Analytical and numerical investigations of the modified Camassa–Holm equation
MAHMOUD A E ABDELRAHMAN1,2 ,∗and ABDULGHANI ALHARBI1
1Department of Mathematics, College of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia
2Department of Mathematics, Faculty of Science, Mansoura University, 35516 Mansoura, Egypt
∗Corresponding author. E-mail: mahmoud.abdelrahman@mans.edu.eg
MS received 28 June 2020; revised 10 February 2021; accepted 22 March 2021
Abstract. In this article, analytical and numerical solutions to the simplified modified Camassa–Holm (MCH) equation by using the Riccati–Bernoulli (RB) sub-ODE method and moving mesh method are obtained. Some new solutions are given. The discrtisation of the presented model is introduced in the form of finite difference operators.
Some 3D graphs for the presented solution are plotted with the aid of the Matlab software for suitable values of parameters. We introduce the comparison between numerical and exact solutions.
Keywords. Modified Camassa–Holm equation; Riccati–Bernoulli sub-ODE method; solitary wave solutions;
moving mesh partial differential equations; moving adaptive scheme.
PACS Nos 02.30.Jr; 02.60.Cb; 04.20.Jb
1. Introduction
Nonlinear partial differential equations (NLPDEs) exhibit various complex phenomena in various fields of applied science, like optics, nuclear physics, chemical reactions, mathematical biosciences, plasma physics, relativity, fluid mechanics, ecology and several other fields [1–12]. The investigation of exact solutions of these NLPDEs will help the scientists to understand these phenomena better. Thus, various methods for obtaining solutions of NLPDEs have been established and developed, like the tanh–sech method [13,14], exp-function method [15,16], homogeneous balance method [17,18], Jacobi elliptic function method [19, 20], Riccati–Bernoulli (RB) sub-ODE method [21,22], sine–cosine method [23,24], F-expansion method [25, 26], extended tanh-method [27,28],(G/G)-expansion method [29,30].
One important NLPDE is the simplified modified Camassa–Holm (MCH) equation which deduced a com- pletely integrable wave equation, specifically CH equa- tion, for water waves by retaining two terms that are usually neglected in the small amplitude, shallow water limit [31,32]:
ut +2δux−ux xt +βu2ux =0, (1.1) whereβ >0 andδ∈R. Tian and Song [33] introduced
the new peaked solitary wave solutions of the MCH equation, (1.1). Wazwaz [31] also gave some compact as well as non-compact solutions for two variants of eq.
(1.1). In this study, the RB sub-ODE method [21] is used to construct travelling wave solutions of the simplified MCH equation. This method provides infinite solutions, which is a very important feature of this method [10,21].
We also apply ther-adaptive mesh technique [34–36]
which employs a monitor function and moving mesh PDEs (MMPDEs) to concentrate and locate adaptively the mesh coupled to eq. (1.1). We trust thatr-adaptive mesh techniques are much better for MCH equation than thehp(orh)-adaptive mesh techniques given in [35,36].
The performance ofr-adaptive techniques redistributes the mesh based on a monitor function which directly follows a solution characteristic (shock waves, large cur- vatures, variations) and consequently resolves precisely the internal layers further.
The MMPDEs which generally have a common form of nonlinear diffusion equation [34,35] could be cou- pled to eq. (1.1). This can be efficiently operated in finite difference or finite element for these prob- lems. Otherwise,h-adaptive techniques are commonly implemented utilising the finite element method. Ther- adaptive techniques were not used as commonly ashp orh-adaptive techniques. They were used in numerous applications, such as convective heat transfer [37], mete- 0123456789().: V,-vol
orological [38,39] and computational fluid mechanics [40,41] problems. Ther-adaptive techniques build the foundation of common goal publicly convenient solvers for 1D PDE systems, for instance, TOMS731 [42] and MOVCOL [43].
