# Solving the Modified Regularized Long Wave Equations via Higher Degree B-Spline Algorithm

Authors: Pshtiwan Othman Mohammed; Manar A. Alqudah; Y. S. Hamed; Artion Kashuri; Khadijah M. Abualnaja
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The current article considers the sextic B-spline collocation methods (SBCM1 and SBCM2) to approximate the solution of the modified regularized long wave (MRLW) equation. In view of this, we will study the solitary wave motion and interaction of higher (two and three) solitary waves. Also, the modified Maxwellian initial condition into solitary waves is studied. Moreover, the stability analysis of the methods has been discussed, and these will be unconditionally stable. Moreover, we have calculated the numerical conserved laws and error norms L2 and L∞ to demonstrate the efficiency and accuracy of the method. The numerical examples are presented to illustrate the applications of the methods and to compare the computed results with the other methods. The results show that our proposed methods are more accurate than the other methods.


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## Body

## 1. Introduction

The regularized long wave (RLW) equation is defined by the following nonlinear partial differential equation [1]:
(1)στ+ση+ζσση−μσηητ=0,where μ and ζ are positive parameters. This equation was first introduced by Peregrine [1] and after that by Benjamin et al. [2] to describe the behavior of the undular bore. It has also a great role in physics science, especially in physics media since it is useful in describing a phenomenon in different disciplines, such as the nonlinear transverse waves in magneto hydrodynamics waves in plasma, ion-acoustic waves in plasma, shallow water, longitudinal dispersive waves in elastic rods, phonon packets in nonlinear crystals, and pressure waves in liquids gas bubbles.There are many analytical methods to obtain the solution of theRLW equation for certain boundary and initial conditions; for example, see [2, 3]. Also, the numerical solutions of the RLW equation has been studied by many researchers via various methods, such as finite difference methods [4, 5], Fourier pseudospectral methods [6], various models of finite element methods including least square, collocation, and Galerkin methods [7–9], mesh-free method [10], and Galerkin finite element methods [11–13].The generalized form of theRLW equation is known as the GMRLW equation which is given by
(2)στ+ση+ζσpση−μσηητ=0,where p is a positive parameter. In the extension of nonlinear dispersive waves, this equation has an important role. There are many numerical methods to investigate its solution. For more details, we advise the reader to visit [14–17]. In the current attempt, we consider a special case of the GMRLW (namely, the MRLW) equation, given by
(3)στ+ση+ζσ2ση−μσηητ=0,subject to the boundary conditions (B.Cs):
(4)σρ1,τ=β1,σρ2,τ=β2,σηρ1,τ=0,σηρ2,τ=0,σηηρ1,τ=0,σηηρ2,τ=0,and the initial condition (I.C.) is taken as
(5)ση,0=fη,where fη is assumed to be localized disturbance inside the given interval. There are many authors who obtained the numerical solution of the MRLW equation; for example, Gardner et al. [18] used the cubic B-spline finite element method, Prenter [19] used variational and spline methods, and Khalifa et al. [20] used finite difference method; in [21], they used the Adomian decomposition method, they also in [22] used the collocation method, and Fazal-i-Haq et al. [23] used the quartic B-Spline collocation method to get an approximate solution of the MRLW equation.In this study, inspired by the abovementioned studies, we use the sextic B-spline collocation methods to approximate the solution of theMRLW equations (3)–(5). The rest of the paper is organized as follows. In Sections 2.1 and 2.2, we discuss the B-spline collocation methods I and II and their stability analysis on the proposed MRLW equation. Section 3 is dedicated to the numerical implementations and comparison of our obtained results with those obtained in the literature: Section 3.1 is for single solitary wave, and Sections 3.2 and 3.3 are for interactions of multiple solitary waves. A conclusion is subsequently given in Section 3.


## 2. The Methods of B-Spline Collocation

Let us partition the finite intervalρ1,ρ2 into a uniform mesh by points ηℓ,ℓ=−3,−2,⋯,J+2 such a way that ρ1=η0<η1<⋯<ηJ−1<ηJ=ρ2, where Δη=ÿ=ρ2−ρ1/J=ηℓ−ηℓ−1. Then, we state the sextic B-spline collocation methods I (SBCM1) and II (SBCM2).


