# Error Analysis of an Implicit Spectral Scheme Applied to the Schrödinger-Benjamin-Ono System

Authors: Juan Carlos Muñoz Grajales
Journal: International Journal of Differential Equations
 (2016)

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## Abstract

We develop error estimates of the semidiscrete and fully discrete formulations of a Fourier-Galerkin numerical scheme to approximate solutions of a coupled nonlinear Schrödinger-Benjamin-Ono system that describes the motion of two fluids with different densities under capillary-gravity waves in a deep water regime. The accuracy of the numerical solver is checked using some exact travelling wave solutions of the system.


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## Body

## 1. Introduction

In this paper we shall study numerically the nonlinear one-dimensional system (named as 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 system describes the motion of two fluids with different densities under capillary-gravity waves in a deep water regime. In this physical phenomenon the long internal wave is described by a wave equation with a dispersive term represented by a nonlocal Hilbert operator, and the short surface wave is described by a Schrödinger type equation. This nonlinear coupled system was derived by Funakoshi and Oikawa [1] in a regime such that the fluid depth of the lower layer is sufficiently large, in comparison with the wavelength of the internal wave. Here u=u(x,t):R×R→C denotes the short wave term and v=v(x,t):R×R→R denotes the long wave term. Furthermore, α, β are positive constants, γ∈R, and H denotes the Hilbert transform defined by(2)Hfx≔p.v.1π∫fyy-xdy.The SBO system also appears in the sonic-Langmuir wave interaction in plasma physics (Karpman [2]), in the capillary-gravity interaction waves (Djordjevic and Redekoop [3], Grimshaw [4]), and in the general theory of water-wave interaction in a nonlinear medium (Benney [5, 6]).An important property of system (1) due to nonlinearity and dispersive effects is that it possesses the so-calledtravelling wave solutions in the form(3)ux,t=eiwpteicx-ct/2ϕx-ct,vx,t=ψx-ct, with wp,c∈R and ϕ, ψ being periodic real-valued functions or smooth functions such that, for each n∈N, ϕ(n)(ξ)→0 and ψ(n)(ξ)→0, as ξ→∞. In this last case, these solutions are calledsolitary waves. Angulo and Montenegro [7] have proved the existence of even solitary wave solutions using the concentration compactness method (Lions [8, 9]) and the theory of symmetric decreasing rearrangements. The existence and stability of a new set of solitary waves were presented 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 a well-posedness theory in the Sobolev space HCs(R)×HRs-1/2(R), with s≥0. When γ=1, Pecher [12] showed the local well-posedness for s>0. More recently, Angulo et al. [13] proved the global well-posedness for s=0 in the case that γ≠1. In the periodic setting, there are only a few results known. 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.In this paper, we shall develop a rigorous analysis of the error of the semidiscrete and fully discrete formulations of a Fourier-Galerkin scheme to approximate solutions of the SBO system (1). The time-stepping method is implemented by using a second-order implicit Crank-Nicholson strategy. The rates of convergence of the semidiscrete and fully discrete schemes are O(Ns+2-r) and O(Ns+2-r+Δt2), respectively, where r>s+2 depends only on the smoothness of the exact solution, Δt is the time step, and N is the number of spatial Fourier modes. The strategy to obtain these rates of convergence follows the one used by Muñoz Grajales [14, 15] and Antonopoulos et al. [16, 17] and Pelloni and Dougalis [18] for other dispersive systems. On the other hand, Rashid and Akram [19] studied the error of an implicit spectral scheme for the SBO system but only in the particular case that γ=0. Furthermore, Funakoshi and Oikawa [1] computed numerically some approximations to travelling wave solutions of the SBO system. However, an analysis of error of a fully discrete spectral numerical scheme for the complete SBO system has not been performed in previous works to the best of the author’s knowledge. This is one motivation for the present study. We point out that system (1) does not have exact solutions in the general case that γ≠0. Therefore, a numerical strategy is very important in order to investigate the properties of the solution space, such as existence of periodic and nonperiodic travelling waves, orbital stability under small initial disturbances, and interactions among these solutions.The accuracy and convergence rate of the Fourier-spectral scheme proposed in this paper are illustrated by using a family of exact solitary wave solutions of system (1) when γ=0. In order to apply this scheme in a nonperiodic setting, we approximate the initial value problem for system (1) with x∈R, by the corresponding periodic Cauchy problem for x∈[0,L], with a large spatial period L. This type of approximation can be justified by the decay of the solutions of the unrestricted problem as x→∞.This paper is organized as follows. In Section2, we introduce notation and functional spaces which will be used in our work. In Section 3, the analytical properties and convergence of the semidiscrete scheme to approximate solutions of the SBO system are investigated. Section 4 deals with the convergence of the fully discrete scheme that we propose for solving the SBO system. Finally in Section 5, to validate the theoretical results, some numerical experiments using a family of analytical and approximate solutions of the SBO system are performed.


