Solitonic Solutions for Homogeneous KdV Systems by Homotopy Analysis Method

Abstract

We study two-component evolutionary systems of a homogeneous KdV equations of second and third order. The homotopy analysis method (HAM) is used for analytical treatment of these systems. The auxiliary parameter h of HAM is freely chosen from the stability region of the h-curve obtained for each proposed system.

1. Introduction

Applications in physics are modeled by nonlinear systems. Very few nonlinear systems have closed form solutions, therefore, many researchers stress their goals to search numerical solutions. Homotopy analysis method (HAM), first proposed by Liao [1], is an elegant method which has proved its effectiveness and efficiency in solving many types of nonlinear equations [2, 3]. Liao in his book [4] proved that HAM is a generalization of some previously used techniques such as the d-expansion method, artificial small parameter method [5], and Adomian decomposition method. Moreover, unlike previous analytic techniques, the HAM provides a convenient way to adjust and control the region and rate of convergence [6]. Recently, new interested applications of the homotopy analysis have been introduced by Abbasbandy and coauthors [7, 8]. Also, in [9] HAM is used to study the effects of thermocapillarity and thermal radiation on flow and heat transfer in a thin liquid film.

In this work, we consider a two-component evolutionary system of a homogeneous KdV equations of third order type (I) and (II) given, respectively, by 𝑢_𝑡=𝑢_𝑥𝑥𝑥+𝑢𝑢_𝑥+𝑣𝑣_𝑥,𝑣_𝑡=−2𝑣_𝑥𝑥𝑥−𝑢𝑣_𝑥,𝑢_𝑡=𝑢_𝑥𝑥𝑥+2𝑣𝑢_𝑥+𝑢𝑣_𝑥,𝑣_𝑡=𝑢𝑢_𝑥. (1.1)(1.2) Also, we study a two-component evolutionary system of a homogeneous KdV equations of second order given by 𝑢_𝑡=−3𝑣_𝑥𝑥,𝑣_𝑡=4𝑢_𝑥𝑥+𝑢². (1.3) In the literature many other direct mathematical methods such as the sine-cosine method, rational sine-cosine method, and extended tanh-coth method [10–14] have been implemented in obtaining different solitonic solutions to these systems. Our main goal in this paper is to see how much accuracy we may gain by applying HAM to such evolutionary systems.

In what follows, we highlight the main features of the homotopy analysis method. More details and examples can be found in [7, 8, 15] and the references therein.