### 2.1. The SBCM1

We know that the sextic B-splines are usually defined onJ+1 nodes over a given interval ρ1,ρ2 with 12 additional nodes outside the interval ρ1,ρ2. The additional nodes may be given as follows:
(6)η−6<η−5<η−4<η−3<η−2<η−1<η0,ηJ<ηJ+1<ηJ+2<ηJ+3<ηJ+4<ηJ+5<ηJ+6.The sextic B-splinesBℓ at the knots ηℓ are defined as [1]
(7)Bℓη=1ℏ6η−ηℓ−36,ηℓ−3,ηℓ−2,η−ηℓ−36−7η−ηℓ−26,ηℓ−2,ηℓ−1,η−ηℓ−36−7η−ηℓ−26+21η−ηℓ−16,ηℓ−1,ηℓ,η−ηℓ−36−7η−ηℓ−26+21η−ηℓ−16−35η−ηℓ6,ηℓ,ηℓ+1,η−ηℓ+46−7η−ηℓ+36+21η−ηℓ+26,ηℓ+1,ηℓ+2,η−ηℓ+46−7η−ηℓ+36,ηℓ+2,ηℓ+3,η−ηℓ+46,ηℓ+3,ηℓ+4,0otherwise,for ℓ=−3,−2,⋯,J+2, and the set B−3,B−2,⋯,BJ+2 of sextic B-splines can be a basis over the interval ρ1,ρ2.The approximate solutionσJη,τ of the GMRLW equation to the exact solution ση,τ will be determined as follows:
(8)σJη,τ=∑ℓ=−3J+2ϑℓτBℓη,where the time dependent parameters ϑℓτ will be determined from the sextic B-spline collocation formula of equation (3).In view of equation (8) and Table 1, the nodal values σi,i=0,1,⋯,5 at the knots ηℓ can be found as
(9)σℓ=ϑℓ+2+57ϑℓ+1+302ϑℓ+302ϑℓ−1+57ϑℓ−2+ϑℓ−3,σ′ℓ=6ÿϑℓ+2+25ϑℓ+1+40ϑℓ−40ϑℓ−1−25ϑℓ−2−ϑℓ−3,σ″ℓ=30ÿ2ϑℓ+2+9ϑℓ+1−10ϑℓ−10ϑℓ−1+9ϑℓ−2+ϑℓ−3,σℓ3=120ÿ3ϑℓ+2+ϑℓ+1−8ϑℓ+8ϑℓ−1−ϑℓ−2−ϑℓ−3,σℓ4=360ÿ4ϑℓ+2−3ϑℓ+1+2ϑℓ−2ϑℓ−1+3ϑℓ−2−ϑℓ−3,σℓ5=720ÿ5−ϑℓ+2+5ϑℓ+1−10ϑℓ+10ϑℓ−1−5ϑℓ−2+ϑℓ−3.Table 1
The sextic B-spline values at the grid points.
ηηℓ−3ηℓ−2ηℓ−1ηℓηℓ+1ηℓ+2Qℓ157302302571ℏQ′ℓ6150−240−240−150−6ℏ2Q″ℓ30270−300−30027030ℏ3Qℓ3120120−960960−120−120ℏ4Qℓ4360−1080720720−1080360ℏ5Qℓ5720236007200−72003600−720Now, we implement the collocation method at the nodesηi,i=0,1,⋯,J. Also, we substitute the nodal variables σℓ and its derivatives at the knots ηi in equation (9) into equation (3); then, we get the following system of nonlinear ordinary differential equations:
(10)ϑℓ+2∘+57ϑℓ+1∘+302ϑℓ∘+302ϑℓ−1∘+57ϑℓ−2∘+ϑℓ−3∘+6dℏϑℓ+2+25ϑℓ+1+40ϑℓ−40ϑℓ−1−25ϑℓ−2−ϑℓ−3−30μℏ2ϑℓ+2∘+9ϑℓ+1∘−10ϑℓ∘−10ϑℓ−1∘+9ϑℓ−2∘+ϑℓ−3∘=0,where d=1+ζzℓ=1+ζϑℓ−3+57ϑℓ−2+302ϑℓ−1+302ϑℓ+57ϑℓ+1+ϑℓ+2p and ∘ denote derivative with respect to time.The unknown parametersϑℓ and ϑℓ∘ are linearly interpolated between n and n+1 (n and n+1 are two time levels) via the the Crank-Nicolson formula and the usual forward difference formula, respectively, as follows:
(11)ϑℓ=ϑℓn+1+ϑℓn2,ϑℓ∘=ϑℓn+1−ϑℓnΔτ,where ϑℓn denotes the parameters at time nΔτ. Then, by making use of equation (11), it follows that
(12)αℓ1ϑℓ−3n+1+αℓ2ϑℓ−2n+1+αℓ3ϑℓ−1n+1+αℓ4ϑℓn+1+αℓ5ϑℓ+1n+1+αℓ6ϑℓ+2n+1=αℓ6ϑℓ−3n+αℓ5ϑℓ−2n+αℓ4ϑℓ−1n+αℓ3ϑℓn+αℓ2ϑℓ+1n+αℓ1ϑℓ+2n,where
(13)αℓ1=ℏ2−3ℏdΔτ−30μ,αℓ2=57ℏ2−75ℏdΔτ−270μ,αℓ3=302ℏ2−120ℏdΔτ+300μ,αℓ4=302ℏ2+120ℏdΔτ+300μ,αℓ5=57ℏ2+75ℏdΔτ−270μ,αℓ6=ℏ2+3ℏdΔτ−30μ.The system (12) consisting of the J+1 equations with J+6 unknown parameters. It can be solved uniquely if we eliminate the parameters ϑ−3n+1,ϑ−2n+1,ϑ−1n+1,ϑJ+1n+1,ϑJ+2n+1 by using the five B.Cs σρ1,τ=β1,σρ2,τ=β2,σηρ1,τ=σηρ2,τ=σηηρ1,τ=0; that is,
(14)ϑ−3n+1+57ϑ−2n+1+302ϑ−1n+1+302ϑ0n+1+57ϑ1n+1+ϑ2n+1=β1,6ℏϑ2n+1+25ϑ1n+1+40ϑ0n+1−40ϑ−1n+1−25ϑ−2n+1−ϑ−3n+1=0,30ℏ2ϑ2n+1+9ϑ1n+1−10ϑ0n+1−10ϑ−1n+1+9ϑ−2n+1+ϑ−3n+1=0,ϑJ−3n+1+57ϑJ−2n+1+302ϑJ−1n+1+302ϑJn+1+57ϑJ+1n+1+ϑJ+2n+1=β2,6ℏϑJ+2n+1+25ϑJ+1n+1+40ϑJ+1n+1−40ϑJ−1n+1−25ϑJ−2n+1−ϑJ−3n+1=0.Consequently, we get a matrix system of dimensionJ+1×J+1, and one can solve it easily by using a variant of the Thomas algorithm.To deal with nonlinearity in (12) at each time step, we carry out the following corrector methods:
(1)
Approximatingϑn+1 by using the following simple corrector:
(15)ϑ∗n+1=ϑn+ϑn+1−ϑn2(2)
As an approximation forϑn+1, we use ϑ∗n+1(3)
Repeating this procedure twice at time stepn+1 to refine ϑn+1(4)
Repeating this procedure at each time step along the execution of the programThe iterative procedureϑℓn in (4) can start by determining the initial parameters ϑℓ0, and it can be determined by making use of the B.Cs (4), I.C. (5), and the following requirements:
(16)σ′Jρ1,0=ϑ20+25ϑ10+40ϑ00−40ϑ−10−25ϑ−20−ϑ−30=0,σ′′Jρ1,0=ϑ20+9ϑ10−10ϑ00−10ϑ−10+9ϑ−20+ϑ−30=0,σJ3ρ1,0=ϑℓ+20+ϑℓ+10−8ϑℓ0+8ϑℓ−10−ϑℓ−20−ϑℓ−30=0,σJηi,0=ϑj−30+57ϑj−20+302ϑj−10+302ϑj0+57ϑj+10+ϑj+20=fηj,j=0,1,⋯,J,σ′Jρ1,0=ϑJ+20+25ϑJ+10+40ϑJ+10−40ϑJ−10−25ϑJ−20−ϑJ−30=0,σ′′Jρ1,0=ϑJ+20+9ϑJ+10−10ϑJ0−10ϑJ−10+9ϑJ−20+ϑJ−30=0.Again, one can solve it by a variant of the Thomas algorithm and the approximate solutionσJiη,τ,i=0,1,⋯,5 that can be obtained from equation (9).Now, we can apply the von Neumann stability method to establish the stability of the scheme (12), but the von Neumann stability method is applicable to linear schemes; so, we shall line arise the nonlinear term σpση by taking σ as a constant value k, and thus the nonlinear term becomes σp=zℓ=kp. Then, by substitution the Fourier mode ϑℓn=p^neiℓφ into our linearized form of equation (12) with writing p^n+1=qp^n, we get
(17)q=A1+iA2A3+iA4,where q is growth factor and
(18)A1=αℓ6cos3φ+αℓ5+αℓ1cos2φ+αℓ4+αℓ2cosφ+αℓ3,A2=−αℓ6sin3φ−αℓ5−αℓ1sin2φ−αℓ4−αℓ2sinφ,A3=αℓ1cos3φ+αℓ2+αℓ6cos2φ+αℓ3+αℓ5cosφ+αℓ4,A4=−αℓ1sin3φ−αℓ2−αℓ6sin2φ−αℓ3−αℓ2sinφ.Thanks to Python software, we obtain same expressions forA12+A22 and A32+A42 in the following form:
(19)A12+A22=A32+A42=sinφ−245ℏ2+195ℏdΔτ+570μ−sin2φ56ℏ2+78ℏdΔτ−240μ+sin3φ−ℏ2−3ℏdΔτ+30μ2+cosφ359ℏ2+45ℏdΔτ+30μ+cos2φ58ℏ2+72ℏdΔτ−300μ+cos3φℏ2+3ℏdΔτ−30μ+302ℏ2−120ℏdΔτ+300μ2,in order for the magnitude of the growth factor that is∣q∣=1, and thus the linearized numerical algorithm for the GMRLW equation will be unconditionally stable.