## 2. Preliminaries

We set(4)L20,L≔f:0,L⟶C,f0=∫0Lfx2dx1/2<∞,with the inner product(5)f,g=∫0Lfxgx¯dx.The space of all functions of class Ck that are L-periodic is denoted by Cperk(0,L), k=0,1,2,…. Further Cper=Cper0(0,L) is the space of all continuous functions of period L.We will denote byP the space of all infinitely differentiable functions that are L-periodic as well as all their derivatives. We say that T:P→C defines a periodic distribution, that is, T∈P′, if T is linear and there exists a sequence (Ψn)n∈N⊂P such that(6)T,φ=limn→∞∫0LΨnxφxdx,∀φ∈P.Let s∈R. The Sobolev space, denoted by Hpers=Hpers(0,L), is defined as(7)Hpers0,L=f∈P′:fs2=L∑n∈Z1+k2sf^n2<∞,where f^:Z→C represents the Fourier transform of f∈P′ defined by(8)f^n=1Lf,e-2πinx/L,n∈Z.In case that f∈Cper, we can rewrite f^(n) as(9)f^n=1L∫0Lfxe-2πinx/Ldx,n∈Z.It can be shown that, for all s∈R, Hpers is a Hilbert space with respect to the inner product 〈·,·〉s defined as follows: (10)f,gs=L∑n∈Z1+n2sf^ng^n¯.In particular, when s=0, we get the Hilbert space denoted by Lper2=Hper0. It is important to note that this space is isometrically isomorphic to L2(0,L). Further we recall that Parseval’s identity holds; that is, for f∈Cper(11)∑n∈Zf^n2=1Lf02,or, equivalently,(12)f,g=L∑n∈Zf^ng^¯n=Lf^,g^.Let N be an even integer and consider the finite dimensional space SN defined by(13)SN=span1Le2πinx/L:-N2≤n≤N2.Remember that the family {1/Le2πinx/L:n∈Z} is an orthonormal and complete system in Lper2(0,L). Let PN be the orthogonal projection PN:Lper2(0,L)→SN on the space SN:(14)PNg≔∑n=-N/2N/2g^nϕn,with(15)ϕnx=e2πinx/L,g^n=1L∫0Lgxϕ¯nxdx.This operator has the following properties (see [20, 21]): For any g∈Lper2(0,L), (16)PNg-g,ϕ=0,∀ϕ∈SN.Furthermore, given integers 0≤s≤α, there exists a constant C independent of N such that, for any g∈Hperα(0,L),(17)PNg-gs≤CNs-αgα.In what follows, for a positive integerk, Ck([0,T],X) denotes the space of k-times continuously differentiable maps from [0,T] onto a Banach space X.


