# A Newton-Type Approach to Approximate Travelling Wave Solutions of a Schrödinger-Benjamin-Ono System

Authors: Juan Carlos Muñoz Grajales
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

We introduce a Newton’s iterative method to approximate periodic and nonperiodic travelling wave solutions of the Schrödinger-Benjamin-Ono system derived by M. Funakoshi and M. Oikawa. We analyze numerically the influence of the model’s parameters on these solutions and illustrate the collision of two unequal-amplitude solitary waves propagating with different speeds computed by using the proposed numerical scheme.


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## Body

## 1. Introduction

This paper is concerned with the nonlinear one-dimensional system (hereafter called the Schrödinger-Benjamin-Ono system (SBO))(1)i∂tu+∂x2u=αvu,∂tv-γH∂x2v=βu2x,for x∈(0,L), t>0 with periodic spatial boundary conditions. This nonlinear dispersive system was derived by Funakoshi and Oikawa [1] to describe the motion of two fluids with different densities under capillary-gravity waves in a deep water regime. The functions u=u(x,t):R×R→C and v=v(x,t):R×R→R denote the short and long wave terms, respectively. The parameters α,β, γ are real numbers, and H denotes the Hilbert transform(2)Hfx≔p.v.1π∫fyy-xdy,where p.v.∫ stands for the integration in the principal value sense. Other physical scenarios, where system (1) with γ=0 arises, are the sonic-Langmuir wave interaction in plasma physics (Karpman [2]), the capillary-gravity interaction waves (Djordjevic and Redekopp [3], Grimshaw [4]), and water-wave interaction in a nonlinear medium (see, e.g., [5, 6]).Due to a balance between dispersive and nonlinear effects, system (1) admits the so-calledtravelling-wave solutions in the form(3)ux,t=eiwteicx-ct/2ϕx-ct,vx,t=ψx-ct,where w,c∈R and ϕ,ψ are periodic real-valued functions or smooth real-valued functions such that, for each n∈N, ϕ(n)(χ)→0 and ψ(n)(χ)→0, as |χ|→∞. In this last case, these solutions are calledsolitary waves. In the literature there are some previous analytical results about periodic and nonperiodic travelling wave solutions to the SBO system. For instance, Angulo and Montenegro [7] established the existence of even solitary wave solutions by employing the concentration compactness method (Lions [8, 9]). Existence and stability of a new family of solitary waves was established in [10] for system (1) and a coupled Schrödinger-KdV model. On the other hand, when |γ|≠1, the nonperiodic initial value problem corresponding to the SBO system has been considered by Bekiranov et al. [11], who proved well-posedness in the Sobolev space HCs(R)×HRs-1/2(R), with s≥0. In the case of |γ|=1, Pecher [12] showed local well-posedness for s>0, and Angulo et al. [13] proved global well-posedness for s=0 and |γ|≠1. However, there are only a few results known for the spatial periodic problem. For instance, assuming that |γ|≠0,1, Angulo et al. [13] showed that system (1) is locally well posed in the Sobolev space Hs(0,2π)×Hs-1/2(0,2π) for s≥1/2 and they further established existence of a smooth branch of periodic travelling wave solutions of the SBO system for γ≠0 and close to 0, and w-c2/4>2π2/L2, where L is the fundamental period.In this paper, we introduce a numerical scheme for approximating the periodic and nonperiodic travelling wave solutions of system (1), with γ≠0 whose existence was established in previous works ([7, 10, 13]). This scheme combines a Fourier