Our motivations for doing this work are: First, we apply the RB sub-ODE technique to find new solu- tions of the nonlinear simplified MCH eq. (1.1). We provide new solutions, which can clarify some interest- ing physical phenomena for physicists, engineers and mathematicians. Indeed, this technique is powerful for solving other types of NPDEs (see [10,21,44,45]). Sec- ondly, we focus on implementing the finite difference scheme on an adaptive moving mesh for solving the 1D equation [46,47]. Extending this method to higher dimensions will be considered elsewhere. These two aspects give us an opportunity to present a new moti- vation that make our analysis better suited. To the best of our knowledge, no previous research work has been presented utilising the techniques proposed in this work for solving the MCH equation.
The rest of the paper is arranged as follows: Section 2gives the description of RB sub-ODE method. In §3, we give vital solutions of the MCH equation. In §4, 3D graphs of some selected solutions are given. Section5 gives a brief overview for MMPDEs and mesh density functions. Moreover, we present the numerical results and compare them with exact solutions. Conclusions are presented in §6.
2. RB sub-ODE method
Here we prescribe RB sub-ODE approach [21] for find- ing travelling wave solutions of NPDEs.
Let us consider a general NPDE in two independent variablesx andt as
G(ϕ, ϕt, ϕx, ϕtt, ϕx x, . . .)=0, (2.1) whereϕ(x,t)is the solution of NPDE eq. (2.1). Utilising the wave transformation
ϕ(x,t)=ϕ(ξ), ξ =k(x +ςt), (2.2) changes eq. (2.1) to the following ODE:
H(ϕ, ϕ, ϕ, ϕ, . . .)=0, (2.3) wheredenotes the derivative ofϕwith respect toξand H is a polynomial inϕ(ξ). We assume that eq. (2.3) has the following solution:
ϕ =aϕ2−n +bϕ+cϕn. (2.4) Equation (2.4) gives
ϕ =ab(3−n)ϕ2−n +a2(2−n)ϕ3−2n+nc2ϕ2n−1
+bc(n+1)ϕn+(2ac+b2)ϕ; (2.5) ϕ =(ab(3−n)(2−n)ϕ1−n
+a2(2−n)(3−2n)ϕ2−2n
+n(2n−1)c2ϕ2n−2+bcn(n+1)ϕn−1
+(2ac+b2))ϕ. (2.6)
The solutions of eq. (2.4) are 1. Atn =1,
ϕ(ξ)=μe(a+b+c)ξ. (2.7)
2. Atn =1;b=0;c=0,
ϕ(ξ)=(a(n−1)(ξ+μ))n−11 . (2.8) 3. Atn =1;b=0;c=0,
ϕ(ξ)= −a
b +μeb(n−1)ξ n−11
. (2.9)
4. Atn =1;a=0;b2−4ac<0, ϕ(ξ) =
−b 2a +
√4ac−b2 2a
×tan
(1−n)√
4ac−b2
2 (ξ +μ)
1−1n
(2.10) and
ϕ(ξ) = −b
2a −
√4ac−b2 2a
×cot
(1−n)√
4ac−b2
2 (ξ +μ)
1−n1
(2.11) 5. Atn =1;a=0;b2−4ac>0,
ϕ(ξ) = −b
2a −
√b2−4ac 2a
×coth
(1−n)√
b2−4ac
2 (ξ +μ)
1−n1
(2.12) and
ϕ(ξ) = −b
2a −
√b2−4ac 2a
×tanh
(1−n)√
b2−4ac
2 (ξ +μ)
1−n1 . (2.13)
6. Atn=1;a =0;b2−4ac =0, ϕ(ξ)=
1
a(n−1)(ξ+μ) − b 2a
1−n1
. (2.14)
Hereμis an arbitrary constant.