## 3. The Semidiscrete Scheme

Let us consider the SBO system(18)i∂tu+∂x2u-αuv=0,x∈0,L,t>0,∂tv-γH∂x2v-β∂xu2=0,subject to the initial conditions u(x,0)=u0(x), v(x,0)=v0(x) and u,vL-periodic complex valued functions in the variable x.The semidiscrete Fourier-Galerkin spectral scheme to solve problem (18) is to find uN,vN∈C([0,T],SN) such that(19)i∂tuN+∂x2uN-αuNvN,ϕ=0,∂tvN-γH∂x2vN-β∂xuN2,ψ=0,uN0=PNu0,vN0=PNv0,for any ϕ,ψ∈SN and 0≤t≤T.Theorem 1.
Lets≥2 be an integer, and let u,v∈C1([0,T],Hperr(0,L)) be the classical solution of problem (18) for some integer r>s+2. Let uN,vN∈C1([0,TN],SN) be the solution of the semidiscrete formulation (19) defined until some maximal time 0<TN<T. Then, for N sufficiently large, uN, vN can be extended to the whole interval [0,T] and there exists a constant C>0, independent of N,t, such that (20)ut-uNts+1+vt-vNts≤CNs+2-r,for any 0≤t≤T.Proof.
Let(21)e1=u-uN=u-PNu+PNu-uN≔ξ1+η1,e2=v-vN=v-PNv+PNv-vN≔ξ2+η2.Observe that by the fact that PN is an orthogonal projection and if s∈Z+ then ∂xsϕ∈SN for any ϕ∈SN, and thus(22)ξl,ϕs=0,ϕ∈SN,l=1,2;that is,(23)u-PNu,ϕs=0,v-PNv,ϕs=0,for any ϕ∈SN. Therefore, by virtue of (17)(24)els≤ξls+ηls≤CNs-r+ηls,l=1,2.Now, by combining the equations satisfied by the pairs (u,v), (uN,vN), we arrive at(25)i∂tu-uN+∂x2u-uN,ϕs=αvu-vNuN,ϕs,∂tv-vNt-γH∂x2v-vN,ψs=β∂xu2-uN2,ψs.As a consequence,(26)i∂te1+∂x2e1,ϕs=αvu-vNuN,ϕs,(27)∂te2-γH∂x2e2,ψs=β∂xu2-uN2,ψs.Letting ϕ=η1 in (26), we obtain (28)i∂te1,η1s+∂x2e1,η1s=αvu-vNuN,η1s.Since e1=ξ1+η1,(29)i∂tξ1+∂tη1,η1s+∂x2ξ1+∂x2η1,η1s=αvu-vNuN,η1s.Note that due to ∂tη1∈SN thus 〈ξ1,∂tη1〉s=0 and(30)∂tξ1,η1s=∂tξ1,η1s+ξ1,∂tη1s=0.As a consequence, ∂tξ1,η1s=0. Furthermore, since ∂x2η1∈SN and using integration by parts, we have(31)∂x2ξ1,η1s=ξ1,∂x2η1s=0.Therefore,(32)i∂tη1,η1s-∂xη12=αvu-vNuN,η1s,i2∂tη1s2-∂xη1s2=αvu-vNuN,η1s.Now taking imaginary part(33)12∂tη1s2=αIm⁡vu-vNuN,η1s.To bound the nonlinear terms, let us observe that(34)αIm⁡vu-vNuN,η1s≤αvu-vNuN,η1s≤αvu-vNuNsη1s≤Cvu-vNuNs2+η1s2,and as the space Hpers is an algebra for s>1/2, (35)vu-vNuNs=uv-vN+vNu-uNs≤usv-vNs+vNsu-uNs≤use2s+vNse1s.
From hypothesis,u,v∈C([0,T],Hperr(0,L)), with r>s+2, and thus there exists a positive constant M>0 such that(36)maxt∈0,Tuts+vts<M.Further suppose that 0<TN<T is the largest value such that(37)maxt∈0,TNuNts+vNts<2M.Because of the embedding to Hpers⊂L∞, for s>1/2, we obtain from (33) and inequality (34)(38)12∂tη1s2≤CN2s-r+η1s2+η2s2.On the other hand, from (27),(39)∂tξ2+∂tη2,ψs=β∂xu2-uN2,ψs+γH∂x2ξ2,ψs+γH∂x2η2,ψs.Letting ψ=η2, (40)∂tξ2,η2s+∂tη2,η2s=β∂xu2-uN2,η2s+γH∂x2ξ2,η2s+γH∂x2η2,η2s.Then, following an analogous procedure as above, we can obtain that(41)∂tξ2,η2s=0,and, taking into account the property of the Hilbert transform operator(42)H∂x2η2,η2s=0,we get that(43)12∂tη2s2=β∂xu2-uN2,η2s+γH∂x2ξ2,η2s.Again, to bound the nonlinear terms, let us note that(44)β∂xu2-uN2,η2s≤C∂xu2-uN2s2+η2s2.But (45)∂xu2-uN2s=∂xuu¯-∂xuNuN¯s≤Cu¯s∂xu-uNs+∂xuNsu¯-uN¯s+∂xu¯su-uNs+uNs∂xu¯-uN¯s≤C∂xξ1s+∂xη1s+ξ1+η1s≤CNs+1-r+η1s+∂xη1s.Therefore, from (43) and H∂x2ξ2s≤ξ2s+2≤Ns+2-r, we arrive at (46)12∂tη2s2≤C∂xu2-uN2s2+η2s2+H∂x2ξ2s2+η2s2≤CN2s+1-r+η1s2+∂xη1s2+η2s2+ξ2s2≤CN2s+1-r+η1s2+η2s2+N2s+2-r+∂xη1s2≤Cη1s2+η2s2+N2s+2-r+∂xη1s2.Letting ϕ=∂tη1 in (26), we have that(47)i∂te1,∂tη1s+∂x2e1,∂tη1s-αvu-vNuN,∂tη1s=0.Then we can replace e1=ξ1+η1 in the previous equation to get(48)i∂tξ1,∂tη1s+i∂tη1,∂tη1s+∂x2ξ1,∂tη1s+∂x2η1,∂tη1s-αvu-vNuN,∂tη1s=0.Therefore, since ∂tη1∈SN and i〈∂tξ1,∂tη1〉s=0 and using integration by parts, it follows that(49)∂x2ξ1,∂tη1s=ξ1,∂x2∂tη1s=0,∂x2η1,∂tη1s=-∂xη1,∂x∂tη1s.As a consequence, (48) implies that(50)i∂tη1s2-12∂t∂xη1s2-αvu-vNuN,∂tη1s=0.Taking imaginary part of the previous equation and using Cauchy-Schwartz and Hölder inequalities, (51)∂tη1s2=αIm⁡vu-uNvN,∂tη1s≤vu-vNuNs∂tη1s≤α14ϵvu-vNuNs2+ϵ∂tη1s2,where ϵ>0. The nonlinear terms in the right-hand side of the last equation can be estimated as in (34), and then, choosing ϵ>0 small enough, we obtain the estimate(52)∂tη1s2≤CN2s-r+η1s2+η2s2.On the other hand, we can take real part of (50) to achieve (53)12∂t∂xη1s2=-αvu-vNuN,∂tη1s≤Cvu-vNuNs∂tη1s≤Cvu-vNuNs2+∂tη1s2≤CN2s-r+η1s2+η2s2+∂tη1s2.Taking into account the previous results, we arrive at (54)∂tη1s2+∂xη1s2+η2s2≤CN2s-r+η1s2+η2s2+∂tη1s2+N2s+2-r+∂xη1s2≤CN2s+2-r+η1s2+η2s2+∂xη1s2.Thus, using Gronwall’s lemma,(55)η1s+1+η2s≤CNs+2-r,for 0≤t≤TN. Observe that, for any 0≤t≤TN, (56)uNts+vNts=uN-PNu+PNu-u+us+vN-PNv+PNv-v+vs≤uN-PNus+PNu-us+us+vN-PNvs+PNv-vs+vs≤η1s+ξ1s+us+η2s+ξ2s+vs≤CNs+2-r+Ns-r+M<2M,for N large enough and r>s+2. This fact contradicts the maximality of TN. Thus the solutions uN,vN of the semidiscrete formulation can be extended for any 0≤t≤T, and inequality (55) is satisfied for any t∈[0,T]. Finally, from (24) the result follows.