spectral discretization together with a Newton-type iteration which is initialized by means of well-known exact solutions for the case γ=0. By using the spectral Crank–Nicolson numerical scheme introduced in [14] for solving system (1), we validate the approximate travelling waves computed and also illustrate the phenomenon of collision of two right-running unequal-amplitude solitary waves propagating with different speeds. The behavior of the colliding solitary waves is found to be similar to solitary wave solutions of other nonintegrable dispersive-type equations, such as the scalar Benjamin equation [15] and the regularized Benjamin-Ono equation [16]. Such numerical study on the travelling wave solutions of the SBO system has not been considered in previous works to the best knowledge of the author. Furthermore, we obtain numerically new periodic travelling wave solutions which are not included in the analytical theory presented in the work by Angulo et al. [13]. These are the main contributions of the present study. Other contribution is to explain the application of a numerical scheme which belongs to the spectral collocation methods to approximate the solutions of integrodifferential equations, such as system (1). These numerical methods have been intensively used in the solution of dispersive-type equations in the last years. We point out that exact solutions for system (1) are not known when γ≠0. Therefore, a numerical strategy is very important in order to investigate the properties of the solution space, such as to establish the parameter regime for existence of periodic and nonperiodic travelling waves, orbital stability under small initial disturbances, and interactions among these solutions, for example. We point out that the constant γ in the SBO system is positive for physical meaning [1]. However, as we mentioned above, in previous works [7, 10–12], the SBO system has been studied from a mathematical point of view, establishing well-posedness and that travelling wave solutions for the SBO system are possible when the parameter γ takes both signs. This is an interesting mathematical fact, and thus we will conduct some numerical experiments which illustrate the family of travelling wave solutions for both γ>0 and γ<0.This paper is organized as follows. In Section2, we present a brief review of well-known exact solutions of the SBO system for γ=0. In Section 3, we explain the numerical methodology used to find approximations to travelling wave solutions of the SBO system. In Section 4, we compute some periodic and nonperiodic travelling wave solutions of the system, analyze the influence of model’s parameters on the geometry of periodic and nonperiodic travelling waves, and explore the collision of two right-going solitary waves of the SBO system. Finally, Section 5 contains the conclusions of our work.


## 2. Computing Exact Travelling Wave Solutions

For the sake of completeness, in this section, we present a brief review of a class of exact travelling wave solutions of system (1) (periodic and non-periodic) which are well known in the literature when γ=0 (see [7, 10, 13] for more details and analytical results on existence of travelling wave solutions for γ≠0). These analytical special cases are important because they can be used as initial points for computing branches of solutions when γ≠0, through Newton’s iteration and numerical continuation, for example.In the nonperiodic case andγ=0, we can construct a family of