2.1 Bäcklund transformation
Ifϕm−1(ξ)andϕm(ξ)(ϕm(ξ) =ϕm(ϕm−1(ξ)))are the solutions of eq. (2.4), then
dϕm(ξ)
dξ = dϕm(ξ) dϕm−1(ξ)
dϕm−1(ξ) dξ
= dϕm(ξ)
dϕm−1(ξ)(aϕ2m−−n1+bϕm−1+cϕnm−1), that is,
dϕm(ξ) aϕm2−n+bϕm+cϕmn
= dϕm−1(ξ)
aϕm2−−n1+bϕm−1+cϕmn−1. (2.15) Integrating eq. (2.15) with respect toξ, we obtain the following Bäcklund transformation of eq. (2.4):
ϕm(ξ)=
−cK1+a K2(ϕm−1(ξ))1−n bK1+a K2+a K1(ϕm−1(ξ))1−n
1−1n , (2.16) where K1 and K2 are arbitrary constants. When we obtain a solution for this equation, we use eq. (2.16) to get infinite solutions of eqs (2.4) and (2.1).
3. Results and discussion
In this section, we extract the exact solutions for eq.
(1.1). Utilising the wave transformation
u(x,t)=q(ξ), ξ =x −wt, (3.1) we obtain the following NODE:
3wq+βq3+(6δ−3w)q =0. (3.2) Substituting eq. (2.5) into eq. (3.2), the following equation is obtained:
3w
ab(3−n)q2−n+a2(2−n)q3−2n+nc2q2n−1 +bc(n+1)qn+(2ac+b2)q
+βq3+(6δ−3w)q =0. (3.3) Settingn =0, eq. (3.3) is degenerated in the following form:
3w(3abq2+2a2q3+bc+(2ac+b2)q)
+βq3+(6δ−3w)q =0. (3.4)
Equating each coefficient ofqi(i = 0,1,2,3)to zero, yield algebraic equations. Solving these equations gives the following solutions of eq. (1.1):
Rational function solutions
The rational solution of eq. (1.1) is
u1(x,t)=(−a(x−wt+μ))−1 , (3.5) wherewandμare arbitrary constants.
Trigonometric function solution
The trigonometric solutions of eq. (1.1) are u2,3(x,t)= ±√
3
2δ−w β
×tan w−2δ
2w (x −wt+μ)
(3.6) and
u4,5(x,t)= ±√ 3
2δ−w
β
×cot w−2δ
2w (x −wt+μ)
, (3.7) whereδ, w, βandμare arbitrary constants.
Hyperbolic function solution
The hyperbolic solutions of eq. (1.1) are u6,7(x,t)= ±√
3
w−2δ β
×tanh 2δ−w
2w (x −wt+μ)
(3.8) and
u8,9(x,t)= ±√ 3
w−2δ β
×coth 2δ−w
2w (x −wt+μ)
, (3.9) whereδ, w, βandμare arbitrary constants.
Remark1. Using eq. (2.16) forui(x,t),i = 1, . . . ,9, once, then eq. (3.2) as well as for eq. (1.1) has an infinite solutions. In sequence, by applying this process again, we get new families of solutions.
u1(x,t)= C3
−aC3(x −wt+μ)±1, (3.10)
u2,3(x,t)= −3(w−β2δ)±C3
√3 2δ−w
β tan(
w−2δ
2w (x −wt+μ)) C3±√
3 2δ−w
β tan(
w−2δ
2w (x −wt+μ)) , (3.11)
u4,5(x,t)= −3(w−β2δ)±C3
√3 2δ−w
β cot(
w−2δ
2w (x −wt+μ)) C3±√
3 2δ−w
β cot(
w−2δ
2w (x−wt+μ)) , (3.12)
u6,7(x,t)= −3(w−β2δ)±C3
√3 w−2δ
β tanh(
2δ−w
2w (x −wt+μ)) C3±√
3 w−2δ
β tanh(
2δ−w
2w (x−wt+μ)) , (3.13)
u8,9(x,t)= −3(w−β2δ)±C3
√3 w−2δ
β coth(
2δ−w
2w (x −wt+μ)) C3±√
3 w−2δ
β coth(
2δ−w
2w (x−wt+μ)) , (3.14)
whereδ, w, β,C3andμare arbitrary constants.