### 2.2. The SBCM2

One can split equation (3) as a system of partial differential equation:
(20)σ−μσηητ+2ζσpση=0,(21)σ−μσηητ+2ση=0.To applied the collocation approach for system (20) and (21), we identify the collocation points with the nodes ηi,i=0,1,⋯,J. If we put the approximation (7) into equations (20) and (21), we can obtain the following system of 1st order ordinary differential equations:
(22)ϑℓ+2∘+57ϑℓ+1∘+302ϑℓ∘+302ϑℓ−1∘+57ϑℓ−2∘+ϑℓ−3∘−30μℏ2ϑℓ+2∘+9ϑℓ+1∘−10ϑℓ∘−10ϑℓ−1∘+9ϑℓ−2∘+ϑℓ−3∘=0+12ζzℓℏϑℓ+2+25ϑℓ+1+40ϑℓ−40ϑℓ−1−25ϑℓ−2−ϑℓ−3,(23)ϑℓ+2∘+57ϑℓ+1∘+302ϑℓ∘+302ϑℓ−1∘+57ϑℓ−2∘+ϑℓ−3∘−30μℏ2ϑℓ+2∘+9ϑℓ+1∘−10ϑℓ∘−10ϑℓ−1∘+9ϑℓ−2∘+ϑℓ−3∘=0+12ℏϑℓ+2+25ϑℓ+1+40ϑℓ−40ϑℓ−1−25ϑℓ−2−ϑℓ−3,where ∘ is defined in (10), and nonlinearity term is zℓ=ϑℓ−3+57ϑℓ−2+302ϑℓ−1+302ϑℓ+57ϑℓ+1+ϑℓ+2p. Approximating the parameters ϑℓ between n and n+1/2 by using the Crank-Nicolson formula and ϑℓ∘ by using the finite difference rule as follows:
(24)ϑℓ=ϑℓn+1/2+ϑℓn4,ϑℓ∘=ϑℓn+1/2−ϑℓnΔτ.Thus, equation (22) becomes
(25)α1ϑℓ−3n+1/2+α2ϑℓ−2n+1/2+α3ϑℓ−1n+1/2+α4ϑℓn+1/2+α5ϑℓ+1n+1/2+α6ϑℓ+2n+1/2=α6ϑℓ−3n+α5ϑℓ−2n+α4ϑℓ−1n+α3ϑℓn+α2ϑℓ+1n+α1ϑℓ+2n,where
(26)α1=ℏ2−3ℏζzℓΔτ−30μ,α2=57ℏ2−75ℏζzℓΔτ−270μ,α3=302ℏ2−120ℏζzℓΔτ+300μ,α4=302ℏ2+120ℏζzℓΔτ+300μ,α5=57ℏ2+75ℏζzℓΔτ−270μ,α6=ℏ2+3ℏζzℓΔτ−30μ.Analogously, in view of the Crank-Nicolson and forward finite difference approaches in time, both parametersϑℓ and ϑℓ∘ are linearly interpolated between two time levels n+1/2 and n+1, respectively, as follows:
(27)ϑℓ=ϑℓn+1+ϑℓn+1/24,ϑℓ∘=ϑℓn+1−ϑℓn+1/2Δτ.Thus, equation (23) becomes
(28)α7ϑℓ−3n+1+α8ϑℓ−2n+1+α9ϑℓ−1n+1+α10ϑℓn+1+α11ϑℓ+1n+1+α12ϑℓ+2n+1=α12ϑℓ−3n+1/2+α11ϑℓ−2n+1/2+α10ϑℓ−1n+1/2+α9ϑℓn+1/2+α8ϑℓ+1n+1/2+α7ϑℓ+2n+1/2,where
(29)α7=ℏ2−30μ−3ℏΔτ,α8=57ℏ2−75ℏΔτ−270μ,α9=302ℏ2+300μ−120ℏΔτ,α10=302ℏ2+120ℏΔτ+300μ,α11=57ℏ2−270μ+75ℏΔτ,α12=ℏ2+3ℏΔτ−30μ.Equations (25) and (28) constitute the numerical algorithms for the GMRLW equation. We can remove the nonlinearity terms occurring in equation (25) by replacing ϑℓ by ϑℓn in zℓ, and thus the equation (25) will be linearized.Therefore, we see that the iterative systems of equations (25) and (28) consist the J+1 equations in the J+6 unknown parameters, and one can solve it by eliminating the parameters ϑ−3j,ϑ−2j,ϑ−1j,ϑJ+1j,ϑJ+2j,j=n+1/2,n+1 by making use of the five B.Cs σρ1,τ=β1,σρ2,τ=β2,σηρ1,τ=σηρ2,τ=σηηρ1,τ=0; then, we can obtain
(30)ϑ−3j+57ϑ−2j+302ϑ−1j+302ϑ0j+57ϑ1j+ϑ2j=β1,6ℏϑ2j+25ϑ1j+40ϑ0j−40ϑ−1j−25ϑ−2j−ϑ−3j=0,30ℏ2ϑ2j+9ϑ1j−10ϑ0j−10ϑ−1j+9ϑ−2j+ϑ−3j=0,ϑJ−3j+57ϑJ−2j+302ϑJ−1j+302ϑJj+57ϑJ+1j+ϑJ+2j=β2,6ℏϑJ+2j+25ϑJ+1j+40ϑJ+1j−40ϑJ−1j−25ϑJ−2j−ϑJ−3j=0.Thus, we get a matrix system of dimensionJ+1×J+1, and we can easily solve it by using a variant of the Thomas algorithm.To deal with nonlinearity in (28) at each time step, we carry out the following corrector procedure:
(31)ϑ∗ij=ϑin+ϑij−ϑin2.This iterative scheme is executed two times by determiningϑ∗ij for ϑij, where j=n+1/2,n+1.We start the time evolution of theϑℓj,j=n+1/2,n+1 using (25) and (28) by calculating initial parameters ϑℓ0. Therefore, the approximate solution (8) must agree with the I.C. at the knots, and this leads to J+1 equations. Also, the further five equations can be obtained by using the derivatives of σJ in (8) at the ends:
(32)σ′Jρ1,0=ϑ20+25ϑ10+40ϑ00−40ϑ−10−25ϑ−20−ϑ−30=0,σ′′Jρ1,0=ϑ20+9ϑ10−10ϑ00−10ϑ−10+9ϑ−20+ϑ−30=0,σJ3ρ1,0=ϑℓ+20+ϑℓ+10−8ϑℓ0+8ϑℓ−10−ϑℓ−20−ϑℓ−30=0,σJηi,0=ϑj−30+57ϑj−20+302ϑj−10+302ϑj0+57ϑj+10+ϑj+20=fηj,j=0,1,⋯,J,σ′Jρ1,0=ϑJ+20+25ϑJ+10+40ϑJ+10−40ϑJ−10−25ϑJ−20−ϑJ−30=0,σ′′Jρ1,0=ϑJ+20+9ϑJ+10−10ϑJ0−10ϑJ−10+9ϑJ−20+ϑJ−30=0.Consequently, the parametersϑi0,i=−3,−2,⋯,J+2 will be determined as solution of a matrix equation.To establish the stability of the scheme (25), we carry out the von Neumann stability scheme by linearizing the nonlinear term σpση by taking σ as a constant k so that σp becomes σp=zℓ=kp. By substitution the Fourier mode of ϑℓn=p^neiℓφ into our linearized form of equation (25) and by writing p^n+1=qp^n in the resulting iterative equation, we can deduce
(33)q=A1+iA2A3+iA4,where
(34)A1=α3+α1+α5cos2φ+α2+α4cosφ+α6cos3φ,A2=α1−α5sin2φ+α2−α4sinφ−α6sin3φ,A3=α4+α2+α6cos2φ+α1cos3φ+α3+α5cosφ,A4=α2−α3sinφ+α6−α2sin2φ−α1sin3φ.Here, note that the von Neumann condition is fulfilled; that is,∣q∣≤1. This affects the difference scheme (25) to be unconditionally stable. In the same way, we can show that the difference equation (28) can be unconditionally stable as well.