exact travelling wave solutions in the form (3) of system (1). In first place, note that, for γ=0, the real-valued functions ϕ,ψ must satisfy the system(4)ϕ′′+c24-wϕ=αϕψ,(5)-cψ′-βϕ2′=0,where the tildes denote differentiation with respect to the variable χ=x-ct.We recall thatϕ,ψ and their derivatives decay to zero at infinity. Thus we can integrate (5) to get(6)ψ=-βcϕ2.After substituting the value of ψ into (4), we obtain(7)ϕ′′+c24-wϕ=-αβcϕ3.By multiplying the previous equation by ϕ′, we get (8)ϕ′ϕ′′=w-c24ϕϕ′-αβcϕ3ϕ′.Therefore,(9)ϕ′22′=w-c24ϕ2′2-αβϕ4′4c,(10)ψ=-βcϕ2.Thus, by integrating (9) and using again the decaying properties of the functions ϕ,ψ, we obtain(11)ϕ′2=w-c24ϕ2-αβ2cϕ4,(12)ψ=-βcϕ2.Looking for solutions of (11) in the form ϕ(χ)=Asechs(Bχ), where A,B,s are real constants to be determined, we have that(13)A2B2s2sechBχ2+2ssinh2Bχ=w-c24A2sech2sBχ-αβ2cA4sech4sBχ,or, equivalently,(14)A2B2s2sech2Bχsinh2Bχ=w-c24A2-αβ2cA4sech2sBχ.Using the identitysinh2(Bχ)=-1+cosh2(Bχ), we arrive at(15)-A2B2s2sech2Bχ+A2B2s2+c24-wA2=-αβ2cA4sech2sBχ.From this equation we can conclude that(16)s=1,A2B2=w-c24A2.On the other hand, evaluating (15) at χ=0,(17)A2B2s2=αβ2cA4.Substituting the value of the constant B in the previous equation, we arrive at(18)A2=2cαβw-c24,B2=w-c24.Therefore,(19)ϕχ=AsechBχ,ψχ=-βcA2sech2Bχ.On the other hand, in the periodic case and with γ=0, we also have that ϕ,ψ must also satisfy system (4)-(5). By integrating (5), we get(20)-cψ=βϕ2+Aψ,where Aψ is a constant. Here we assume that Aψ=0. Thus, after substituting this expression for ψ into (4), we arrive at(21)ϕ′′+c24-wϕ=-αβcϕ3.Therefore, we obtain(22)ϕ′2′=w-c24ϕ2′-αβϕ4′2c.Then integrating the last equation, we arrive at(23)ϕ′2=w-c24ϕ2-αβϕ42c+Bϕ=αβ2c-ϕ4+2cw-c2/4αβϕ2+2cBϕαβ,where Bϕ is a constant. Suppose that w>c2/4, α,β,c are positive real numbers and Bϕ is negative. Furthermore, let η1,η2, -η1,-η2 (η1>0,  η2>0) be the roots of the polynomial of the right side of the previous equations; then we can write(24)ϕ′2=αβ2cη12-ϕ2ϕ2-η22,and thus(25)ϕ′2=αβ2c-ϕ4-η12η22+η12ϕ2+ϕ2η22=αβ2c-ϕ4+η12+η22ϕ2-η12η22.Therefore,(26)η12+η22=2cw-c2/4αβ,-η12η22=2cBϕαβ.Without less of generality, we may assume that η1>η2, and consider the change of variable ζ=ϕ/η1. Substituting (24), we get(27)η12ζ′2=αβ2cη12-η12ζ2η12ζ2-η22=αβ2cη121-ζ2η12ζ2-η22η12.Letk2=1-η22/η12=η12-η22/η12. Thus(28)ζ′2=αβη122c1-ζ2ζ2+k2-1.Now let us define ζ2=1-k2sin2χ. Then(29)2ζζ′=-2k2sin⁡χcos⁡χχ′,and thus(30)ζ′2=k4sin2χcos2χχ′21-k2sin2χ.Substituting this expression into (28), we obtain(31)k4sin2χcos2χχ′21-k2sin2χ=αβ2cη12k2sin2χk2-k2sin2χ.We can simplify the previous equation to get(32)χ′2=αβη122c1-k2sin2χ,χ′=αβ2cη11-k2sin2χ,dχ1-k2sin2χ=η1αβ2cdξ.Assuming that ζ(0)=1, then χ(0)=0 and integrating the previous equation(33)∫χ0χξdt1-k2sin2t=∫0ξη1αβ2cdξ=η1αβ2cξ=lξ, where l=η1αβ/2c. Using from the definition of the Jacobian elliptic function sn(u;k), we get sinχ=sn(lξ;k) and hence ζ(ξ)=1-k2sn2(lξ;k)≡dn(lξ;k). In terms of the original variables ϕ,ψ, we obtain the dnoidal wave solutions corresponding to system (11)-(12):(34)ϕξ=ϕξ;η1,η2=η1dnlξ;kψξ=ψξ;η1,η2=-η12βcdn2lξ;k,where(35)k2=η12-η22η12,η12+η22=2cw-c2/4αβ,0<η2<η1.Since the fundamental period of the dnoidal function dn is 2K(k), where(36)Kk≡∫0π/2dt1-k2sin2t,it follows that ϕ has fundamental period Tϕ given by(37)Tϕ=2Kl=2Kη12cαβ.We point out that, due to the factor eic(x-ct)/2 in the component u(x,t) of the travelling wave solution (3), we have the condition(38)Tϕ=2Kη12cαβ=4πcm,for some m∈N.