## 4. The Fully Discrete Scheme

The fully discrete Crank-Nicholson scheme to discretize system (18) consists in finding a sequence {uNn,vNn} of elements of SN×SN, such that, for all n=1,2,…,M-1, we have(57)iuNn+1-uNnΔt+∂x2uNn+1+uNn2-αuNn+1+uNn2vNn+1+vNn2,ϕ=0,vNn+1-vNnΔt-γH∂x2vNn+1+vNn2-β∂xuNn+1+uNn22,ψ=0,for all ϕ,ψ∈SN and subject to uN0=PN(u0), vN0=PN(v0). Here Δt is a time step chosen together with a positive integer M such that MΔt=T. Furthermore we define tn=nΔt, n=0,1,2,…,M.Theorem 2.
Lets≥2 be an integer, and let (u,v)∈C3([0,T],Hperr(0,L)) be a classical solution of system (18) with r>s+2 integer. Let (uN,vN)∈C1([0,T],SN) be the solution of the semidiscrete formulation (19) and let {(uNn,vNn)} be the solution of the fully discrete scheme (57) such that uNns+vNns≤B, for some sufficiently large constant B>0, independent of n, N, and Δt. If uN0=uN(0), vN0=vN(0), then, with the assumption that u0,v0∈Hperr(0,L), there exists a constant C independent of N and Δt such that if N is large enough and Δt sufficiently small, we have that(58)max0≤n≤Mutn-uNns+1+vtn-vNns≤CNs+2-r+Δt2.Proof.
For a functionf, let us introduce the following notation:(59)ftn=1Δtfn+1-fn,f^n=12fn+1+fn,where fn≔f(tn). Thus the fully discrete formulation (57) can be rewritten as(60)iutn+∂x2u^Nn-αu^Nnv^Nn,ϕs=0,vtn-γH∂x2v^Nn-β∂xu^Nn2,ψs=0,uN0=PNu0,vN0=PNv0.Observe that(61)utn,2u^ns=1Δtun+1s2-uns2.We also introduce the notation (62)e1n=un-uNn=un-PNun+PNun-uNn≔ξ1n+η1n,e2n=vn-vNn=vn-PNvn+PNvn-vNn≔ξ2n+η2n,where un≔u(tn),vn≔v(tn). Observe that 〈ξln,ϕ〉s=0, l=1,2, for any ϕ∈SN, and, furthermore,(63)elns≤ξlns+ηlns≤CNs-r+ηlns,l=1,2.Combining the equations satisfied by (u,v), (uN,vN), one can get(64)ie1tn,ϕs+∂x2e^1n,ϕs-αu^nv^n-u^Nnv^Nn,ϕs=τ1n,ϕs,(65)e2tn,ψs-γH∂x2e^2n,ψs-β∂xu^n2-u^Nn2,ψs=τ2n,ψs,where (66)τ1n≔iutn-i∂tun+∂x2u^n-∂x2un+αunvn-u^nv^n,τ2n≔vtn-∂tvn-γH∂x2v^n+γH∂x2vn+β∂xun2-u^n2.In order to estimate the quantities τ1n, τ2n, let us rewrite them as (67)τ1n=iun+1-unΔt-i∂tu+∂x2un+1+un2-∂x2un+αunvn-αun+1+un2vn+1+vn2,τ2n=vn+1-vnΔt-∂tvn-γH∂x2vn+1+vn2+γH∂x2vn+β∂xun2-un+1+un22.Therefore(68)τ1n=iun+Δt∂tun+1/2Δt2∂t2un-un+OΔt3Δt-i∂tu+∂x2un+Δt∂tun+1/2Δt2∂t2un+un+OΔt32-∂x2un+αunvn-α4un+Δt∂tun+12τ2∂t2un+un+OΔt3vn+Δt∂tvn+12Δt2∂t2vn+vn+OΔt3.From this result, we can deduce that τ1ns≤Cτ2. Analogously, (69)τ2n=vn+Δt∂tvn+1/2Δt2∂t2vn-vn+OΔt3Δt-∂tvn-γH∂x2vn+Δt∂tvn+1/2Δt2∂t2vn+vn+OΔt32+γH∂x2vn+β∂xun2-un+Δt∂tun+un+OΔt324.As a consequence, we also have that τ2ns≤Cτ2.