## 2.1. The SBCM1

We know that the sextic B-splines are usually defined onJ+1 nodes over a given interval ρ1,ρ2 with 12 additional nodes outside the interval ρ1,ρ2. The additional nodes may be given as follows:
(6)η−6<η−5<η−4<η−3<η−2<η−1<η0,ηJ<ηJ+1<ηJ+2<ηJ+3<ηJ+4<ηJ+5<ηJ+6.The sextic B-splinesBℓ at the knots ηℓ are defined as [1]
(7)Bℓη=1ℏ6η−ηℓ−36,ηℓ−3,ηℓ−2,η−ηℓ−36−7η−ηℓ−26,ηℓ−2,ηℓ−1,η−ηℓ−36−7η−ηℓ−26+21η−ηℓ−16,ηℓ−1,ηℓ,η−ηℓ−36−7η−ηℓ−26+21η−ηℓ−16−35η−ηℓ6,ηℓ,ηℓ+1,η−ηℓ+46−7η−ηℓ+36+21η−ηℓ+26,ηℓ+1,ηℓ+2,η−ηℓ+46−7η−ηℓ+36,ηℓ+2,ηℓ+3,η−ηℓ+46,ηℓ+3,ηℓ+4,0otherwise,for ℓ=−3,−2,⋯,J+2, and the set B−3,B−2,⋯,BJ+2 of sextic B-splines can be a basis over the interval ρ1,ρ2.The approximate solutionσJη,τ of the GMRLW equation to the exact solution ση,τ will be determined as follows:
(8)σJη,τ=∑ℓ=−3J+2ϑℓτBℓη,where the time dependent parameters ϑℓτ will be determined from the sextic B-spline collocation formula of equation (3).In view of equation (8) and Table 1, the nodal values σi,i=0,1,⋯,5 at the knots ηℓ can be found as
(9)σℓ=ϑℓ+2+57ϑℓ+1+302ϑℓ+302ϑℓ−1+57ϑℓ−2+ϑℓ−3,σ′ℓ=6ÿϑℓ+2+25ϑℓ+1+40ϑℓ−40ϑℓ−1−25ϑℓ−2−ϑℓ−3,σ″ℓ=30ÿ2ϑℓ+2+9ϑℓ+1−10ϑℓ−10ϑℓ−1+9ϑℓ−2+ϑℓ−3,σℓ3=120ÿ3ϑℓ+2+ϑℓ+1−8ϑℓ+8ϑℓ−1−ϑℓ−2−ϑℓ−3,σℓ4=360ÿ4ϑℓ+2−3ϑℓ+1+2ϑℓ−2ϑℓ−1+3ϑℓ−2−ϑℓ−3,σℓ5=720ÿ5−ϑℓ+2+5ϑℓ+1−10ϑℓ+10ϑℓ−1−5ϑℓ−2+ϑℓ−3.Table 1
The sextic B-spline values at the grid points.
ηηℓ−3ηℓ−2ηℓ−1ηℓηℓ+1ηℓ+2Qℓ157302302571ℏQ′ℓ6150−240−240−150−6ℏ2Q″ℓ30270−300−30027030ℏ3Qℓ3120120−960960−120−120ℏ4Qℓ4360−1080720720−1080360ℏ5Qℓ5720236007200−72003600−720Now, we implement the collocation method at the nodesηi,i=0,1,⋯,J. Also, we substitute the nodal variables σℓ and its derivatives at the knots ηi in equation (9) into equation (3); then, we get the following system of nonlinear ordinary differential equations:
(10)ϑℓ+2∘+57ϑℓ+1∘+302ϑℓ∘+302ϑℓ−1∘+57ϑℓ−2∘+ϑℓ−3∘+6dℏϑℓ+2+25ϑℓ+1+40ϑℓ−40ϑℓ−1−25ϑℓ−2−ϑℓ−3−30μℏ2ϑℓ+2∘+9ϑℓ+1∘−10ϑℓ∘−10ϑℓ−1∘+9ϑℓ−2∘+ϑℓ−3∘=0,where d=1+ζzℓ=1+ζϑℓ−3+57ϑℓ−2+302ϑℓ−1+302ϑℓ+57ϑℓ+1+ϑℓ+2p and ∘ denote derivative with respect to time.The unknown parametersϑℓ and ϑℓ∘ are linearly interpolated between n and n+1 (n and n+1 are two time levels) via the the Crank-Nicolson formula and the usual forward difference formula, respectively, as follows:
(11)ϑℓ=ϑℓn+1+ϑℓn2,ϑℓ∘=ϑℓn+1−ϑℓnΔτ,where ϑℓn denotes the parameters at time nΔτ. Then, by making use of equation (11), it follows that
(12)αℓ1ϑℓ−3n+1+αℓ2ϑℓ−2n+1+αℓ3ϑℓ−1n+1+αℓ4ϑℓn+1+αℓ5ϑℓ+1n+1+αℓ6ϑℓ+2n+1=αℓ6ϑℓ−3n+αℓ5ϑℓ−2n+αℓ4ϑℓ−1n+αℓ3ϑℓn+αℓ2ϑℓ+1n+αℓ1ϑℓ+2n,where
(13)αℓ1=ℏ2−3ℏdΔτ−30μ,αℓ2=57ℏ2−75ℏdΔτ−270μ,αℓ3=302ℏ2−120ℏdΔτ+300μ,αℓ4=302ℏ2+120ℏdΔτ+300μ,αℓ5=57ℏ2+75ℏdΔτ−270μ,αℓ6=ℏ2+3ℏdΔτ−30μ.The system (12) consisting of the J+1 equations with J+6 unknown parameters. It can be solved uniquely if we eliminate the parameters ϑ−3n+1,ϑ−2n+1,ϑ−1n+1,ϑJ+1n+1,ϑJ+2n+1 by using the five B.Cs σρ1,τ=β1,σρ2,τ=β2,σηρ1,τ=σηρ2,τ=σηηρ1,τ=0; that is,
(14)ϑ−3n+1+57ϑ−2n+1+302ϑ−1n+1+302ϑ0n+1+57ϑ1n+1+ϑ2n+1=β1,6ℏϑ2n+1+25ϑ1n+1+40ϑ0n+1−40ϑ−1n+1−25ϑ−2n+1−ϑ−3n+1=0,30ℏ2ϑ2n+1+9ϑ1n+1−10ϑ0n+1−10ϑ−1n+1+9ϑ−2n+1+ϑ−3n+1=0,ϑJ−3n+1+57ϑJ−2n+1+302ϑJ−1n+1+302ϑJn+1+57ϑJ+1n+1+ϑJ+2n+1=β2,6ℏϑJ+2n+1+25ϑJ+1n+1+40ϑJ+1n+1−40ϑJ−1n+1−25ϑJ−2n+1−ϑJ−3n+1=0.Consequently, we get a matrix system of dimensionJ+1×J+1, and one can solve it easily by using a variant of the Thomas algorithm.To deal with nonlinearity in (12) at each time step, we carry out the following corrector methods:
(1)
Approximatingϑn+1 by using the following simple corrector:
(15)ϑ∗n+1=ϑn+ϑn+1−ϑn2(2)
As an approximation forϑn+1, we use ϑ∗n+1(3)
Repeating this procedure twice at time stepn+1 to refine ϑn+1(4)
Repeating this procedure at each time step along the execution of the programThe iterative procedureϑℓn in (4) can start by determining the initial parameters ϑℓ0, and it can be determined by making use of the B.Cs (4), I.C. (5), and the following requirements:
(16)σ′Jρ1,0=ϑ20+25ϑ10+40ϑ00−40ϑ−10−25ϑ−20−ϑ−30=0,σ′′Jρ1,0=ϑ20+9ϑ10−10ϑ00−10ϑ−10+9ϑ−20+ϑ−30=0,σJ3ρ1,0=ϑℓ+20+ϑℓ+10−8ϑℓ0+8ϑℓ−10−ϑℓ−20−ϑℓ−30=0,σJηi,0=ϑj−30+57ϑj−20+302ϑj−10+302ϑj0+57ϑj+10+ϑj+20=fηj,j=0,1,⋯,J,σ′Jρ1,0=ϑJ+20+25ϑJ+10+40ϑJ+10−40ϑJ−10−25ϑJ−20−ϑJ−30=0,σ′′Jρ1,0=ϑJ+20+9ϑJ+10−10ϑJ0−10ϑJ−10+9ϑJ−20+ϑJ−30=0.Again, one can solve it by a variant of the Thomas algorithm and the approximate solutionσJiη,τ,i=0,1,⋯,5 that can be obtained from equation (9).Now, we can apply the von Neumann stability method to establish the stability of the scheme (12), but the von Neumann stability method is applicable to linear schemes; so, we shall line arise the nonlinear term σpση by taking σ as a constant value k, and thus the nonlinear term becomes σp=zℓ=kp. Then, by substitution the Fourier mode ϑℓn=p^neiℓφ into our linearized form of equation (12) with writing p^n+1=qp^n, we get
(17)q=A1+iA2A3+iA4,where q is growth factor and
(18)A1=αℓ6cos3φ+αℓ5+αℓ1cos2φ+αℓ4+αℓ2cosφ+αℓ3,A2=−αℓ6sin3φ−αℓ5−αℓ1sin2φ−αℓ4−αℓ2sinφ,A3=αℓ1cos3φ+αℓ2+αℓ6cos2φ+αℓ3+αℓ5cosφ+αℓ4,A4=−αℓ1sin3φ−αℓ2−αℓ6sin2φ−αℓ3−αℓ2sinφ.Thanks to Python software, we obtain same expressions forA12+A22 and A32+A42 in the following form:
(19)A12+A22=A32+A42=sinφ−245ℏ2+195ℏdΔτ+570μ−sin2φ56ℏ2+78ℏdΔτ−240μ+sin3φ−ℏ2−3ℏdΔτ+30μ2+cosφ359ℏ2+45ℏdΔτ+30μ+cos2φ58ℏ2+72ℏdΔτ−300μ+cos3φℏ2+3ℏdΔτ−30μ+302ℏ2−120ℏdΔτ+300μ2,in order for the magnitude of the growth factor that is∣q∣=1, and thus the linearized numerical algorithm for the GMRLW equation will be unconditionally stable.