## 3. Methodology for Approximation of Travelling Wave Solutions of the Full SBO System

Forγ≠0, we recall that if (u,v) is a travelling wave solution in the form(39)ux,t=eiwteicx-ct/2ϕx-ct,vx,t=ψx-ct,of system (1), then ϕ,ψ are real-valued functions which must satisfy the following equations:(40)ϕ′′+c24-wϕ-αϕψ=0,γHψ′+cψ=-βϕ2.In first place, we are interested in finding approximations to even solutions (ϕ,ψ) with period 2l, l>0 of system (1); let us introduce truncated cosine expansions for u and v:(41)ϕx≈ϕ0+∑n=1N/2ϕncos⁡nπlx,ψx≈ψ0+∑n=1N/2ψncos⁡nπlx,where(42)ϕ0=1l∫0lϕxdx=12l∫02lϕxdx,ϕn=2l∫0lϕxcos⁡nπxldx=22l∫02lϕxcos⁡nπxldx,and let us introduce analogous expressions for the coefficients ψn. By substituting expressions (41) into (40), evaluating them at the N/2+1 collocation points(43)xj=2lj-1N,j=1,…,N2+1,and using the property of the Hilbert transform(44)Heikx=isignkeikx,k∈Z,we obtain the system of N+2 nonlinear equations,(45)-∑n=1N/2nπl2ϕncos⁡nπxjl+c24-w∑n=0N/2ϕncos⁡nπxjl-α∑n=0N/2ϕncos⁡nπxjl∑n=0N/2ψncos⁡nπxjl=0,-γ∑n=1N/2nπlψncos⁡nπxjl+c∑n=0N/2ψncos⁡nπxjl=-β∑n=0N/2ϕncos⁡nπxjl2,for j=1,…,N/2+1, which can be written in the form(46)Fϕ0,ϕ1,…,ϕN/2,ψ0,ψ1,…,ψN/2=0,where the N+2 coefficients ϕn,ψn are the unknowns. We point out that the property (44) of the Hilbert transform makes the numerical solution of system (40) with the spectral collocation proposed in the present paper easier. This is the motivation for the choice of this numerical strategy.Nonlinear system (46) can be solved by Newton’s iteration. Computation of the cosine series in (41) and the integrals in (42) is performed using the FFT (Fast Fourier Transform) algorithm. The Jacobian of the vector field F:RN+2→RN+2 is approximated by the second-order accurate formula(47)Ji,jFx≈Fix+hej-Fix-hej2h,j=1,…,N+2,where ej=(0,…,1,…,0) and h=0.01. We stop Newton’s iteration when the relative error between two successive approximations and the value of the vector field F are smaller than 10-12. We use the travelling wave solutions given in Section 2, when γ=0, for initializing the Newton’s iterative scheme explained above.


## 4. Description and Discussion about the Numerical Results

In the following, we present some numerical experiments performed by using the numerical scheme described in the previous section.Experiment Set 1 (Periodic Waves). In the first numerical experiment we set γ=α=β=1, c=0.5, and w=1.5, and the fundamental period is L=4π/c. The iterative Newton’s procedure is initialized with(48)ϕ0x=2cos2πxL,ψ0x=cos2πxL.The result of this computer simulation is presented in Figure 1. To check that we have computed really a periodic travelling wave of the SBO system (1), we run the spectral Crank-Nicolson numerical solver to approximate the solutions to this system, introduced by the author in [14], with time step Δt=10-3, and N=28 FFT points; the spatial computational domain is the interval [0,L] with L=4π/c≈25.13, and the initial values are(49)ux,0=eicx/2ϕx,vx,0=ψx,with ϕ,ψ being the profiles displayed in Figure 1. The result of this computer simulation at t=7 is displayed in Figure 2, superimposed with the expected position of the travelling wave given by (39). We observe a good accordance with a maximum error of 10-3 between the profiles of the modulus of components u and v, corroborating that, in fact, we have an approximation of a travelling wave solution to system (1). The same verification was performed successfully for other periodic travelling waves computed for different values of the parameter w, displayed in Figures 3, 4, and 5 for w=0.5, w=0.1, and w=0.06, respectively.Figure 1
Approximations of the functionsϕ,ψ in the travelling wave solution (39) with 9 Newton’s iterations. The model’s parameters are α=β=γ=1, w=1.5, and wave speed c=0.5 and the fundamental period is L≈25.13.Figure 2
Evolution of the travelling wave solution (39) (with ϕ,ψ as in Figure 1) of the SBO system at t=7. In solid line, |u|,v computed with the numerical solver introduced in [14]. In pointed line, |u|,v for the approximate travelling wave solution (39).Figure 3