4. Graphs for the solutions
In this section we give 3D graphics for some selected solutions, that is, figures1–3.
5. Numerical results on an adaptive moving mesh Consider eq. (1.1), whereδ andβ are parameters. We introduce the numerical solution of eq. (1.1, where the boundary conditions
ux(a,t)=ux(b,t)=0, ux x(a,t)=ux x(b,t)=0, (5.1)
Figure 1. The 3D graph of solution u = u1(x,t) with w = 0.6, δ = 0.3, β = −1.5, μ = 2, y = 0, 0 ≤ t ≤ 3 and−3≤x≤3.
with the initial condition eq. (3.8) at t = 0, utilis- ing finite difference methods. The numerical scheme is implemented on an adaptive moving mesh. Here, we use the exact solution eq. (3.8) with fixed timet = 6 to compare with the adaptive moving mesh as well as a uniform mesh. In order to transform the computational domain depending onξto the physical domain depend- ing onx, we apply the following transformation:
x =x(ξ,t):c≡ [0,1] →p ≡ [a,b], t>0. cis the computational domain andpis the physical domain. Hence, the solution is degenerated as follows:
u(x,t)=u(x(ξ,t),t).
Figure 2. The 3D graph of solution u = u2(x,t) with w = 2.9, δ = 0.8, β = −0.5, μ = 2, 0 ≤ t ≤ 3 and
−3≤x≤3.
Figure 3. The 3D graph of solution u = u6(x,t) with w = 1.9, δ = 1.8, β = −0.7, μ =2, 0 ≤ t ≤ 3 and
−3≤x≤3.
A moving mesh associated with the solutionshis Jh(ts):xj(ξ)=x(ξj,t), j =1, . . . ,N +1, (5.2) where the boundary nodes are
x1 =a, xN+1 =b. (5.3)
A uniform mesh on the computational domain is pre- scribed in the following way:
Jhc(t):ξj = (j−1)
N , j=1, . . . ,N +1. (5.4) After some processing, we have the following transfor- mations:
ux = uξ
xξ, ut = ut −uξ
xξ xt. (5.5)
Using the above, eq. (1.1) can be written as Gt − uξ
xξ xt = −Qξ
xξ , (5.6)
G=u− 1 xξ
uξ xξ
ξ, (5.7)
Q =2δu+βu3/3. (5.8)
The boundary conditionsu(a,t)=α1,u(b,t)=α2are replaced by the following ODE:
Gt,1 =0, Gt,N+1=0, (5.9) whereα1, α2are constants as the problem required.
We obtain the equidistributing coordinate transforma- tionx =x(ξ,t)by solving MMPDEs [48,49]. Utilising the MMPDE6,
MMPDE6: xt,ξξ = −1
τ(ρˆxξ)ξ, (5.10) described in [34,35]. A semi-discretisation scheme util- ising the centred finite differences to discretise the spatial derivatives is employed. Here,ρ(ˆ x,t)is a mesh density function (defined below) andτ > 0 is a user- specified relaxation parameter.τ is the time-scale over which the mesh responds to changes in the monitor func- tionρ(ˆ x,t). Whenτis smaller, the mesh responds more quickly to changes in ρ(ˆ x,t). Also, the mesh moves slowly when value ofτ is large. The boundary condi- tionsx(ξ1,t) =a andx(ξN+1,t) =b are replaced by their ODE form:
xt,1=0, xt,N+1=0. (5.11) The initial condition
x(ξ,t=0)=a+(j−1)b−a
N , j =1,2, . . . ,N+1 (5.12) symbolises a uniform initial mesh on the physical domain p ≡ [a,b]. Finally, the monitor function
ˆ
ρ(x,t)is given by
Arc-length monitor function: ˆρ(x,t)=
1+αu2x. (5.13) Curvature monitor function: ˆρ(x,t)=
1+αu2x x. (5.14)
Here,α is a user-defined parameter. Ifuis not smooth, the discretised monitor function computed as above can change abruptly and slow down the computation. To get a smoother mesh and also make the MMPDEs easier to integrate, it is a common practice in the most standard moving mesh methods to smooth the monitor function.