2. Survey of Homotopy Analysis Method

To illustrate the basic ideas of this method, we consider the following nonlinear system of differential equations:𝑁_𝑗𝑢₁(𝑥,𝑡),𝑢₂(𝑥,𝑡),⋯𝑢_𝑚(𝑥,𝑡)=0,𝑗=1,…,𝑛, (2.1) where 𝑁_𝑗 are nonlinear operators, 𝑡 is an independent variable, 𝑢_𝑖(𝑡) are unknown functions. By means of generalizing the traditional homotopy method, Liao construct the zeroth-order deformation equation as follows: 1−𝑞𝐿_𝑗𝜙_𝑖𝑥,𝑡,𝑞−𝑢_𝑖,0(𝑥,𝑡)=𝑞ℎ𝐻(𝑥,𝑡)𝑁_𝑗𝜙₁𝑥,𝑡,𝑞,…,𝜙_𝑚𝑥,𝑡,𝑞, (2.2) where 𝑖=1,…,𝑚,𝑗=1,…,𝑛, 𝑞𝜖[0,1] is an embedding parameter,𝐿_𝑗 is a general differential linear operator, 𝑢_𝑖,0(𝑥,𝑡) are initial guesses of 𝑢_𝑖(𝑥,𝑡),𝜙_𝑖(𝑥,𝑡,𝑞) are unknown functions, and ℎ and 𝐻(𝑥,𝑡) are auxiliary parameter and auxiliary function, respectively. It is important note that, one has great freedom to choose auxiliary objects such as ℎ and 𝐿_𝑗 in HAM; this freedom plays an important role in establishing the key stone of validity and flexibility of HAM as shown in this paper. obviously, when 𝑞=0 and 𝑞=1, both 𝜙_𝑖(𝑥,𝑡,0)=𝑢_𝑖,0(𝑥,𝑡),𝜙_𝑖(𝑥,𝑡,1)=𝑢_𝑖(𝑥,𝑡),𝑖=1,…,𝑚, (2.3) thus as 𝑞 increasing from 0 to 1, the solutions of 𝜙_𝑖(𝑥,𝑡,𝑞) change from the initial guesses 𝑢_𝑖,0(𝑥,𝑡) to the solutions 𝑢_𝑖(𝑥,𝑡). Expanding 𝜙_𝑖(𝑥,𝑡,𝑞) in Taylor series with respect to 𝑞, one has 𝜙_𝑖𝑥,𝑡,𝑞=𝑢_𝑖,0(𝑥,𝑡)+^+∞_𝑘=1𝑢_𝑖,𝑘(𝑥,𝑡)𝑞^𝑘,𝑖=1,…,𝑚, (2.4) where 𝑢_𝑖,𝑘()=1𝑥,𝑡𝜕𝑘!^𝑘𝜙_𝑖𝑥,𝑡,𝑞𝜕𝑞^𝑘_𝑞=0,𝑖=1,…,𝑚, (2.5) if the auxiliary linear operators, the initial guesses, the auxiliary parameter ℎ, and the auxiliary function are so properly chosen, then the series (2.4) converges at 𝑞=1, then one has 𝜙_𝑖(𝑥,𝑡,1)=𝑢_𝑖,0(𝑥,𝑡)+^+∞_𝑘=1𝑢_𝑖,𝑘(𝑥,𝑡),𝑖=1,…,𝑚, (2.6) which must be one of the solutions of the original nonlinear equations, as proved by liao. Define the vector ^→𝑢_𝑖,𝑛(𝑥,𝑡)=𝑢_𝑖,0(𝑥,𝑡),𝑢_𝑖,1(𝑥,𝑡),…,𝑢_𝑖,𝑛(𝑥,𝑡),𝑖=1,…,𝑚. (2.7) Differentiating (2.2), 𝑘 times with respect to the embedding parameter 𝑞 and then setting 𝑞=0 and finally dividing them by 𝑘!, we have the so-called kth-order deformation equation 𝐿_𝑗𝑢_𝑖,𝑘(𝑥,𝑡)−𝜒_𝑘𝑢_{𝑖,𝑘−1}(𝑥,𝑡)=ℎ𝑅_𝑗,𝑘^→𝑢_{𝑖,𝑘−1}(𝑥,𝑡),𝑖=1,…,𝑚,𝑗=1,…,𝑛, (2.8) subject to the initial conditions 𝐿_𝑗(0)=0, (2.9) where 𝑅_𝑗,𝑘^→𝑢_{𝑖,𝑘−1}()=1𝑥,𝑡()!𝜕𝑘−1^𝑘−1𝑁_𝑗𝜙₁𝑥,𝑡,𝑞,𝜙₂𝑥,𝑡,𝑞,…,𝜙_𝑚𝑥,𝑡,𝑞𝜕𝑞^𝑘−1_𝑞=0,𝜒_𝑚=0,𝑚≤1;𝜒_𝑚=1,𝑚>1. (2.10) It should be emphasized that 𝑢_𝑖,𝑘(𝑥,𝑡) for 𝑚≥1 is governed by the linear equation (2.8) under the linear boundary conditions that come from original problem, which can be easily solved by symbolic computation software such as Mathematica. When ℎ=−1, and 𝐻(𝑥,𝑡)=1 (2.2) becomes 1−𝑞𝐿_𝑗𝜙_𝑖𝑥,𝑡,𝑞−𝑢_𝑖,0(𝑥,𝑡)+𝑞𝑁_𝑗𝜙₁𝑥,𝑡,𝑞,…,𝜙_𝑚𝑥,𝑡,𝑞=0,𝑖=1,…,𝑚;𝑗=1,…,𝑛, (2.11) which is used mostly in the homotopy perturbation method [16–20].

For the following numerical examples, we use 𝐿_𝑗=𝜕/𝜕𝑡; 𝐻(𝑥,𝑡)=1.