## 2.2. The SBCM2

One can split equation (3) as a system of partial differential equation:
(20)σ−μσηητ+2ζσpση=0,(21)σ−μσηητ+2ση=0.To applied the collocation approach for system (20) and (21), we identify the collocation points with the nodes ηi,i=0,1,⋯,J. If we put the approximation (7) into equations (20) and (21), we can obtain the following system of 1st order ordinary differential equations:
(22)ϑℓ+2∘+57ϑℓ+1∘+302ϑℓ∘+302ϑℓ−1∘+57ϑℓ−2∘+ϑℓ−3∘−30μℏ2ϑℓ+2∘+9ϑℓ+1∘−10ϑℓ∘−10ϑℓ−1∘+9ϑℓ−2∘+ϑℓ−3∘=0+12ζzℓℏϑℓ+2+25ϑℓ+1+40ϑℓ−40ϑℓ−1−25ϑℓ−2−ϑℓ−3,(23)ϑℓ+2∘+57ϑℓ+1∘+302ϑℓ∘+302ϑℓ−1∘+57ϑℓ−2∘+ϑℓ−3∘−30μℏ2ϑℓ+2∘+9ϑℓ+1∘−10ϑℓ∘−10ϑℓ−1∘+9ϑℓ−2∘+ϑℓ−3∘=0+12ℏϑℓ+2+25ϑℓ+1+40ϑℓ−40ϑℓ−1−25ϑℓ−2−ϑℓ−3,where ∘ is defined in (10), and nonlinearity term is zℓ=ϑℓ−3+57ϑℓ−2+302ϑℓ−1+302ϑℓ+57ϑℓ+1+ϑℓ+2p. Approximating the parameters ϑℓ between n and n+1/2 by using the Crank-Nicolson formula and ϑℓ∘ by using the finite difference rule as follows:
(24)ϑℓ=ϑℓn+1/2+ϑℓn4,ϑℓ∘=ϑℓn+1/2−ϑℓnΔτ.Thus, equation (22) becomes
(25)α1ϑℓ−3n+1/2+α2ϑℓ−2n+1/2+α3ϑℓ−1n+1/2+α4ϑℓn+1/2+α5ϑℓ+1n+1/2+α6ϑℓ+2n+1/2=α6ϑℓ−3n+α5ϑℓ−2n+α4ϑℓ−1n+α3ϑℓn+α2ϑℓ+1n+α1ϑℓ+2n,where
(26)α1=ℏ2−3ℏζzℓΔτ−30μ,α2=57ℏ2−75ℏζzℓΔτ−270μ,α3=302ℏ2−120ℏζzℓΔτ+300μ,α4=302ℏ2+120ℏζzℓΔτ+300μ,α5=57ℏ2+75ℏζzℓΔτ−270μ,α6=ℏ2+3ℏζzℓΔτ−30μ.Analogously, in view of the Crank-Nicolson and forward finite difference approaches in time, both parametersϑℓ and ϑℓ∘ are linearly interpolated between two time levels n+1/2 and n+1, respectively, as follows:
(27)ϑℓ=ϑℓn+1+ϑℓn+1/24,ϑℓ∘=ϑℓn+1−ϑℓn+1/2Δτ.Thus, equation (23) becomes
(28)α7ϑℓ−3n+1+α8ϑℓ−2n+1+α9ϑℓ−1n+1+α10ϑℓn+1+α11ϑℓ+1n+1+α12ϑℓ+2n+1=α12ϑℓ−3n+1/2+α11ϑℓ−2n+1/2+α10ϑℓ−1n+1/2+α9ϑℓn+1/2+α8ϑℓ+1n+1/2+α7ϑℓ+2n+1/2,where
(29)α7=ℏ2−30μ−3ℏΔτ,α8=57ℏ2−75ℏΔτ−270μ,α9=302ℏ2+300μ−120ℏΔτ,α10=302ℏ2+120ℏΔτ+300μ,α11=57ℏ2−270μ+75ℏΔτ,α12=ℏ2+3ℏΔτ−30μ.Equations (25) and (28) constitute the numerical algorithms for the GMRLW equation. We can remove the nonlinearity terms occurring in equation (25) by replacing ϑℓ by ϑℓn in zℓ, and thus the equation (25) will be linearized.Therefore, we see that the iterative systems of equations (25) and (28) consist the J+1 equations in the J+6 unknown parameters, and one can solve it by eliminating the parameters ϑ−3j,ϑ−2j,ϑ−1j,ϑJ+1j,ϑJ+2j,j=n+1/2,n+1 by making use of the five B.Cs σρ1,τ=β1,σρ2,τ=β2,σηρ1,τ=σηρ2,τ=σηηρ1,τ=0; then, we can obtain
(30)ϑ−3j+57ϑ−2j+302ϑ−1j+302ϑ0j+57ϑ1j+ϑ2j=β1,6ℏϑ2j+25ϑ1j+40ϑ0j−40ϑ−1j−25ϑ−2j−ϑ−3j=0,30ℏ2ϑ2j+9ϑ1j−10ϑ0j−10ϑ−1j+9ϑ−2j+ϑ−3j=0,ϑJ−3j+57ϑJ−2j+302ϑJ−1j+302ϑJj+57ϑJ+1j+ϑJ+2j=β2,6ℏϑJ+2j+25ϑJ+1j+40ϑJ+1j−40ϑJ−1j−25ϑJ−2j−ϑJ−3j=0.Thus, we get a matrix system of dimensionJ+1×J+1, and we can easily solve it by using a variant of the Thomas algorithm.To deal with nonlinearity in (28) at each time step, we carry out the following corrector procedure:
(31)ϑ∗ij=ϑin+ϑij−ϑin2.This iterative scheme is executed two times by determiningϑ∗ij for ϑij, where j=n+1/2,n+1.We start the time evolution of theϑℓj,j=n+1/2,n+1 using (25) and (28) by calculating initial parameters ϑℓ0. Therefore, the approximate solution (8) must agree with the I.C. at the knots, and this leads to J+1 equations. Also, the further five equations can be obtained by using the derivatives of σJ in (8) at the ends:
(32)σ′Jρ1,0=ϑ20+25ϑ10+40ϑ00−40ϑ−10−25ϑ−20−ϑ−30=0,σ′′Jρ1,0=ϑ20+9ϑ10−10ϑ00−10ϑ−10+9ϑ−20+ϑ−30=0,σJ3ρ1,0=ϑℓ+20+ϑℓ+10−8ϑℓ0+8ϑℓ−10−ϑℓ−20−ϑℓ−30=0,σJηi,0=ϑj−30+57ϑj−20+302ϑj−10+302ϑj0+57ϑj+10+ϑj+20=fηj,j=0,1,⋯,J,σ′Jρ1,0=ϑJ+20+25ϑJ+10+40ϑJ+10−40ϑJ−10−25ϑJ−20−ϑJ−30=0,σ′′Jρ1,0=ϑJ+20+9ϑJ+10−10ϑJ0−10ϑJ−10+9ϑJ−20+ϑJ−30=0.Consequently, the parametersϑi0,i=−3,−2,⋯,J+2 will be determined as solution of a matrix equation.To establish the stability of the scheme (25), we carry out the von Neumann stability scheme by linearizing the nonlinear term σpση by taking σ as a constant k so that σp becomes σp=zℓ=kp. By substitution the Fourier mode of ϑℓn=p^neiℓφ into our linearized form of equation (25) and by writing p^n+1=qp^n in the resulting iterative equation, we can deduce
(33)q=A1+iA2A3+iA4,where
(34)A1=α3+α1+α5cos2φ+α2+α4cosφ+α6cos3φ,A2=α1−α5sin2φ+α2−α4sinφ−α6sin3φ,A3=α4+α2+α6cos2φ+α1cos3φ+α3+α5cosφ,A4=α2−α3sinφ+α6−α2sin2φ−α1sin3φ.Here, note that the von Neumann condition is fulfilled; that is,∣q∣≤1. This affects the difference scheme (25) to be unconditionally stable. In the same way, we can show that the difference equation (28) can be unconditionally stable as well.