A simple but powerful smoothing scheme, where the following weighted averaging is its basic ingredient is [34,50],
Figure 4. (a) Time evolution of u(x,t)to the wave solution and (b) the corresponding mesh trajectories x(ξ,t)utilising the moving adaptive method for N =2000 (initialx ≈0.01), MMPDE6 and arc-length monitor function. The timet lies between 2 and 6 with step 2. The parameter values are:δ=3,w=10−1,β= −7×10−1andμ=2. The monitor function smoothing parameters are p=2,γ =2 andτ =10−2.
ˆ ρj :=
j+p k=j−pρˆk2
γ 1+γ
|k−j|
j+p k=j−p
γ 1+γ
|k−j| , j =1, . . . ,N+1. (5.15) Here the smoothing index p is a non-negative integer andγ ∈IR is a positive smoothing parameter.
5.1 Numerical results
We employ the non-uniform mesh scheme (eqs (5.6) and (5.7)) for discretising eq. (1.1). This problem is derived
from the boundary conditions given by eq. (5.1) and the initial condition is eq. (3.8) att =0.
We seek the numerical solution of this problem to determine the evolution ofu(x,t)utilising the finite dif- ference technique and the method of lines. In the whole results presented in sequel, the parameter values are:
δ=3,w=10−1,β = −7×10−1andμ=2.The mon- itor function smoothing parameters are p = 2,γ = 2 to perform the numerical results in a completely unified way.
Figures4and5compare the travelling wave solution (eq. (3.8)) and the presented numerical solution.
6. Conclusions
In this study, the RB sub-ODE technique is applied to extract the exact travelling wave solutions for the sim- plified MCH equation. Moreover, the application of the
moving mesh method for the simplified MCH equation is also introduced. The exact and numerical solutions are successfully computed. Indeed, the moving mesh method is very useful for finding numerical solutions to many NLPDEs. The initial condition for the simplified
Figure 5. (a)u(x,t =6), (b) arc-length monitor function and (c)x(ξ,t =6)obtained using the moving adaptive scheme for N =1000 (initialx≈0.01), MMPDE6 and arc-length monitor function. The timet lies between 2 and 6 with step 2.
The parameter values are:δ =3,w =10−1,β = −7×10−1andμ=2. The monitor function smoothing parameters are p=2,γ =2 andτ =10−2.
Figure 5. Continued.
MCH equation is obtained from the analytical solu- tion used. We conclude that the methods presented here can be used for many other NLPDEs arising in natural sciences.
References
[1] S Z Hassan, N A Alyamani and M A E Abdelrahman, Eur. Phys. J. Plus134, 425 (2019)
[2] A R Alharbi and M B Almatrafi,Int. J. Math. Comput.
Sci.15, 367 (2020)
[3] H G Abdelwahed,J. Taibah Univ. Sci.14(1), 777 (2020) [4] H G Abdelwahed, J. Taibah Univ. Sci. 14(1), 1363
(2020)
[5] A R Alharbi, M B Almatrafi and M A E Abdelrahman, Phys. Scr.95(4), 045215 (2020)
[6] M K Sharaf, E K El-Shewy and M A Zahran,J. Taibah Univ. Sci.14(1), 1416 (2020)
[7] H Triki, C Bensalem, A Biswas, S Khan, Q Zhou, S Adesanya, S P Moshokoa and M Belic,Opt. Commun.
437, 392 (2019)
[8] M Younis, S Ali and S A Mahmood,Nonlinear Dyn.81, 1191 (2015)
[9] H Bulut, T A Sulaiman and H M Baskonus,Optik163, 49 (2018)
[10] M A E Abdelrahman and M A Sohaly,Indian J. Phys.