Approximations of the functionsϕ,ψ in the travelling wave solution (39) with 10 Newton’s iterations. The model’s parameters are α=β=γ=1, w=0.5, and wave speed c=0.5 and the fundamental period is L≈25.13.Figure 4
Approximations of the functionsϕ,ψ in the travelling wave solution (39) with 11 Newton’s iterations. The model’s parameters are α=β=γ=1, w=0.1, and wave speed c=0.5 and the fundamental period is L≈25.13.Figure 5
Approximations of the functionsϕ,ψ in the travelling wave solution (39) with 23 Newton’s iterations. The model’s parameters are α=β=γ=1, w=0.06, and wave speed c=0.5 and the fundamental period is L≈25.13.We remark that Angulo et al. in [13] established, by using the concentration compactness method (Lions [8, 9]), the existence of periodic travelling wave solutions of system (1), provided that w-c2/4>2π2/L2, where L is the fundamental period. Thus, the numerical experiment in Figure 5 with w=0.06, c=0.5, that is, w-c2/4<0, shows that travelling wave solutions of the SBO system may also exist outside the parameter regime considered in [13]. In Figure 6, we display another numerical simulation with w=0.06, c=1, where we also obtain an approximate travelling wave solution for w-c2/4<0, by initializing the Newton’s iteration with(50)ϕ0x=4cos2πxL,ψ0x=cos2πxL.In Figure 7, we show new periodic travelling waves of system (1) for γ≠0 computed by using as initial step for Newton’s iteration the periodic solutions obtained in (34), corresponding to the case γ=0. In this experiment we use the model’s parameters α=β=1, γ=1, c=1.5, and w=1.5. Given the parameters α,β,c, the values of the parameters η1,η2,k are approximated numerically satisfying the relationships obtained in Section 2:(51)η1=2cw-c2/4αβ-η22,2Kη12-η22/η12η12cαβ-4πc=0.In this case, the results are η1≈1.6729, η2≈0.1175, and k≈0.9975. The resulting fundamental period of the travelling wave is L=4π/c≈8.37.Figure 6
Approximations of the functionsϕ,ψ in the travelling wave solution (39) with 18 Newton’s iterations. The model’s parameters are α=β=γ=1, w=0.06, and wave speed c=1 and the fundamental period is L≈12.57.Figure 7
Approximations of the functionsϕ,ψ in the travelling wave solution (39) with 8 Newton’s iterations. The model’s parameters are α=β=γ=1, w=1.5, and wave speed c=1.5 and the fundamental period is L≈8.38.In Figure8, we show the computation of other periodic travelling wave of the SBO system with γ=1≠0, using as initial step in Newton’s scheme the periodic solution (34) with the parameters α=β=0.5, c=2, and w=2. Again, the parameters η1,η2,k are approximated numerically and the results are η1≈3.9328, η2≈0.7297, and k≈0.9826. Thus the resulting fundamental period of the travelling wave is L=4π/c≈6.28. We point out that, in this numerical simulation, the travelling wave is faster (c=2) than the one computed in the previous numerical experiment (c=1.5).Figure 8
Approximations of the functionsϕ,ψ in the travelling wave solution (39) with 6 Newton’s iterations. The model’s parameters are α=β=γ=1, w=2, and wave speed c=2 and the fundamental period is L≈6.28.It is important to remark that the first purpose of the previous numerical experiments is to show that the proposed Newton-pseudospectral scheme can be used to approximate with good accuracy and low computational cost, travelling-wave solutions of the SBO system. This fact was corroborated for every travelling wave computed (see, e.g., Figure2), where we see that the corresponding wave components u,v translate with the expected speed without changing their shape, and neither dissipation nor dispersion was observed during the time evolution. In second place, we obtained numerically new periodic travelling wave solutions of the SBO system in a parameter’s regime not covered by the theoretical results in previous works, suggesting that the regime of existence of such type of solutions of the SBO system can be extended to the case w-c2/4<0.Experiment Set 2 (Solitary Waves). In second place, we wish to compute approximations of solitary wave solutions of the SBO system. The following numerical simulations have the goal of showing that the proposed numerical