## 3. Numerical Calculations

Here, numerical tests are presented to demonstrate the performance of our proposed algorithm for single and interactions of multiple solitary waves. Also, the modified MaxwellianI.C.s are pointed out to generate a train of solitary waves. Furthermore, the accuracy of the presented schemes is measured in terms of the following discrete error norms L2 and L∞:
(35)L2=ℏ∑i=0Jσiexact−σJi2,L∞=σexact−σJ∞=maxiσiexact−σJi.The conservation properties of theMRLW equation related to energy, mass, and momentum can be determined by finding the three basic invariants [24, 25]:
(36)χ1=∫ρ1ρ2σdx,χ2=∫ρ1ρ2σ2+μση2dx,χ3=∫ρ1ρ2σ4−6ζμση2dx.


### 3.1. Single Solitary Wave

Letη0 be any arbitrary constant. Then, the exact solution of the solitary wave of the MRLW equation is given as follows [20]:
(37)ση,τ=6λζsechλμλ+1η−λ+1τ−η0.The modified MaxwellianI.C. is defined by
(38)ση,0=6λζsechλμλ+1η−η0,and the B.Cs can be concluded from the exact equation.We chooseζ=μ=1,λ=0.1,ℏ=0.125,Δτ=0.1,η0=0,−40≤η≤60 so that we can compare our results with results in [20, 23]. The program is executed up to times τ=10 to find error norms and the invariants χ1,χ2,χ3 at different times, and the results are given in Table 2. From Table 2, one can observe that the predicted error norms L2 and L∞ are smaller than those obtained in [20, 23], and also the invariants χ1,χ2, and χ3 are sanely in good agreement with their exact values. The solutions at τ=0,20 and the motion of the solitary wave along with the interval −40≤η≤60 with 0≤τ≤20 to the right are illustrated in Figure 1. Moreover, the error variations are demonstrated for the proposed algorithms SBCM1 and SBCM2 in Figure 2 at time τ=10. Consequently, we can observe from Table 2 that the results obtained by the SBCM2 are more accurate than those obtained by the SBCM1.Table 2
Error norms and invariants for the single solitary wave for the above parameters.
TimeMethodL2×104L∞×104χ1χ2χ3Analytical7.8098752.1298870.1302500SBCM10.00.07.8097022.1298860.130251SBCM20.00.07.8097022.1298860.1302511SBCM10.0008990.0004977.8097252.1298870.130251SBCM20.0019820.0015887.8097252.1298870.1302512SBCM10.0004810.0006817.8097442.1298870.130250SBCM20.0009860.0009667.8097442.1298870.1302503SBCM10.0009790.0003817.8097672.1298870.130250SBCM20.0010810.0021247.8097672.1298870.1302504SBCM10.0025190.0009947.8097832.1298870.130250SBCM20.0069570.0040667.8097832.1298870.1302505SBCM10.0077380.0057447.8098002.1298870.130250SBCM20.0098860.0087597.8098002.1298870.1302506SBCM10.0099560.0089647.8098192.1298870.130250SBCM20.0110420.0124647.8098192.1298870.1302507SBCM10.0100090.0099797.8098412.1298870.130250SBCM20.0212440.0100127.8098412.1298870.1302508SBCM10.0136920.0102107.8098552.1298870.130250SBCM20.0338970.0212527.8098552.1298870.1302509SBCM10.0243500.0137767.8098682.1298870.130250SBCM20.0398730.0288067.8098682.1298870.13025010SBCM10.0268790.0109687.8098742.1298870.130251SBCM20.0354710.0263987.8098742.1298870.13025110[20]6.9828001.9952407.8093202.1298800.13031510[23]0.0486740.0336117.8079482.1298870.130251Figure 1
Plot illustration for a single solitary wave solution atτ=0 and 20.Figure 2
Absolute error distribution atτ=10 for SBCM1 and SBCM2.