On the other hand, to estimate nonlinear terms, let us observe that(70)u^nv^n-u^Nnv^Nns≤Cun+1sη2n+1s+vNn+1sη2n+1s+un+1sη2ns+vNnsη1n+1s+unsη2n+1s+vNn+1sη1ns+vnsη1ns+uNnsη2ns,∂xu^n2-u^N2s=∂xun+1+un22-∂xuNn+1+uNn22s=14∂xun+1¯+un¯η1n+1+η1n+uNn+1+uNnη1n+1¯+η1n¯s≤C∂xη1n+1+η1ns+η1n+1+η1ns.Now letting ϕ=2η^1n in (64), one can get(71)ie1tn,2η^1ns+∂x2e^1n,2η^1ns-αu^nv^n-u^Nnv^Nn,2η^1ns=τ1n,2η^1ns. We recall that e1n=ξ1n+η1n. Therefore, (71) leads to(72)iΔtη1n+1s2-η1ns2=12∂xη1n+1+η1n2s2+αu^nv^n-u^Nnv^Nn,2η^1ns+τ1n,2η^1ns.Form hypothesis, u,v∈C([0,T],Hperr). For this reason, there exists a constant B>0 such that u(t)s+v(t)s≤B, for all t∈[0,T]. Then, by taking imaginary part of (72), one obtains (73)η1n+1s2-η1ns2≤CΔtu^nv^n-u^Nnv^Nns2+η^1ns2+τ1ns2+η^1ns2≤CΔtη2n+1s2+η1n+1s2+η2ns2+η1ns2+Δt4.Thus, we arrive at(74)η1n+1s21-CΔt≤CΔtη2ns2+η2n+1s2+Δt4+1+CΔtη1ns2,η1n+1s2≤1+CΔt1-CΔtη1ns2+CΔt1-CΔtη2n+1s2+η2ns2+Δt4.As a consequence of this,(75)η1n+1s2≤Cη1ns2+CΔtη2n+1s2+η2ns2+CΔt5.Then summing up the previous equation for n=0 to m and taking into account that n=T/Δt, it follows that(76)∑n=0mη1n+1s2≤C∑n=0mη1ns2+CΔt∑n=0mη2n+1s2+η2ns2+CΔt4.Therefore,(77)η1m+1s2≤Cη10s2+Δt∑n=0m+1η2ns2+Δt4.
Analogously, lettingψ=2η^2n in (65), one can get(78)e2tn,2η^2ns-γH∂x2e^2n,2η^2ns-β∂xu^n2-u^Nn2,2η^2s=τ2n,2η^2ns.We recall that e2n=ξ2n+η2n. Therefore, (78) leads to(79)1Δtη2n+1s2-η2ns2=β∂xu^n2-u^Nn2,2η^2ns+τ2n,2η^2ns.Then, by taking imaginary part of (79), one obtains(80)η2n+1s2-η2ns2≤CΔt∂xu^n2-u^Nn2sη^2ns+τ2nsη^2ns≤CΔt∂xη1n+1s2+∂xη1ns2+η2n+1s2+η1n+1s2+η1ns2+η2ns2+Δt4.Thus, we arrive at(81)η2n+1s21-CΔt≤CΔt∂xη1n+1s2+∂xη1ns2+η1n+1s2+η1ns2+Δt4+1+CΔtη2ns2,η2n+1s2≤1+CΔt1-CΔtη2ns2+CΔt1-CΔt∂xη1n+1s2+∂xη1ns2+η1n+1s2+η1ns2+Δt4.As a consequence of this,(82)η2n+1s2≤Cη2ns2+CΔt∂xη1n+1s2+∂xη1ns2+η1n+1s2+η1ns2+CΔt5.Then, summing up the previous equation for n=0 to m, it follows that(83)∑n=0mη2n+1s2≤C∑n=0mη2ns2+CΔt∑n=0m∂xη1n+1s2+∂xη1ns2+η1n+1s2+η1ns2+CΔt4.Therefore,(84)η2m+1s2≤Cη20s2+Δt∑n=0m+1∂xη1ns2+Δt∑n=0m+1η1ns2+Δt4.In order to estimate the terms ∂xη1ns, let ϕ=η1tn in (64) to get(85)ie1tn+∂x2e^1n-αu^nv^n-u^Nnv^Nn,η1tns=τ1n,η1tns.Taking imaginary part of the resulting equation, one deduces that(86)η1tns2=αIm⁡u^nv^n-u^Nnv^Nn,η1tns+Im⁡τ1n,η1tns.Therefore,(87)η1tns2≤Cη2n+1s2+η1n+1s2+η2ns2+η1ns2+CΔt4.Letting ψ=2η1tn in (64),(88)ie1tn,2η1tns+∂x2e^1n,2η1tns=αu^nv^n-u^Nnv^Nn,2η1tns+τ1n,2η1tns,and thus, simplifying the equation above and taking real part of the resulting equation, we obtain that(89)-1Δt∂xη1n+1s2-∂xη1ns2=2αReu^nv^n-u^Nnv^Nn,η1tns+2Reτ1n,η1tns.Therefore, using the estimate for η1ts obtained above, one realizes that (90)1Δt∂xη1n+1s2-∂xη1ns2≤Cη1n+1s2+η1n2+η2n+1s2+η2ns2+η1tns2+CΔt4≤Cη1n+1s2+η1ns2+η2n+1s2+η2ns2+CΔt4.Summing up n=0,…,m, we arrive at(91)∂xη1m+1s2≤C∂xη10s2+Δt∑n=0m+1η1ns2+η2ns2+CΔt4.We have from the previous results that(92)Em≔η1m+1s2+∂xη1m+1s2+η2m+1s2=η1m+1s+12+η2m+1s2≤Cη10s2+Δt∑n=0m+1η2ns2+Δt4+Cη20s2+Δt∑n=0m+1∂xη1ns2+Δt∑n=0m+1η1ns2+Δt4+C∂xη10s2+Δt∑n=0m+1η1ns2+η2ns2+Δt4≤Cη10s2+∂xη10s2+η20s2+Δt∑n=0mη1ns2+η2ns2+∂xη1ns2+Δt4+CΔt∂xη1m+1s2+CΔtη1m+1s2+CΔtη2m+1s2.We have that(93)Em1-CΔt≤Cη10s2+∂xη10s2+η20s2+Δt∑n=0mη1ns2+η2ns2+∂xη1ns2+Δt4,Em≤Cη10s+12+η20s2+CΔt4+CΔt∑n=1mEn-1.Finally, since η10s=η20s=0,(94)Em≤CΔt4+CΔt∑n=1mEn-1,and the result follows from Gronwall’s lemma.


