A Steady-State Trio for Bretherton Equation

Z. Naturforsch. 2016; 71(12)a: 1099–1104

Zhao Niu, Zeng Liu and Jifeng Cui* A Steady-state Trio for Bretherton Equation

DOI 10.1515/zna-2016-0302 solution, and Darwish and Ramady [4] through algebraic Received August 15, 2016; accepted September 8, 2016; previously method to obtain travelling wave solutions. Recently, a published online October 8, 2016 generalised Bretherton equation

242 Abstract: To investigate if steady-state resonant solution ∂∂ψψ∂ ψ mn ++αβ+=δψ γψ , (3) ∂∂24∂ 2 exist for any system of weakly interacting waves in a dis- txx persive medium, a trio is considered in the Bretherton equation based on the homotopy analysis method (HAM). where α, β, δ, γ, m, and n are nonzero constants, was pro- Time-independent spectrum was found when all compo- posed. Romeiras [5] used both truncated Painleve expan- nents were travelling in the same direction. Within the sion method and an algebraic method to investigate the trio, the amplitude of longer component is larger than that existence of periodic and solitary travelling waves solu- of shorter one. As the difference of wave number between tions, Kudryashov et al. [6] applied Backlund transforma- components in trio increases or the nonlinearity of whole tion to search several new families of simply periodic and system increases, the amplitudes of all components tends elliptic solutions, Esfahani [7] used trigonometric func- to increase simultaneously. These findings are helpful tion method and He’s semi-inverse method to find trave- to enrich and deepen our understanding about resonant ling wave solutions, and Akbar et al. [8] applied improved solutions in any dispersive medium, especially for a two- (G′/G)-expansion method to construct travelling wave dimensional scenario. solutions. Note the original Bretherton equation (1) was pro- Keywords: Bertherton Equation; Homotopy Analysis posed to investigate the mathematical basis for resonance: Method; Steady-State Resonant Solutions. to clarify some aspects of the mechanism whereby energy is transferred between different wave numbers [1]. When resonance condition is satisfied, the perturbation results 1 Introduction contain secular terms and the perturbation theory breaks down due to singularities in the transfer functions [9], so The Bretherton equation most researchers considered the resonance from the point

242 of view of initial/boundary-value problems (the spectrum ∂∂ψψ∂ ψ 2 +++=ψψ, (1) changes with time during the wave evolution). ∂∂tx24∂x2 By means of the homotopy analysis method (HAM) [10–12], an analytic approximation method for highly was introduced by Bretherton [1] as a model of a disper- nonlinear problems, the multiple solutions of steady-state sive wave systems to investigate the resonant nonlinear resonant waves were obtained by Liao [13] from the point interaction between three linear modes. Later, a modified of view of boundary-value problems (the spectrum keeps Bretherton equation unchanged with time during the wave evolution) in deep 242 ∂∂ψψ∂ ψ 3 water. The same approach was extended further by Xu +++=ψψ, (2) ∂∂24∂ 2 txx et al. [14] to special quartet in finite water depth, by Liu and Liao [15] to general resonant sets in deep water, and was studied by Kudryashov [2] and Berloff and Howard [3] by Liao et al. [16] to nearly resonant waves. Especially, through truncated Painleve expansion to search periodic the multiple steady-state resonant waves were observed experimentally (Liu et al. [17]) in a basin at State Key Labo- *Corresponding author: Jifeng Cui, College of Science, Inner ratory of Ocean Engineering, with excellent agreement to Mongolia University of Technology, Hohhot 010051, China, their theoretical predictions given by means of the HAM. E-mail: [email protected] In this article, the object is to investigate the exist- Zhao Niu: CCCC Second Harbor Engineering Company LTD, ence of steady-state resonant solutions for the Brether- Wuhan 430040, China Zeng Liu: School of Naval Architecture and Ocean Engineering, ton equation (1). The phenomenon of resonance happens Huazhong University of Science and Technology, Wuhan 430074, for any system with weakly interacting waves in a disper- China sive medium, whereas steady-state resonant solutions 1100 Z. Niu et al.: A Steady-State Trio for Bretherton Equation

∂∂22ψψ∂∂24ψψ has only been considered for water wave system up to L[]ψω=+222ωω ++ω k4 112222∂∂ξξ 1 4 now. So taking a trio as an example, we further applied ∂∂ξξ1212 ∂ξ1 the HAM to the Bretherton equation (1) to check if the ∂∂44ψψ∂∂44ψψ ++46kk32kk23++4kk k4 steady-state resonant sets could be find for other nonlin- 12 3212 2312 2 4 ∂∂ξξ12 ∂∂ξξ12 ∂∂ξξ12 ∂ξ2 ear system. ∂∂22ψψ∂2 ψ ++kk222,kk++ψ (7) 112222∂∂ξξ ∂∂ξξ1212