### 3.2. Two Solitary Waves

Now, we study the interaction of two solitary waves having different amplitudes, which is the sum of two modified MaxwellianI.C.:
(39)ση,0=∑j=126λjζsechλjμλj+1η−ηj.Here, we take the parametersζ=6,λ1=4,λ2=1,η1=25,η2=55,ℏ=0.2,Δτ=0.025,μ=1,η0=0,0≤η≤200 to concur with those used in [22, 23, 26]. The program is executed up to time τ=20, and the values of invariants χ1,χ2,χ3 are shown in Table 3 and compared with those obtained in [22, 23, 26] at time τ=20. On the other hand, Figure 3 illustrates the interaction of solitary waves at the times τ=0 and τ=15, respectively.Table 3
Two solitary waves invariants for the above parameters.
TimeMethodχ1χ2χ3Analytical11.46769814.62924322.8804660SBCM111.46769814.62925822.880465SBCM211.46769814.62926022.8804772SBCM111.46769814.62961622.882143SBCM211.46769814.62877722.8838034SBCM111.46769814.62994622.886081SBCM211.46769814.62755122.8886796SBCM111.46769814.62984122.886133SBCM211.46769914.62777322.8890578SBCM111.46769814.62980022.889777SBCM211.46769914.62891622.87373610SBCM111.46769914.62921522.881137SBCM211.46769914.62816822.88366312SBCM111.46769914.62927522.885653SBCM211.46769914.62855722.87153114SBCM111.46769914.62906722.885269SBCM211.46770014.62856222.88881616SBCM111.46769914.62955422.885142SBCM211.46770014.62660922.87498918SBCM111.46769914.62990322.886686SBCM211.46770014.62942122.87722820SBCM111.46770014.62928722.885799SBCM211.46770114.62919022.87480920[22]11.467714.629222.880920[23]11.46770114.58308922.69651020[26]11.466114.624922.8631Figure 3
Plot illustrations for interaction of 2 solitary waves atτ=0 and τ=15.
(a)
τ=0(b)
τ=15


### 3.3. Three Solitary Waves

Here, we study theMRLW equation with the modified Maxwellian I.C. and different amplitudes:
(40)ση,0=∑j=136λjζsech10λjμλj+1η−ηj.Numerical experiments are executed for the parametersζ=6,ÿ=0.2,μ=1,Δτ=0.025,λ1=0.03,λ2=0.02,λ3=0.01,η1=18,η2=48,η3=88 in the region −40≤η≤180 in order to see an interaction of three solitary waves takes place. The program is executed up to time τ=45. Table 4 compares our obtained values of the invariants of the three solitary waves by SBCM2 with those obtained by [23]. It is clear from the table that our obtained results of the invariants remain almost the same during the computer run, and they are found to be very close to the results given in [23]. In addition, these are all in good agreement with their analytical results.Table 4
Invariants for3 solitary waves for ζ=6,μ=1,ℏ=0.2,Δτ=0.025,λ1=0.03,λ2=0.02,λ3=0.01,η1=18,η2=48,η3=88,−40≤η≤180.
Our SBCM2[23]Timeχ1χ2χ3χ1χ2χ30.09.5182510.9041300.007863049.5177050.9041290.007863040.19.5180160.9041300.007863129.5175800.9041300.007862720.29.5181930.9041300.007863239.5175850.9041300.007862860.39.5181010.9041300.007863389.5175900.9041300.007862930.49.5181300.9041300.007863509.5175940.9041300.007862940.59.5181450.9041300.007863659.5175980.9041300.007862960.69.5181590.9041300.007863889.5176020.9041300.007862970.79.5181770.9041300.007863009.5176060.9041300.007862980.89.5181850.9041300.007863009.5176100.9041300.007862990.99.5181970.9041300.007863009.5176130.9041300.007863001.09.5182530.9041300.007863019.5176160.9041300.00786300Also, numerical experiments are carried out for the parametersℏ=0.2,μ=1,k=0.025,λ1=4,λ2=1,λ3=0.25,η1=15,η2=45,η3=60,0≤η≤250. The computation is done until time τ=45 to find numerical results of the invariants χ1,χ2,χ3. The result values of the invariants of the proposed SBCM2 algorithm together with the values of the invariants obtained in [26] are documented in Table 5.Table 5
Invariants for3 solitary waves for μ=1,ℏ=0.2,Δτ=0.025,λ1=4,λ2=1,λ3=0.25,η1=15,η2=45,η3=60,0≤η≤250.
Our SBCM1[26]Timeχ1χ2χ3χ1χ2χ3014.98010515.82178922.99210014.980115.837523.0081514.98888215.82954722.91455314.979915.836523.00361014.98678215.82693322.93509514.985015.845323.02071514.98621315.82600522.94008914.980915.836722.99862014.98480515.82538322.95498614.979015.834022.99272514.98390715.82442622.96220514.978015.832322.98763014.98324315.82333722.97721814.977715.831122.98273514.98163715.82249822.99904514.977815.829922.97794014.98014415.82106622.99797414.979515.829122.97284514.98000815.82178122.99685514.953415.829022.9649In addition, we demonstrate the interaction of three solitary waves at timesτ=1,5 and τ=10, respectively, in Figure 4 and consequently, we can see that at time τ=0,5, the three solitary waves interact and then at times τ=20,35, the three solitary waves separate and emerging unchanged.Figure 4
Plot illustrations for interaction of 3 solitary waves atτ=5, τ=20, and τ=35.
(a)
τ=0(b)
τ=5(c)
τ=20(d)
τ=35