## 5. Numerical Experiments

The purpose of this section is to present some numerical simulations using the numerical scheme described in the previous section. We point out that the technique and procedure used here for the numerical simulations are similar to the work by Muñoz Grajales in [15] about a Fourier-Galerkin numerical scheme applied to a 1D Benney-Luke-Paumond equation.We recall that any function inSN can be written as (95)u=∑j=-N/2N/2u^jϕj,with (96)u^j=1L∫0Luxϕjx¯dx.As a consequence, we have that scheme (57) can be written equivalently as(97)iu^jn+1-u^jnΔt=wj2u^jn+1+u^jn2+α2Pjun+1vn+1+Pjunvn,v^jn+1-v^jnΔt=-γiwjwjv^jn+1+v^jn2+βiwj2Pjun+12+Pjun2,subject to u^j0=u^0jv^j0=v^0j and where (98)wj=2πjL,j=-N2,…,0,…,N2,and Pj[·] denotes the operator(99)Pjg=1L∫0Lgxe-iwjxdx.Additionally, let u(n), v(n) be the approximations of the unknowns u(x,t), v(x,t), respectively, at time tn=nΔt, where Δt represents the time step of the method and u^jn, v^jn are the approximations to the Fourier transforms of the functions u and v, respectively, with respect to the variable x, evaluated at the time nΔt.Observe that, in the scheme given in (97), the unknowns u^jn+1, v^jn+1 and u(n+1), v(n+1) at tn=(n+1)Δt must be computed by iteration. To do this, we use the iterative process (100)u^jn+1,k+1=u^jn-Δtiwj2u^jn/2-Δtiα/2Pjun+1,kvn+1,k+Pjunvn1+Δtiwj2/2,v^jn+1,k+1=v^jn-Δtγiwjwjv^jn/2+Δtiwjβ/2Pjun+1,k2+Pjun21+Δtiγwj2/2,k=1,2,….To compute the initial values u^jn+1,0, v^jn+1,0, un+1,0, and vn+1,0 for this iteration, we use the explicit scheme(101)iu^jn+1,0-u^jnΔt=wj2u^jn+αPjunvn,v^jn+1,0-v^jnΔt=-γiwjwjv^jn+βiwjPjun2.In all of the numerical experiments presented, we used double precision in Matlab R2016b on a Mac platform. The approximation of the Fourier-type integral appearing in the definition of the operator Pj[·] is performed through the well-known Fast Fourier Transform (FFT) algorithm.