2 Mathematical Formulae L [cos()mnξξ12+=]cλξmn,1os()mn+ ξ2 , (8)

2.1 Solution Procedure where λω=−1(mk ++nk )(24mk +−nk )(mn+ ω )2 Consider the nonlinear interaction of a steady-state mn,121212 =−ωω22()mn+ ω (9) periodic oscillation. Write mn,12 ξ =−kx σ ti, = 1, 2, ii i (4) L is an eigenvalue of with the property λ1,0 ≡ λ0,1 ≡ 0. When the linear resonance condition where ki is the wave number and σi denotes actual wave frequency. Due to the nonlinearity, the actual wave fre- mk12+=nk km01, ωω+=n 20ω (10) quency σ (k , A ), which is larger than the linear fre- i i m,n is satisfied by a free component with frequency quency ω (k ), depends on both the wave number k and i i i 24 ω00=−1,kk+ 0 it holds ωm,n = ω0 = mω1 + nω2, so λm,n ≡ 0. amplitude Am,n. And for Bretherton equation (1), the 24 For simplicity, a trio is considered in this article and the linear frequency ωii=−1.kk+ i The “steady-state resonant solutions” linear resonance condition reads means amplitude Am,n and actual wave −= ωω−=ω frequency σi are constants (time independent) when kk12k01, 20, (11) resonance condition is satisfied. So the solution can be so the eigenvalue holds λ ≡ λ ≡ λ ≡ 0. Based on the expressed in the form 1,0 0,1 1,−1 property of auxiliary linear operator L, we define the +∞ +∞ initial guess as shown in (a11). The constants A in (5) φ ξξ=+ξξ m,n (,12 )c∑∑Ammn,1os()n 2 . (5) mn=−∞=−∞ can now be obtained order by order in the framework of HAM. For details, refer to Appendix.

The original initial/boundary-value problem governed by (1) can now be transformed into a boundary-value 2.2 Results Verification one by means of the new variables ξ1 and ξ2. In the new coordinate system (ξ , ξ ), the governing equation is 1 2 Let us consider the case of ε = σ /ω = 1.01 when k = 1.5. The defined in (6), where N is a nonlinear operator defined i i 1 corresponding values of k = 1.03764 and k = 0.4623651 on ψ. 2 0 are determined by the resonant condition (11). Figure 1 ∂∂22ψψ∂∂24ψψ shows the averaged residual squares at different orders N []ψσ=+222σσ ++σ k4 112222∂∂ξξ 1 4 ∂∂ξξ1212 ∂ξ1 of approximation defined in (12),

4444 2 ∂∂ψψ∂∂ψψ − ++46kk32kk23++4kk k4 ππ2 55m 1 π 12∂∂ξξ3212∂∂ξξ2312∂∂ξξ 2 ∂ξ4 Π = Rm, n , (12) 12 12 12 2 mi∑∑ ∑  25 mn==00i = 0 55 222  22∂∂ψψ∂ ψ 2 ++kk20kk++ψψ−=, (6) 11∂∂ξξ2222∂∂ξξ 1212 which suggest us the optimal convergence-control parameter c = −1.4. Using this optimal convergence-control The steady-state resonant solutions for periodic oscilla- 0 parameter, the residual squares of the governing equation tions in (1) can be obtained in a similar way as Liao [13] indeed decreases quickly as the order of approximation did, so the approach is briefly described below. increases, as shown in Figure 2. Based on the linear part of the governing equation (6), we choose the following auxiliary linear operator (7), which has the property 1 It is assumed that k2 > k0. Z. Niu et al.: A Steady-State Trio for Bretherton Equation 1101

Π_2 Table 1: [m, m] Homotopy-Pade approximation of A1,0, A0,1, and A1,−1 –3 _3 Π with k = 1.5 and ε = 1.01. Π_4 1 Π_5 –4 n |A1,0| |A0,1| |A1,−1| 1 0.026909 0.044490 0.049736 –5 2 0.027388 0.045053 0.050313 3 0.027419 0.045093 0.050353

m 4 0.027421 0.045096 0.050356 Π

10 –6 5 0.027421 0.045096 0.050356 g lo

–7 until the results of Homotopy-Pade approximation agree to four significant figures. –8