## 3.1. Single Solitary Wave

Letη0 be any arbitrary constant. Then, the exact solution of the solitary wave of the MRLW equation is given as follows [20]:
(37)ση,τ=6λζsechλμλ+1η−λ+1τ−η0.The modified MaxwellianI.C. is defined by
(38)ση,0=6λζsechλμλ+1η−η0,and the B.Cs can be concluded from the exact equation.We chooseζ=μ=1,λ=0.1,ℏ=0.125,Δτ=0.1,η0=0,−40≤η≤60 so that we can compare our results with results in [20, 23]. The program is executed up to times τ=10 to find error norms and the invariants χ1,χ2,χ3 at different times, and the results are given in Table 2. From Table 2, one can observe that the predicted error norms L2 and L∞ are smaller than those obtained in [20, 23], and also the invariants χ1,χ2, and χ3 are sanely in good agreement with their exact values. The solutions at τ=0,20 and the motion of the solitary wave along with the interval −40≤η≤60 with 0≤τ≤20 to the right are illustrated in Figure 1. Moreover, the error variations are demonstrated for the proposed algorithms SBCM1 and SBCM2 in Figure 2 at time τ=10. Consequently, we can observe from Table 2 that the results obtained by the SBCM2 are more accurate than those obtained by the SBCM1.Table 2
Error norms and invariants for the single solitary wave for the above parameters.
TimeMethodL2×104L∞×104χ1χ2χ3Analytical7.8098752.1298870.1302500SBCM10.00.07.8097022.1298860.130251SBCM20.00.07.8097022.1298860.1302511SBCM10.0008990.0004977.8097252.1298870.130251SBCM20.0019820.0015887.8097252.1298870.1302512SBCM10.0004810.0006817.8097442.1298870.130250SBCM20.0009860.0009667.8097442.1298870.1302503SBCM10.0009790.0003817.8097672.1298870.130250SBCM20.0010810.0021247.8097672.1298870.1302504SBCM10.0025190.0009947.8097832.1298870.130250SBCM20.0069570.0040667.8097832.1298870.1302505SBCM10.0077380.0057447.8098002.1298870.130250SBCM20.0098860.0087597.8098002.1298870.1302506SBCM10.0099560.0089647.8098192.1298870.130250SBCM20.0110420.0124647.8098192.1298870.1302507SBCM10.0100090.0099797.8098412.1298870.130250SBCM20.0212440.0100127.8098412.1298870.1302508SBCM10.0136920.0102107.8098552.1298870.130250SBCM20.0338970.0212527.8098552.1298870.1302509SBCM10.0243500.0137767.8098682.1298870.130250SBCM20.0398730.0288067.8098682.1298870.13025010SBCM10.0268790.0109687.8098742.1298870.130251SBCM20.0354710.0263987.8098742.1298870.13025110[20]6.9828001.9952407.8093202.1298800.13031510[23]0.0486740.0336117.8079482.1298870.130251Figure 1
Plot illustration for a single solitary wave solution atτ=0 and 20.Figure 2
Absolute error distribution atτ=10 for SBCM1 and SBCM2.


## 3.2. Two Solitary Waves

Now, we study the interaction of two solitary waves having different amplitudes, which is the sum of two modified MaxwellianI.C.:
(39)ση,0=∑j=126λjζsechλjμλj+1η−ηj.Here, we take the parametersζ=6,λ1=4,λ2=1,η1=25,η2=55,ℏ=0.2,Δτ=0.025,μ=1,η0=0,0≤η≤200 to concur with those used in [22, 23, 26]. The program is executed up to time τ=20, and the values of invariants χ1,χ2,χ3 are shown in Table 3 and compared with those obtained in [22, 23, 26] at time τ=20. On the other hand, Figure 3 illustrates the interaction of solitary waves at the times τ=0 and τ=15, respectively.Table 3
Two solitary waves invariants for the above parameters.
TimeMethodχ1χ2χ3Analytical11.46769814.62924322.8804660SBCM111.46769814.62925822.880465SBCM211.46769814.62926022.8804772SBCM111.46769814.62961622.882143SBCM211.46769814.62877722.8838034SBCM111.46769814.62994622.886081SBCM211.46769814.62755122.8886796SBCM111.46769814.62984122.886133SBCM211.46769914.62777322.8890578SBCM111.46769814.62980022.889777SBCM211.46769914.62891622.87373610SBCM111.46769914.62921522.881137SBCM211.46769914.62816822.88366312SBCM111.46769914.62927522.885653SBCM211.46769914.62855722.87153114SBCM111.46769914.62906722.885269SBCM211.46770014.62856222.88881616SBCM111.46769914.62955422.885142SBCM211.46770014.62660922.87498918SBCM111.46769914.62990322.886686SBCM211.46770014.62942122.87722820SBCM111.46770014.62928722.885799SBCM211.46770114.62919022.87480920[22]11.467714.629222.880920[23]11.46770114.58308922.69651020[26]11.466114.624922.8631Figure 3
Plot illustrations for interaction of 2 solitary waves atτ=0 and τ=15.
(a)
τ=0(b)
τ=15


## 3.3. Three Solitary Waves

Here, we study theMRLW equation with the modified Maxwellian I.C. and different amplitudes:
(40)ση,0=∑j=136λjζsech10λjμλj+1η−ηj.Numerical experiments are executed for the parametersζ=6,ÿ=0.2,μ=1,Δτ=0.025,λ1=0.03,λ2=0.02,λ3=0.01,η1=18,η2=48,η3=88 in the region −40≤η≤180 in order to see an interaction of three solitary waves takes place. The program is executed up to time τ=45. Table 4 compares our obtained values of the invariants of the three solitary waves by SBCM2 with those obtained by [23]. It is clear from the table that our obtained results of the invariants remain almost the same during the computer run, and they are found to be very close to the results given in [23]. In addition, these are all in good agreement with their analytical results.Table 4
Invariants for3 solitary waves for ζ=6,μ=1,ℏ=0.2,Δτ=0.025,λ1=0.03,λ2=0.02,λ3=0.01,η1=18,η2=48,η3=88,−40≤η≤180.
Our SBCM2[23]Timeχ1χ2χ3χ1χ2χ30.09.5182510.9041300.007863049.5177050.9041290.007863040.19.5180160.9041300.007863129.5175800.9041300.007862720.29.5181930.9041300.007863239.5175850.9041300.007862860.39.5181010.9041300.007863389.5175900.9041300.007862930.49.5181300.9041300.007863509.5175940.9041300.007862940.59.5181450.9041300.007863659.5175980.9041300.007862960.69.5181590.9041300.007863889.5176020.9041300.007862970.79.5181770.9041300.007863009.5176060.9041300.007862980.89.5181850.9041300.007863009.5176100.9041300.007862990.99.5181970.9041300.007863009.5176130.9041300.007863001.09.5182530.9041300.007863019.5176160.9041300.00786300Also, numerical experiments are carried out for the parametersℏ=0.2,μ=1,k=0.025,λ1=4,λ2=1,λ3=0.25,η1=15,η2=45,η3=60,0≤η≤250. The computation is done until time τ=45 to find numerical results of the invariants χ1,χ2,χ3. The result values of the invariants of the proposed SBCM2 algorithm together with the values of the invariants obtained in [26] are documented in Table 5.Table 5
Invariants for3 solitary waves for μ=1,ℏ=0.2,Δτ=0.025,λ1=4,λ2=1,λ3=0.25,η1=15,η2=45,η3=60,0≤η≤250.
Our SBCM1[26]Timeχ1χ2χ3χ1χ2χ3014.98010515.82178922.99210014.980115.837523.0081514.98888215.82954722.91455314.979915.836523.00361014.98678215.82693322.93509514.985015.845323.02071514.98621315.82600522.94008914.980915.836722.99862014.98480515.82538322.95498614.979015.834022.99272514.98390715.82442622.96220514.978015.832322.98763014.98324315.82333722.97721814.977715.831122.98273514.98163715.82249822.99904514.977815.829922.97794014.98014415.82106622.99797414.979515.829122.97284514.98000815.82178122.99685514.953415.829022.9649In addition, we demonstrate the interaction of three solitary waves at timesτ=1,5 and τ=10, respectively, in Figure 4 and consequently, we can see that at time τ=0,5, the three solitary waves interact and then at times τ=20,35, the three solitary waves separate and emerging unchanged.Figure 4
Plot illustrations for interaction of 3 solitary waves atτ=5, τ=20, and τ=35.
(a)
τ=0(b)
τ=5(c)
τ=20(d)
τ=35


## 4. Conclusion

The main results of the article can be summarized as follows:(i)
The sextic B-spline collocation methods are presented to approximate a new solution of theMRLW equation(ii)
The unconditionally stability of the methods is derived(iii)
The operations are established by calculating both error normsL2 and L∞(iv)
The numerical applications are demonstrated through examples ofMRLW equations with the modified Maxwellian I.C.s



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