### 5.1. Convergence Rate in Space

To verify the numerical properties of the fully discrete scheme proposed in the present paper, we use a technique similar to that in [15]. In particular, we want to validate the spectral order of convergence in space of the Crank-Nicholson Fourier-Galerkin numerical scheme considered. In Figure 1, we present a simulation where a small time step Δt=1e-5 is fixed, L=20, and the number of points in space is gradually increased. We use the analytical travelling wave solution of the SBO system for γ=0, given by(102)ux,t=Aeiwpteicx-ct/2sech⁡Bx-ct-a0,vx,t=-A2βcsech2⁡Bx-ct-a0,where(103)A=2cαβwp-c24,B=wp-c24,with α=β=1, c=0.5, wp=1.5, and a0=10. We start with N=26 and then N is increased by 2 until we get N=29. For every value of N, we compute the numerical solution until reaching the time t=1. From the results in Figure 1, we see that the fully discrete method (57) has spectral accuracy in space (as established in Theorem 2), and the error decreases very rapidly approximately as N-6.07. We point out that this decay rate is faster than that in pure finite difference methods.Figure 1
Plot of the decimal logarithm of the maximum error againstlog10N. The time step is fixed at Δt=1e-5. We see that the plot is approximately a line with slope -6.07.


### 5.2. Convergence Rate in Time

In this section, we wish to validate numerically the rate of convergence in time for the numerical scheme (57). This numerical simulation is performed using the solitary wave solution (102) with the same model’s parameters as in the previous numerical experiment. Furthermore, we set N=212 (Δx=L/N≈5e-3) with the objective that the error in space does not dominate the total error. By starting with Δt=1/22 and decreasing the time step by 1/2 until Δt=1/26, the numerical solution is computed until reaching t=1. The results are presented in Figure 2, from where we can observe that the error in time of the numerical scheme is of order 2, in perfect agreement with Theorem 2.Figure 2
Plot of the decimal logarithm of the maximum error againstlog10Δt. The number of points in space is fixed at N=212. We see that the plot is approximately a line with slope 2.


### 5.3. Numerical Results for the Full SBO System

Finally, we illustrate the numerical scheme (57) in the case that γ=1≠0, where no analytical solution is available in the literature. We use the approximation to a travelling wave solution of the SBO system given by(104)ux,t=eiwpteicx-ct/2ϕx-ct,vx,t=ψx-ct,with ϕ, ψ being periodic real-valued functions computed through a Newton-iterative procedure, together with a collocation-Fourier method applied to the system (105)ϕ′′+-wp+c24ϕ=αϕψ,-γHψ′-cψ=βϕ2,for the parameters α=β=γ=1, c=0.5, and wp=1.5. In Figure 3, we show the plot of the approximate functions ϕ,ψ, with period L=4π/c obtained by using Newton’s procedure mentioned above, with starting point (106)ϕ0x=2cos⁡2πxL,ψ0x=cos⁡2πxL.Figure 3
Approximations of the functionsϕ,ψ in the travelling wave solution (104).We run the numerical solver (57) using the approximate travelling wave (104) at t=0 as initial data and numerical parameters N=29, Δt=1e-3, and L=4π/c≈25.13. The result is presented in Figure 4. We observe that the numerical and the expected profile coincide at t=5 with a maximum error of 2e-4, showing that the scheme proposed captures the nonlinear and dispersive characteristics of the solutions of the SBO system (1). Other numerical experiments with travelling wave solutions of the full SBO system were performed obtaining analogous results.Figure 4
Evolution of a travelling wave solution of the SBO system att=5. In solid line, u, v computed with the numerical solver (57). In pointed line, u, v for the approximate travelling wave solution (104).


### 5.4. Checking Energy Conservation

It is easy to see that the quantity(107)Et=∫0Lux,t2dxmust be conserved in time for the function u(x,t) in system (1). In the numerical experiment presented in Figure 5, we corroborate that the fully discrete scheme (57) further conserves approximately the discrete version of E(t) given by(108)Ct=Δx∑k=0N-1ukΔx,t2.The initial conditions are given by(109)ux,0=vx,0=cos⁡πx,and the period is L=2. We set other numerical parameters as α=β=γ=1 and Δt=1e-3, and the number of FFT points is N=28. Other numerical simulations conducted using the fully discrete scheme (57) with different initial conditions showed similar results.Figure 5
In this numerical experiment we see that the quantityC(t) is close to 1, as long as time evolves. This shows that the numerical solver (57) conserves approximately the L2-norm of the function u(x,t) in system (1).