solver is not restricted to the periodic case, and it can also be further employed to obtain approximations to solitary wave solutions of the SBO system on the real line, provided that the spatial computational domain is chosen large enough so that the solution does not reach the computational boundaries.In this case, the functionsϕ and ψ and their derivatives must decay to zero at infinity. Thus, we run Newton’s iteration to approximate the cero of the vector field given in (46) by using as initial step the solitary wave (19) corresponding to γ=0. Here we set α=β=0.4, γ=-1, c=0.5, w=1.5, and N=28 FFT points. We point out that Newton’s scheme explained in the previous section is designed for searching spatial periodic solutions of the SBO system. However, due to the decay properties at infinity of a solitary wave solution, we can also use Newton’s procedure adopting a spatial computational domain as the interval [0,L] with L large enough. In this case we set L=120. The result of this computer simulation is presented in Figure 9 and the corresponding verification that it corresponds really to a solitary wave solution of the SBO system is presented in Figure 10. In this validation we use the numerical parameters Δt=10-3 and N=210 FFT points for the spatial discretization in the spectral numerical scheme introduced in [14]. The maximum error observed at t=40 between the two wave profiles is roughly 10-4. Other solitary wave solutions of the SBO system were computed for different values of the parameter w and the results are displayed in Figures 11, 12, and 13, for w=0.5, w=0.1, and w=0.07, respectively. The parameters α,β,c,γ are the same as before. In the experiments in Figure 11, we use L=120, N=28, and for the experiments in Figures 12 and 13, we set L=400, N=28, and L=800, N=210, respectively. We recall that the wave component v(x,t) of a travelling solution of the SBO system is given here by v(x,t)=ψ(x-ct). Thus, observe that, in all solitary waves computed in this paper, the component v(x,t) is negative. This is because solution (19) (corresponding to γ=0) was adopted as the starting point for Newton’s iteration in the case of solitary waves.Figure 9
Approximations of the functionsϕ,ψ in the solitary wave solution (39) with 6 Newton’s iterations. The model’s parameters are α=β=0.4, γ=-1, w=1.5, and wave speed c=0.5.Figure 10
Evolution of a solitary wave solution (39) (with ϕ,ψ as in Figure 9) of the SBO system at t=40. In solid line, |u|,v computed with the numerical solver introduced in [14]. In pointed line, |u|,v for the approximate solitary wave solution (39).Figure 11
Approximations ofϕ,ψ in the solitary wave solution (39) with 6 Newton’s iterations. The model’s parameters are α=β=0.4, γ=-1, w=0.5, and wave speed c=0.5.Figure 12
Approximations of the functionsϕ,ψ in the solitary wave solution (39) with 5 Newton’s iterations. The model’s parameters are α=β=0.4, γ=-1, w=0.1, and wave speed c=0.5.Figure 13
Approximations of the functionsϕ,ψ in the solitary wave solution (39) with 5 Newton’s iterations. The model’s parameters are α=β=0.4, γ=-1, w=0.07, and wave speed c=0.5.Experiment Set 3 (Influence of Model’s Parameters). The previous numerical experiments illustrated the shape of some travelling wave solutions of the SBO system, not computed in previous works, to the knowledge of the author. In the next set of experiments, we wish to initiate the analysis of the dependence of the these travelling wave solutions (periodic and non periodic case) on the parameters appearing in the SBO system.In Figure14, we compare the computed periodic travelling waves for the SBO system for γ=0.6,0.8,1.0. The rest of model’s parameters are fixed at α=β=1, c=0.5, and w=0.5. From these computer simulations, we can see that the amplitudes of both components ϕ,ψ of the computed travelling wave increase when the parameter γ decreases. In contrast, in Figure 15, we explore the effect of the frequency w on the shape of the computed travelling wave. The parameters α=β=1, c=0.5, and γ=0.5 are left fixed. Notice that, in this case, the amplitude of the wave