3. Two-Component Evolutionary System of Order 3: Type I

In this section, we consider system (1.1) 𝑢_𝑡=𝑢_𝑥𝑥𝑥+𝑢𝑢_𝑥+𝑣𝑣_𝑥,𝑣_𝑡=−2𝑣_𝑥𝑥𝑥−𝑢𝑣_𝑥 (3.1) subject to 𝑢(𝑥,0)=−6csch²𝑥√2,𝑣()𝑥𝑥,0=6csch√2. (3.2) For application of the homotopy analysis method, we choose the initial approximations as 𝑢₀()(),𝑣𝑥,𝑡=𝑢𝑥,0₀(𝑥,𝑡)=𝑣(𝑥,0). (3.3) Employing HAM with the mentioned parameters in Section 2, we have the following zero-order deformation equations: 1−𝑞𝐿_𝑗𝜙_1𝑡−𝑢(𝑥,0)=𝑞ℎ𝜙_1𝑡−𝜙_{1𝑥𝑥𝑥}−𝜙₁𝜙_1𝑥−𝜙₂𝜙_2𝑥,1−𝑞𝐿_𝑗𝜙_2𝑡−𝑣(𝑥,0)=𝑞ℎ𝜙_2𝑡+2𝜙_{2𝑥𝑥𝑥}+𝜙₁𝜙_2𝑥. (3.4) Subsequently solving the Nth order deformation equations, we obtain 𝑢₁(𝑥,𝑡)=3ℎ𝑡√2csch⁴𝑥√2sinh√,𝑣2𝑥₁(𝑥,𝑡)=−3ℎ𝑡√2coth⁴𝑥√2𝑥csch√2,𝑢₂(𝑥,𝑡)=−3ℎ𝑡csch⁴𝑥√22ℎ𝑡+ℎ𝑡cosh√2𝑥−√2(1+ℎ)sinh√,𝑣2𝑥₂()=3𝑥,𝑡4ℎ𝑡csch³𝑥√23ℎ𝑡+ℎ𝑡cosh√2𝑥−2√2(1+ℎ)sinh√2𝑥. (3.5) We use an 10-term approximation and set 𝑢_HAM(𝑥,𝑡)=¹⁰_𝑖=0𝑢_𝑖(),𝑣𝑥,𝑡_HAM(𝑥,𝑡)=¹⁰_𝑖=0𝑣_𝑖(𝑥,𝑡). (3.6) Comments and illustrations upon the obtained results can be observed in Figures 1 and 2.

(a)

(b)

Figure 1 

(a) The h curve of system (1.1) for 𝑢(3,1) obtained from the 10th order HAM. (b) The h curve of system (1.1) for 𝑣(3,1) obtained from the 10th order HAM.

(a)

(b)

Figure 2 

(a) The obtained 𝑢HAM for system (1.1) using 10-term approximation. (b) The obtained 𝑣HAM for system (1.1) using 10-term approximation. ℎ=−1.1.

4. Two-Component Evolutionary System of Order 3: Type II

In this section, we consider system (1.2) 𝑢_𝑡=𝑢_𝑥𝑥𝑥+2𝑣𝑢_𝑥+𝑢𝑣_𝑥,𝑣_𝑡=𝑢𝑢_𝑥 (4.1) subject to 𝑢()𝑥𝑥,0=−tanh√3𝑣()1𝑥,0=−6−12tanh²𝑥√3. (4.2) For application of the homotopy analysis method, we choose the initial approximations as 𝑢₀()()𝑣𝑥,𝑡=𝑢𝑥,0₀(𝑥,𝑡)=𝑣(𝑥,0). (4.3) Employing HAM with mentioned parameters in Section 2, we have the following zero-order deformation equations: 1−𝑞𝐿_𝑗𝜙_1𝑡−𝑢(𝑥,0)=𝑞ℎ𝜙_1𝑡−𝜙_{1𝑥𝑥𝑥}−2𝜙₂𝜙_1𝑥−𝜙₁𝜙_2𝑥,1−𝑞𝐿_𝑗𝜙_2𝑡−𝑣(𝑥,0)=𝑞ℎ𝜙_2𝑡−𝜙₁𝜙_1𝑥. (4.4) Subsequently solving the Nth order deformation equations, we obtain 𝑢₁(𝑥,𝑡)=−ℎ𝑡√3sech²𝑥√3,𝑣₁(𝑥,𝑡)=−ℎ𝑡√3sech²𝑥√3𝑥tanh√3,𝑢₂(𝑥,𝑡)=−ℎ𝑡3sech³𝑥√3√3(1+ℎ)cosh𝑥√3𝑥−ℎ𝑡sinh√3,𝑣₂(𝑥,𝑡)=ℎ𝑡6sech⁴𝑥√3−2ℎ𝑡+ℎ𝑡cosh2𝑥√3−√3(1+ℎ)sinh2𝑥√3, (4.5) and so on. We use an 10-term approximation and set 𝑢_HAM(𝑥,𝑡)=¹⁰_𝑖=0𝑢_𝑖(),𝑣𝑥,𝑡_HAM(𝑥,𝑡)=¹⁰_𝑖=0𝑣_𝑖(𝑥,𝑡). (4.6) Comments and illustrations upon the obtained results can be observed in Figures 3 and 4.