93, 903 (2019)
[11] M A E Abdelrahman, M A Sohaly and A R Alharbi,J.
Taibah Univ. Sci.13(1), 834 (2019)
[12] M A E Abdelrahman and N F Abdo, Phys. Scr. 95, 045220 (2020)
[13] W Malfliet and W Hereman,Phys. Scr.54, 563 (1996) [14] A M Wazwaz,Appl. Math. Comput.154, 714 (2004) [15] J H He and X H Wu,Chaos Solitons Fractals30, 700
(2006)
[16] H Aminikhad, H Moosaei and M Hajipour, Numer.
Methods Partial Differ. Eqs.26, 1427 (2009) [17] E Fan and H Zhang,Phys. Lett. A246, 403 (1998) [18] M L Wang,Phys. Lett. A213, 279 (1996)
[19] C Q Dai and J Zhang,Chaos Solitons Fractals27, 1042 (2006)
[20] E Fan and J Zhang,Phys. Lett. A305, 383 (2002) [21] X F Yang, Z C Deng and Y Wei,Adv. Diff. Equ.1, 117
(2015)
[22] M A E Abdelrahman,Nonlinear Eng.7(4), 279 (2018) [23] A M Wazwaz,Comput. Math. Appl.50, 1685 (2005) [24] C Yan,Phys. Lett. A224, 77 (1996)
[25] Y J Ren and H Q Zhang,Chaos Solitons Fractals27, 959 (2006)
[26] J L Zhang, M L Wang, Y M Wang and Z D Fang,Phys.
Lett. A350, 103 (2006)
[27] E Fan,Phys. Lett. A277, 212 (2000)
[28] A M Wazwaz,Appl. Math. Comput.187, 1131 (2007) [29] M L Wang, J L Zhang and X Z Li,Phys. Lett. A372,
417 (2008)
[30] S Zhang, J L Tong and W Wang,Phys. Lett. A372, 2254 (2008)
[31] A M Wazwaz,Appl. Math. Comput.163(3), 1165 (2005) [32] R Camassa and D Holm,Phys. Rev. Lett.71, 1661 (1993) [33] L Tian and X Song,Chaos Solitons Fractals 19, 621
(2004)
[34] W Huang and R D Russell,The adaptive moving mesh methods(Springer, 2011)
[35] C J Budd, W Huang and R D Russell,Acta Numer.18, 111 (2009)
[36] V D Liseikin,Grid generation methods(Springer, New York, NY, USA, 2009)
[37] H D Ceniceros and T Y Hou,J. Comput. Phys.172(2), 609 (2001)
[38] E Walsh,Moving mesh methods for problems in meteo- rology, Ph.D. thesis (University of Bath, 2010) [39] C J Budd, M J P Cullen and E J Walsh,J. Comput. Phys.
236, 247 (2013)
[40] T Tang,Contemp. Math.383, 141 (2005)
[41] L Yibao, D Jeong and J Kim, Meccanica 49, 239 (2014)
[42] J G Blom and P A Zegeling,ACM Trans. Math. Softw.
20, 194 (1994)
[43] W Huang and R D Russell,Appl. Numer. Math.20, 101 (1996)
[44] M A E Abdelrahman and M A Sohaly,Results Phys.9, 344 (2018)
[45] S Z Hassan and M A E Abdelrahman, Pramana – J.
Phys.91: 67 (2018)
[46] A R Alharbi, M A E Abdelrahman and M B Almatrafi, Comput. Model. Eng. Sci.122(2), 743 (2020)
[47] M A E Abdelrahman, M B Almatrafi and A R Alharbi, Symmetry12, 429 (2020)
[48] A R Alharbi and S Naire,J. Comput. Appl. Math.319(4), 365 (2017)
[49] A R Alharbi,Numerical solution of thin-film flow equa- tions using adaptive moving mesh methods, Ph.D. thesis (Keele University, 2016)
[50] W Huang, Y Ren and R D Russell,J. Comput. Phys.
113(2), 279 (1994)