components ϕ,ψ results in being proportional to the value of the frequency w. In the case of solitary waves of the SBO system, we conducted similar numerical experiments to analyze the effect of the model’s parameters γ,w on the amplitude of the computed profiles. The results are shown in Figure 16 (with α=β=1,  c=0.5, w=0.5, and  γ=-0.1,-0.7,-1.0) and Figure 17 (with α=β=1,  c=0.5, γ=-1, and w=0.4,0.6,1.0). In contrast to the case of periodic waves, in this case the behavior of the solitary wave components ϕ,ψ differs from each other. For instance, notice that the amplitude of the component ϕ decreases when γ increases. Meanwhile, the component ψ decreases its amplitude when γ decreases. Furthermore, the amplitude of both components ϕ,ψ is directly proportional to the frequency w.Figure 14
Comparison of periodic travelling wave solutions of the SBO system for different values of the parameterγ.Figure 15
Comparison of periodic travelling wave solutions of the SBO system for different values of the frequencyw.Figure 16
Comparison of solitary wave solutions of the SBO system for different values of the parameterγ.Figure 17
Comparison of solitary wave solutions of the SBO system for different values of the frequencyw.To analyze the effect of the parameterα on the shape of travelling waves of system SBO, we conduct additional numerical experiments setting β=1, γ=0.6, c=w=0.5, and α=0.3,0.4,0.6,1.0. The resulting profiles computed with the Newton procedure are displayed in Figure 18. Observe that the amplitude of the computed travelling waves is inversely proportional to the parameter α. On the other hand, fixing α=1, γ=0.6, and c=w=0.5, we study in Figure 19 the influence of the parameter β. In these numerical experiments, we take β=2.8,2.9,3.0,3.1 and see that the corresponding travelling wave solution is more sensitive to changes of the parameter β (which alters the first derivative term (u2)x), in contrast to previous simulations varying the other parameters in the SBO system.Figure 18
Comparison of periodic travelling wave solutions of the SBO system for different values of the parameterα.Figure 19
Comparison of periodic travelling wave solutions of the SBO system for different values of the parameterβ.Experiment Set 4 (Solitary Wave Collisions). In this set of experiments we wish to explore the interaction between two solitary waves of system (1) with different speeds of propagation and amplitudes. As mentioned above, this phenomenon has not been considered in previous works on the SBO system, to the best of the author’s knowledge. To do this, we compute the evolution of the initial profile(52)ux,0=eic1x/2ϕ1x+eic2x-20/2ϕ2x-20,vx,0=ψ1x+ψ2x-20,where ϕ1,ψ1 are the solutions of (40) with wave speed c=c1=0.5 and ϕ2,ψ2 are the solutions corresponding to wave speed c=c2=2. The model’s parameters for computing the profiles ϕi,ψi,  i=1,2 are taken as α=β=0.4, γ=-1, and w=1.5, and the spatial domain is the interval [0,L] with L=120. We point out that the profiles ϕ2, ψ2 have been moved to the right a distance of 20 units, so that the initial separation between the solitary waves is 20 units. Furthermore, the profile with a higher velocity c2=2 is located behind the one with the lower velocity c1=0.5. The numerical parameters are Δt=10-3 and N=28 FFT points in this computer simulation. The results of the interaction of the two solitary waves are displayed in Figures 20 and 21, and in the contour plots shown in Figures 22 and 23, we can see the trajectory of each solitary wave in the x-t plane before and after the collision. When t≈10 the larger solitary wave begins to shrink and the smaller solitary wave begins to grow. This process continues until the wave profiles have swapped identities, after which they separate. We always observe two local maxima in both components |u(x,t)| and |v(x,t)| during the entire interaction. When t≈13, the two solitary waves start to separate from each other. From these numerical experiments, we see that, before collision, the solitary wave with higher velocity c2=2 moves towards the one with slower velocity c1=0.5, with