWWW & FB. Caesarea, Israel,, pp [9] J. Westhuis and R. H. M. Huijsmans, Unstable Bichromatic Wavegroups, MARIN Report No Wageningen, the Netherlands,. [] J. Westhuis, E. van Groesen and R. H. M. Huijsmans, Experiments and Numerics of Bichromatic Wave Groups. Journal of Waterway, Port, Coastal & Ocean Engineering, 7 (), no. 6, [] J. Westhuis, The Numerical Simulation of Nonlinear Waves in a Hydrodynamic Model Test Basin, PhD thesis, Faculty of Mathematical Sciences, University of Twente, Enschede, The Netherlands,. [] A. Slunyaev, Freak wave events and the wave phase coherence, The European Physical Journal Special Topics, 85 (), [] M. Onorato, D. Proment, and A. Toffoli, Freak waves in crossing seas, The European Physical Journal Special Topics, 85 (), [4] Xueping Cheng, Jianyong Wang, and Jinyu Li, Controllable rogue waves in coupled nonlinear Schrödinger equations with varying potentials and nonlinearities, Nonlinear Dynamics, 77 (4), [5] Victor I. Shrira and Vladimir V. Geogjaev, What makes the Peregrine soliton so special as a prototype of freak waves?, Journal of Engineering Mathematics, 67 (), -. [6] E. Cahyono, Analytical Wave Codes for Predicting Surface Waves in a Laboratory Basin, PhD thesis, Faculty of Mathematical Sciences, University of Twente, Enschede, the Netherlands,. [7] E. van Groesen, E. Cahyono, and A. Suryanto, Uni-directional model for narrow- and broadband pulse propagation in second order nonlinear media,

12 Edi Cahyono et al. Optical and Quantum Electronics, 4 (), [8] M. Ramli, S. Munzir, T. Khairuman and V. Halfiani, Amplitude increasing formula of bichromatic wave propagation based on fifth order side band solution of Korteweg de Vries equation, Far East Journal of Mathematical Sciences, 9 (4), no., [9] Q. Feng and B. Zheng, Traveling wave solutions for the fifth-order kdv equation and the BBM equation by (G /G) -expansion method, WSEAS Transactions on Mathematics, 9 (), no., -. [] C. Wen and B. Zheng, A New Fractional Sub-equation Method for Fractional Partial Differential Equations, WSEAS Transactions on Mathematics, (), no. 5, [] B. Zheng, The Riccati sub-ode method for fractional differential-difference equations, WSEAS Transactions on Mathematics, (4), 9 -. [] C. M. Bender and S. Boettcher, A new perturbative approach to nonlinear partial differential equations, Journal of Mathematical Physics, (99), [] G. B. Whitham, Linear and Nonlinear Waves, New York: J. Wiley 974. [4] H. Wang and C. Xiang, Jacobi elliptic function solutions for the modified Korteweg de Vries equation, Journal of King Saud University Science, 5 (), [5] A.H. Bhrawy, M.A. Abdelkawy, and Anjan Biswas, Cnoidal and snoidal wave solutions to coupled nonlinear wave equations by the extended Jacobi s elliptic function method, Communications in Nonlinear Science and Numerical Simulation, 8 (), Received: June 9, 5; Published: September, 5