(a)

(b)

Figure 3 

(a) The h curve of system (1.2) for 𝑢(5,1) obtained from the 10th order HAM. (b) The h curve of system (1.2) for 𝑣(5,1) obtained from the 10th order HAM.

(a)

(b)

Figure 4 

(a) The obtained 𝑢HAM for system (1.2) using 10-term approximation. (b) The obtained 𝑣HAM for system (1.2) using 10-term approximation. ℎ=−1.3.

5. Two-Component Evolutionary System of Order 2

In this section, we consider system (1.3) 𝑢_𝑡=−3𝑣_𝑥𝑥,𝑣_𝑡=4𝑢_𝑥𝑥+𝑢² (5.1) subject to 𝑢()1𝑥,0=−8−18tan²𝑥2√3,𝑣()1𝑥,0=−4√3𝑥tan2√3. (5.2) We proceed in the same manner and choose the initial approximations as 𝑢₀()(),𝑣𝑥,𝑡=𝑢𝑥,0₀(𝑥,𝑡)=𝑣(𝑥,0). (5.3) The zero-order deformation equations are 1−𝑞𝐿_𝑗𝜙_1𝑡−𝑢(𝑥,0)=𝑞ℎ𝜙_1𝑡+𝜙_2𝑥𝑥,1−𝑞𝐿_𝑗𝜙_2𝑡−𝑣(𝑥,0)=𝑞ℎ𝜙_2𝑡−𝜙²₁−4𝜙_1𝑥𝑥. (5.4) Subsequently, we obtain 𝑢₁(𝑥,𝑡)=ℎ𝑡8√3sec²𝑥2√3𝑥tan2√3,𝑣₁(𝑥,𝑡)=−ℎ𝑡24sec²𝑥2√3,𝑢₂(𝑥,𝑡)=ℎ𝑡96sec⁴𝑥2√3𝑥−2ℎ𝑡+ℎ𝑡cos√3+2√3(1+ℎ)sin𝑥√3,𝑣₂(𝑥,𝑡)=ℎ𝑡144sec³𝑥2√3()𝑥−61+ℎcos2√3+√𝑥3ℎsin2√3, (5.5) and so on. We use an 11-term approximation and set 𝑢_HAM(𝑥,𝑡)=¹⁰_𝑖=0𝑢_𝑖(),𝑣𝑥,𝑡_HAM(𝑥,𝑡)=¹⁰_𝑖=0𝑣_𝑖(𝑥,𝑡). (5.6) The obtained h-curve and the HAM solutions of 𝑢(𝑥,𝑡) and 𝑣(𝑥,𝑡) are given in Figures 5 and 6.

(a)

(b)

Figure 5 

(a) The h curve of system (1.3) for 𝑢(3,1) obtained from the 10th order HAM. (b) The h curve of system (1.3) for 𝑣(3,1) obtained from the 10th order HAM.

(a)

(b)

Figure 6 

(a) The obtained 𝑢HAM for system (1.3) using 10-term approximation. (b) The obtained 𝑣HAM for system (1.3) using 10-term approximation. ℎ=−0.7.

6. Discussion and Concluding Remarks

In this paper, we obtained soliton solutions for homogeneous KdV systems of second and third order by means of homotopy analysis method. The convergence region for the obtained, approximation, is determined by the parameter h as shown in Figures 1, 3, and 5, respectively, for systems ((1.1), (1.2), and (1.3)).

Homotopy analysis method provides us a convenient freely chosen with parameter h in contrast to the Homotopy perturbation method, where h is assumed to be −1. In this work, h has been chosen to be −1.1,−1.3,and−0.7, respectively, for systems ((1.1), (1.2), and (1.3)). It is worth noting that the choice of the parameter ℎ in this paper is considered based on the obtained stability region of the ℎ-curve for each system; for example, in system (1.1) the stability region of ℎ falls between −1.5 and −0.7 and we considered the midpoint of this interval.

Finally, the absolute errors for the obtained approximate solution of system (1.1) are given in Tables 1 and 2.

Table 1 

Absolute errors regarding 𝑢(𝑥,𝑡) for system (1.1) with 𝑁=10.

Table 2 

Absolute errors regarding 𝑣(𝑥,𝑡) for system (1.1) with 𝑁=10.