## 5.1. Convergence Rate in Space

To verify the numerical properties of the fully discrete scheme proposed in the present paper, we use a technique similar to that in [15]. In particular, we want to validate the spectral order of convergence in space of the Crank-Nicholson Fourier-Galerkin numerical scheme considered. In Figure 1, we present a simulation where a small time step Δt=1e-5 is fixed, L=20, and the number of points in space is gradually increased. We use the analytical travelling wave solution of the SBO system for γ=0, given by(102)ux,t=Aeiwpteicx-ct/2sech⁡Bx-ct-a0,vx,t=-A2βcsech2⁡Bx-ct-a0,where(103)A=2cαβwp-c24,B=wp-c24,with α=β=1, c=0.5, wp=1.5, and a0=10. We start with N=26 and then N is increased by 2 until we get N=29. For every value of N, we compute the numerical solution until reaching the time t=1. From the results in Figure 1, we see that the fully discrete method (57) has spectral accuracy in space (as established in Theorem 2), and the error decreases very rapidly approximately as N-6.07. We point out that this decay rate is faster than that in pure finite difference methods.Figure 1
Plot of the decimal logarithm of the maximum error againstlog10N. The time step is fixed at Δt=1e-5. We see that the plot is approximately a line with slope -6.07.


## 5.2. Convergence Rate in Time

In this section, we wish to validate numerically the rate of convergence in time for the numerical scheme (57). This numerical simulation is performed using the solitary wave solution (102) with the same model’s parameters as in the previous numerical experiment. Furthermore, we set N=212 (Δx=L/N≈5e-3) with the objective that the error in space does not dominate the total error. By starting with Δt=1/22 and decreasing the time step by 1/2 until Δt=1/26, the numerical solution is computed until reaching t=1. The results are presented in Figure 2, from where we can observe that the error in time of the numerical scheme is of order 2, in perfect agreement with Theorem 2.Figure 2
Plot of the decimal logarithm of the maximum error againstlog10Δt. The number of points in space is fixed at N=212. We see that the plot is approximately a line with slope 2.


## 5.3. Numerical Results for the Full SBO System

Finally, we illustrate the numerical scheme (57) in the case that γ=1≠0, where no analytical solution is available in the literature. We use the approximation to a travelling wave solution of the SBO system given by(104)ux,t=eiwpteicx-ct/2ϕx-ct,vx,t=ψx-ct,with ϕ, ψ being periodic real-valued functions computed through a Newton-iterative procedure, together with a collocation-Fourier method applied to the system (105)ϕ′′+-wp+c24ϕ=αϕψ,-γHψ′-cψ=βϕ2,for the parameters α=β=γ=1, c=0.5, and wp=1.5. In Figure 3, we show the plot of the approximate functions ϕ,ψ, with period L=4π/c obtained by using Newton’s procedure mentioned above, with starting point (106)ϕ0x=2cos⁡2πxL,ψ0x=cos⁡2πxL.Figure 3
Approximations of the functionsϕ,ψ in the travelling wave solution (104).We run the numerical solver (57) using the approximate travelling wave (104) at t=0 as initial data and numerical parameters N=29, Δt=1e-3, and L=4π/c≈25.13. The result is presented in Figure 4. We observe that the numerical and the expected profile coincide at t=5 with a maximum error of 2e-4, showing that the scheme proposed captures the nonlinear and dispersive characteristics of the solutions of the SBO system (1). Other numerical experiments with travelling wave solutions of the full SBO system were performed obtaining analogous results.Figure 4
Evolution of a travelling wave solution of the SBO system att=5. In solid line, u, v computed with the numerical solver (57). In pointed line, u, v for the approximate travelling wave solution (104).


## 5.4. Checking Energy Conservation

It is easy to see that the quantity(107)Et=∫0Lux,t2dxmust be conserved in time for the function u(x,t) in system (1). In the numerical experiment presented in Figure 5, we corroborate that the fully discrete scheme (57) further conserves approximately the discrete version of E(t) given by(108)Ct=Δx∑k=0N-1ukΔx,t2.The initial conditions are given by(109)ux,0=vx,0=cos⁡πx,and the period is L=2. We set other numerical parameters as α=β=γ=1 and Δt=1e-3, and the number of FFT points is N=28. Other numerical simulations conducted using the fully discrete scheme (57) with different initial conditions showed similar results.Figure 5
In this numerical experiment we see that the quantityC(t) is close to 1, as long as time evolves. This shows that the numerical solver (57) conserves approximately the L2-norm of the function u(x,t) in system (1).


## 6. Conclusions

In this paper, we developed a rigorous analysis of the error of the semidiscrete and fully discrete formulations of a Fourier-Galerkin scheme to approximate solutions of the SBO system (1). The time-stepping method was implemented by using a second-order implicit Crank-Nicholson strategy. To the best of our knowledge, a complete error analysis of a fully discrete scheme for the SBO system in the general case γ≠0 has not been developed in previous works. The resulting accuracy and convergence rate O(Ns+2-r+Δt2) (r>s+2) of the numerical solver considered were illustrated by using a family of exact solitary wave solutions of system (1) with γ=0. Numerical experiments with the complete SBO system with γ≠0 were also presented by using some approximations of solitary wave solutions computed using a Newton-collocation scheme combined with a collocation-Fourier method.



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