–9 –3 –2.5 –2 –1.5 –1 –0.5 0 c0 3 Results Analysis

Π Figure 1: Log10 m versus the convergence-control parameter c0 First of all, the resonance condition (10) is analyzed for with k = 1.5 and ε = 1.01. Dash-dot-dot line: 2nd-order approxima- 1 six resonant components, with the corresponding reso- tion; Dash-dot line: 3rd-order approximation; Dashed line: 4th-order nance curves shown in Figure 3. Note that the other five approximation; and Solid line: 5th-order approximation. resonant curves overlap with the one we defined for the

trio in (11). This means when the wave number k1 changes along the resonance curve of trio (11), the resonance con- –4 dition (10) may be exactly or nearly satisfied by other quartet, quintet, sextet, or octet. Remember Liu and Liao –5 [15] found that the near-resonant components as a whole –6 contain more and more wave energy as the nonlinearity

–7 of the resonant wave system increases. So for steady-state trio (11), other high-order exact or near resonant compo- –8 nents may be involved into the resonance and the total m

Π energy would be distributed among more components. 10 –9 g lo –10 k =k -k 5 0 1 2 –11 k0=k 1-2k2

k0=2k1-k2 –12 k0=2k1-2k2 k =2k -3k 4 0 1 2 –13 k0=3k1-4k2

–14 024681012 m 3

Figure 2: Log10Πm versus the approximation order m with k1 = 1.5 2 k and ε = 1.01. 2

To increase the convergent rate of the series solution, 1 we apply the Homotopy-Pade approximation and the corresponding [n, n] approximation of A1,0, A0,1, and A1,−1 are shown in Table 1. It is found from Table 1 that as the approx- 0 11.5 22.5 33.5 44.5 5 imation order n increases, the values of [n, n] Homotopy- k1 Pade approximation of A1,0, A0,1, and A1,−1 gradually tend to constants. Hereinafter all the following calculations stop Figure 3: Linear resonance curves satisfying resonance condition (10). 1102 Z. Niu et al.: A Steady-State Trio for Bretherton Equation

Figure 4 shows the amplitude of two primary com- a A 0.08 1,0 A0,1 ponents A1,0 and A0,1 and the second-order resonant com- A1,–1 ponent A1,−1 for σi/ωi = 1.0001, 1.001, 1.01, respectively. In all three cases, the amplitudes of three components keep 0.06 increasing as k1 changes along the resonance curve of trio

(11). Besides, the amplitude of resonant component A1,−1 is bigger than the amplitudes of second primary component A , which is then bigger than the amplitude of first m, n 0.04

0,1 A primary component A0,1. And the difference among amplitudes of three components is enlarged for bigger values of k1. So, when components of a trio traveling in the same 0.02 direction, the amplitude of longer component (with a smaller wave number) is larger than the amplitude of shorter component (with a bigger wave number). 0 The amplitudes of other high-order components are 1.522.533.544.5 k1 shown in Tables 2–4 for σ /ω = 1.0001, 1.001, 1.01, respec- i i A b 1,0 tively. Within each table, the values of |A |, |A |, |A |, A 1,−2 2,−1 2,−2 0.1 0,1 A1,–1 |A2,−3|, and |A3,−4| increase for bigger value of k1, which means that the amplitudes of all components increase simultaneously when the difference of wave number 0.08 between long and short components in a resonant trio σ ω increases. Besides, as i/ i increases from 1.0001 to 1.01, 0.06 n

the amplitudes of all five high-order components keeps m, increasing, too. So for a steady-state trio, the near-reso- A nant components do become increasingly important as 0.04 the nonlinearity of whole system increases.

0.02

0 4 Conclusions and Discussions 1.41.6 1.822.22.4 2.6 k1 c A In this article, we considered a trio in Bretherton equation 0.1 1,0 A0,1 to check if steady-state resonant solution exist not only for A1,–1 water wave system, but for any other system with weakly interacting waves in a dispersive medium. For all compo- 0.08 nents travelling in the same direction, time-independent spectrum (with constant amplitude and actual wave fre- 0.06

quency) is obtained for the first time in a resonant set. n m, Within the trio, it is found that the amplitude of longer A component is larger than the amplitude of shorter compo- 0.04 nent. When the difference of wave number between long and short components in a trio increases, the amplitudes 0.02 of all components increase simultaneously. Besides, the high-order near-resonant components become increas- 0 ingly important as the nonlinearity of whole system 1.4 1.44 1.48 1.52 1.56 1.61.64 increases. k1 Theoretically speaking, the number of singularities Figure 4: Wave amplitude in trio (11): (a) σ /ω = 1.0001; (b) caused by linear resonance condition become infinity i i σi/ωi = 1.001; (c) σi/ωi = 1.01. Solid line: |A1,0| (first primary compo- when components are travelling in one direction. Plenty nent); Dashed line: |A0,1| (second primary component); and Dash-dot nd of high-order resonant components would join in the line: |A1,−1| (2 -order resonant component). Z. Niu et al.: A Steady-State Trio for Bretherton Equation 1103