the expected speeds, and then one passes through another. We point out that, after collision, some small dispersive trailing waves are generated around the pulses. Then these tails separate from the solitary waves, lengthen, and become weaker progressively as time evolves. Further observe that the tails are the strongest for the wave component v, which is affected by the Hilbert operator in the second equation in the SBO system. We obtained similar results in other numerical simulations performed with different pairs of solitary wave solutions of the SBO system. This behavior is very different from that observed in other well-known integrable dispersive models, such as the Korteweg-de Vries equation [17], the Benjamin-Bona-Mahony equation [18], and the Benjamin-Ono equation [19], where the collision of two superimposed solitary-waves is elastic (i.e., clean, without forming dispersive tails). For the Benjamin-Ono equation, see also the numerical study of solitary wave collisions in the work by Thomeé and Vasudeva Murthy [20] and Pelloni and Dougalis [21]. We remark that this phenomenon has already been observed by Kalisch and Bona [22], who studied the interaction of solitary-wave solutions of the Benjamin-Ono equation, the regularized Benjamin-Ono equation, and the Benjamin equation, which are dispersive models where the Hilbert transform is also present. In accordance with that work, this behavior suggests that the SBO system is not integrable; that is, it does not have exact solitary-wave solutions, similar to the case of the Benjamin equation and the regularized Benjamin-Ono equation.Figure 20
View in thex-t plane of the collision of two solitary waves of the SBO system with velocities c=0.5 and c=2. Observe that the colliding waves have different initial amplitude.Figure 21
View in the planex-t of the collision of two solitary waves of the SBO system with velocities c=0.5 and c=2. Observe that the colliding waves have different initial amplitude.Figure 22
Wave-crest trajectory of collision of the two solitary waves of different speed and amplitude shown in Figure20.Figure 23
Wave-crest trajectory of collision of the two solitary waves of different speed and amplitude shown in Figure21.We point out that the parameter space for system (1) is large and thus, in this paper, we have presented only a limited number of numerical experiments to illustrate the geometry of travelling wave solutions of the SBO system, in a parameter regime where no exact solution is known, and to initiate the study of interactions between these permanent wave states. Further numerical and analytical studies are necessary to determine the complete parameter space for existence of travelling waves to study important analytical properties, such as orbital stability, and to develop an exhaustive discussion of solitary wave interactions for the SBO system.


## 5. Conclusions

In this paper, we introduced a numerical scheme for approximating the travelling wave solutions (periodic and nonperiodic) of system (1), whose existence was established in previous works [7, 10, 13]. This scheme combines a Fourier spectral discretization together with a Newton-type iteration, using as initial step well-known exact solutions when γ=0. Using the spectral Crank-Nicholson numerical scheme introduced by the author in [14] for approximating time evolution of solutions of the SBO system, we also illustrated the collision of two right-running unequal-amplitude solitary waves propagating with different speeds. The behavior of the colliding solitary waves was found to be similar to solitary waves of other well-known nonintegrable dispersive-type equations, such as the scalar regularized Benjamin-Ono equation and the Benjamin equation, for instance. As mentioned in the work by Kalisch and Bona [22], this phenomenon suggests that the SBO system is not integrable. Furthermore, we obtained numerically new periodic travelling wave solutions which are not included in the analytical theory presented in the work by Angulo et al. [13].



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