Table 2: Amplitudes of four high-order components in trio (11) for steady-state resonant water waves, especially when all σ /ω = 1.0001. i i wave components traveling in one direction. k |A | |A | |A | |A | 1 1,−2 2,−1 2,−2 2,−3 Acknowledgements: We are grateful to the Science 3.2 0.0001 Research Project of Inner Mongolia University of Technol- 3.4 0.0001 0.0001 ogy (Approval No. ZD201613) and the Science Research 3.6 0.0002 0.0002 Project of Huazhong University of Science & Techonology 3.8 0.0004 0.0003 4 0.0008 0.0006 0.0004 (Approved No. 0118140077 and 2006140115) for financial 4.1 0.0011 0.0009 0.0005 support. 4.2 0.0015 0.0012 0.0007 4.3 0.0021 0.0018 0.0009 0.0001 4.36 0.0027 0.0022 0.0010 0.0001 Appendix

Let ψ0(ξ1, ξ2) denote a initial guess solution of ψ(ξ1, ξ2), Table 3: Amplitudes of four high-order components in trio (11) for L an auxiliary linear operator, and q ∈ [0, 1] an embed- σ /ω = 1.001. i i ding parameter with no physical meaning, respectively.

Then construct a family of solution Ψ(ξ1, ξ2; q) in q by the k |A | |A | |A | |A | 1 1,−2 2,−1 2,−2 2,−3 zeroth-order deformation equation 1.7 0.0001 −−LNΨξ ξψξξ = Ψξ ξ 1.85 0.0001 (1 qq)[ (,12 ; )(01, 20)] cq [(12, ; q)], (a1) 2 0.0003 0.0001 where c ≠ 0 is the “convergence-control parameter" with 2.2 0.0006 0.0003 0.0002 0 2.4 0.0016 0.0008 0.0005 no physical meaning. When q = 0, we have 2.5 0.0026 0.0014 0.0007 0.0001 Ψξ(, ξψ; 0) = (,ξξ ). (a2) 2.6 0.0050 0.0028 0.0012 0.0002 12 01 2 2.61 0.0053 0.0030 0.0013 0.0003 When q = 1, we have 2.62 0.0058 0.0033 0.0013 0.0003

2.63 0.0063 0.0036 0.0014 0.0003 Ψξ(,12 ξψ; 1) = (,ξξ12 ). (a3) 2.64 0.0068 0.0040 0.0015 0.0004 2.65 0.0075 0.0044 0.0016 0.0004 So, as q increases from 0 to 1, Ψ(ξ1, ξ1; q) varies from the

initial guess ψ0 to the unknown ψ(ξ1, ξ2). Assume the L auxiliary linear operator , the initial guess ψ0(ξ1, ξ2) and the convergence-control parameter c are properly chosen Table 4: Amplitudes of five high-order components in trio (11) for 0 so that the Maclaurin series of Ψ(ξ1, ξ2; q) σi/ωi = 1.01. +∞ Ψξ(, ξψ; qq)(=+ξξ, )(ψξ, ξ ),m k1 |A1,−2| |A2,−1| |A2,−2| |A2,−3| |A3,−4| 12 01 21∑ m 2 (a4) m = 1 1.42 0.0013 0.0001 0.0005 0.0004 ∂mΨξ ξ 1.46 0.0020 0.0002 0.0004 0.0003 1 (,12 ; q) is convergent at q = 1, where ψξm(,12 ξ ).= m 1.5 0.0030 0.0004 0.0005 0.0004 m! ∂q ∂mΨξ ξ q = 0 1.54 0.0046 0.0007 0.0006 0.0006 1 (,12 ; q) ψξm(,12 ξ ).= m Based on (a3) and (a4), we have 1.58 0.0072 0.0012 0.0009 0.0010 m! ∂q q = 0 1.6 0.0094 0.0017 0.0011 0.0014 0.0001 1.61 0.0110 0.0021 0.0012 0.0017 0.0001 +∞ ψξ(, ξψ)(=+ξξ, )(ψξ, ξ ). 1.62 0.0133 0.0027 0.0014 0.0022 0.0002 12 01 21∑ m 2 (a5) m = 1 Substituting the Maclaurin series (a4) into the zeroth- order deformation equation (a1) and equating the like resonant interaction. So it is hard to even image the exist- power of q, we can derive the high-order deformation ence of steady-state solutions in two-dimensional system equation when resonance mechanism is triggered. We acknowledge LL[]ψχ=+cR []ψ , (a6) the Bretherton equation (1) is a simple system compared mm01−−mm1 with fully nonlinear water wave equation when consid- where χ1 = 0 and χm = 1 for m > 1, and Rm − 1 is determined ering the resonant interaction, whereas we, too, believe by the known approximations ψi (i = 0, 1, 2, …, m − 1). The the work in this article helps for further investigation in special solution of ψm is 1104 Z. Niu et al.: A Steady-State Trio for Bretherton Equation

∗−L 1 ψχmm=+[]cR01−−mmψ 1 , (a7) can be obtained now and further calculation for ψi (i > 1) can be carried on in a similar way. where L−1 is the inverse operator of the auxiliary linear operator L, which has the property

cos(mnξξ+ ) L−1 12 References [cos()mnξξ12+=]. (a8) λmn, [1] F. P. Bretherton, J. Fluid Mech. 20, 457 (1964). Note when the resonance condition (11) is satisfied, the [2] N. A. Kudryashov, Phys. Lett. A 155, 269 (1991). auxiliary linear operator (7) has three eigenvalues being [3] N. G. Berloff and L. N. Howard, Stud. Appl. Math. 100, 195 (1998). zero: [4] A. A. Darwish and A. Ramady, Chaos, Solitons Fract. 33, 1263 (2007). λλλ== = 1,00,1 1,−1 0, (a9) [5] F. J. Romeiras, Appl. Math. Comput. 215, 1791 (2009). [6] N. A. Kudryashov, D. I. Sinelshchikov, and M. V. Demina, Phys. which means cos(ξ ), cos(ξ ), and cos(ξ − ξ ) are all 1 2 1 2 Lett. A 375, 1074 (2011). ψ primary solutions. Thus, the common solution of m reads [7] A. Esfahan, Commun. Theor. Phys. 55, 381 (2011).

∗−1,00,1 1, 1 [8] M. A. Akbar, N. H. M. Ali, and E. M. E. Zayed, Commun. Theor. ψψ=+AAcos(ξξ)c++os()A cos(ξξ− ), (a10) mmmm12m 12 Phys. 57, 173 (2012). [9] P. A. Madsen and D. R. Fuhrman, J. Fluid Mech. 698, 304 (2012). ∗ 1,00,1 where ψm is a special solution defined in (a7), and AAmm, , [10] S. J. Liao, Proposed homotopy analysis techniques for the and A1,−1 are three constants to be determined. solution of nonlinear problem, PhD thesis, Shanghai Jiao Tong m University (1992). To start the iteration for ψm, we define the initial guess [11] S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC, Boca Raton 2003. ψξ=+AA1,00cos( )c,1 os()ξξ+−A1,−1 cos( ξ ), 00 10 20 12 (a11) [12] S. J. Liao, Homotopy Analysis Method in Nonlinear Differential Equations, Springer and Higher Education, Heidelberg 2012. 1,00,1 1,−1 where AA00, , and A0 are three constants, which are [13] S. J. Liao, Commun. Nonlinear Sci. Numer. Simul. 16, 1274 (2011). [14] D. L. Xu, Z. L. Lin, S. J. Liao, and M. Stiassnie, J. Fluid Mech. 710, determined by enforcing the coefficients of terms cos(ξ1), 379 (2012). cos(ξ2), and cos(ξ1 – ξ2) on the right-hand side of (a6) equal [15] Z. Liu and S. Liao, J. Fluid Mech. , 664 (2014). to 0, otherwise, secular terms will appear at the first-order 742 [16] S. J. Liao, D. L. Xu, and M. Stiassnie, J. Fluid Mech. 794, 175 ψ approximation 1. In this way, we solve three coupled, (2016). nonlinear algebraic equations about the unknown con- [17] Z. Liu, D. L. Xu, J. Li, T. Peng, A. Alsaedi, et al., J. Fluid Mech. 1,00,1 1,−1 stants AA00, , and A0 . The detailed expression for ψ0 763, 1 (2015).