Solitary wave solutions with non-monotone shapes for the modified Kawahara equation

Accepted Manuscript Solitary wave solutions with non-monotone shapes for the modified Kawahara equation Tchavdar T. Marinov, Rossitza S. Marinova

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S0377-0427(17)30416-8 http://dx.doi.org/10.1016/j.cam.2017.08.027 CAM 11279

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Journal of Computational and Applied Mathematics

Received date : 28 June 2017 Revised date : 30 August 2017 Please cite this article as: T.T. Marinov, R.S. Marinova, Solitary wave solutions with non-monotone shapes for the modified Kawahara equation, Journal of Computational and Applied Mathematics (2017), http://dx.doi.org/10.1016/j.cam.2017.08.027 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Solitary Wave Solutions with Non-Monotone Shapes for the Modified Kawahara Equation Tchavdar T. Marinova , Rossitza S. Marinovab,∗ a Department

of Natural Sciences, Southern University at New Orleans, 6801 Press Drive, New Orleans, LA 70126, U.S.A. b Department of Mathematical and Physical Sciences, Concordia University of Edmonton, 7128 Ada Boulevard, Edmonton, AB T5B 4E4, CANADA

Abstract The propagation of stationary solitary waves of the non-linear modified Kawahara equation is considered. The asymptotic boundary conditions admit the trivial solution along with the solitary wave type solution, which is a bifurcation problem. The bifurcation is treated by reformulating the problem into a problem for identification of an unknown coefficient from over–posed boundary data, in which the trivial solution is excluded. Making use of the method of variational imbedding, the inverse problem for the coefficient identification is reformulated as a higher–order boundary value problem. This approach to solving the fifth–order modified Kawahara equation is allowing identification of non–trivial solutions. The obtained boundary–value problem is solved by means of an iterative difference scheme, which is thoroughly validated. New traveling wave solutions with monotone and non-monotone shapes are obtained for different values of the problem parameters. Keywords: Modified Kawahara equation; Solitary wave solution; Inverse problem; Coefficient identification; Numerical solution; Non-monotone shape

1. Introduction The modified Kawahara equation was formulated first by Kawahara in [14]. This nonlinear equation has attracted attention due to the numerous applications in science and engineering. It plays an important role in the theory of fluid mechanics, optical fibers, biology, solid state physics, chemical kinematics, chemical physics, and geochemistry. Lots of methods deal with obtaining exact and approximate analytic solutions of the modified Kawahara equation, see Biswas in [4], Yusufoˆ glu and Bekir in [21], Jabbari and Kheiri in [12], Araruna, Capistrano-Filho, and Doronin, ∗ Corresponding

author Email addresses: [email protected] (Tchavdar T. Marinov), [email protected] (Rossitza S. Marinova)

Preprint submitted to Elsevier

August 28, 2017

G.G. in [1]. Bekir, G¨ uner, and Bilgil in [2] studied variable-coefficient modified Kawahara equations using solitary wave ansatz method to find dark optical as well as bright optical soliton solutions. Other authors study solitonic solutions by using a variety of powerful methods such as the variational iteration method (VIM) proposed by He in [10], Jin in [13], mesh-free method based on the collocation with radial basis functions used by Zarebnia and Aghili in [22]. In the work of Polat, Kaya and Tutalar in [17], an Adomian decomposition method is proposed for numerical approximation of traveling–wave solutions of the modified Kawahara equation, and the solutions are compared with known analytical solutions. Since not all solutions of the modified Kawahara equation are analytic, there is a need of studying the equation numerically. The present work is employing a special numerical approach to solving the modified Kawahara equation. Stationary localized waves are considered in the frame moving to the right. The original ill–posed problem has a non–unique solution. To cope with this issue, the bifurcation problem is reformulated into a problem for identification of an unknown coefficient from over–posed boundary data, in which the trivial solution is excluded. This approach to solving the modified fifth-order Kawahara equation is original allowing identification of the non-trivial solutions, a variation of which was proposed and used in [16]. The numerical solutions obtained via the developed numerical method is compared with an analytical solution determined by Sirendaoreji in [18] by direct algebraic method. The convergence of the difference scheme is illustrated with several examples. After constructing and validating the numerical method, next step is to use it to find solutions, which cannot be obtained by other means. More specifically, we aim to investigate the solitary waves with non-monotone shapes, which cannot be obtained analytically. Similar approach is presented and used for identification of stationary solitary waves on an infinite elastic rod on elastic foundation, see the work [6] of Christov, Marinov and Marinova. 1.1. Localized Waves Problem The modified Kawahara equation is a non–linear partial differential equation in the form ut + u2 ux + puxxx − quxxxxx = 0, (1) where p > 0 and q > 0 are given constants. The motivation of this work is to devise an approach for obtaining new solitonic solutions of the stationary waves in the moving frame ξ = x − ct. After integrating with respect to ξ and taking into account the localized character of the investigated solutions, the equation (1) reduces to the following nonlinear ordinary differential equation L(u) = −cu +

u3 + puξξ − quξξξξ = 0 . 3

(2)

Solitary waves (also called solitons) are localized within a region and maintain their shape while travel with a constant speed. 2

1.2. Asymptotic behavior of the tails We are looking for solutions of the equation (2) with u → 0 when ξ → ∞. Then u3  u in the tails and the linearized version of the equation (2) coincides with its linear part or reduces to quξξξξ − puξξ + cu = 0 .

(3)

Equation (3) possesses harmonic solutions of the type ekξ . The corresponding dispersion relation reads i p 1 h p ± p2 − 4cq . (4) qk 4 − pk 2 + c = 0 =⇒ k 2 = 2q Equation (4) shows that the asymptotic tails of the localized wave can be either monotonic, purely oscillatory or damped oscillatory, depending on whether k 2 is real positive, real negative or complex. 2. Introducing Unknown Coefficient About a century ago Hadamard introduced the concept of well-posed problems in the theory of differential equations. According to Hadamard [9], a problem is well-posed if three requirements are met: a solution exists, it is unique, and depends continuously on the data. Otherwise the problem is ill-posed. The clear intention of Hadamard was that “ill-posed” means “of no physical interest.” We know today that such problems arise in a fundamental way in modelling of complex physical systems. The definition of Hadamard does not cover all of them, and is pertinent only to the stability of a solution. Usually, the well-posed problems for differential equations can be discretized and solved successfully by stable numerical methods. However, if the problem is not well-posed, it is necessary to be re-formulated prior numerical treatment. Common approach to the solution of ill-posed inverse problems is based on the popular regularization methods [15, 19], requiring to find a minimum with respect to a parameter (called a regularization parameter) of a properly constructed functional. The first regularization procedure was proposed by Tikhonov [20] in 1963. His idea is that the solution should not fit the data exactly but the problem is transformed into a minimization problem with a parameter which determines how well the solution fits the data. The work of Tikhonov spurred significant activity, aiming at the creation of regularizing procedures (see [8, 7, 11, 19]) for problems that are incorrect in the sense of Hadamard, e.g. for smoothing the data in order to avoid the instability due to pollution of the data. Such an approach has an important implication for the practice. At the same time, the very notion of replacing the ill-formulated (e.g. ill-specified and inverse) or ill-posed problem by a well-formulated one, is of no lesser importance. Indeed, if one succeeds in doing so, one arrives at a problem that is also correct in the sense of Hadamard and then it is not susceptible to pollution of the data. For an exposition of the inverse problems, the reader is referred to [3]. 3

The considered problem is ill-posed according to the definition of Hadamard because equation (2) possesses two solutions for the given boundary conditions u(ξ) → 0 when ξ → ±∞. In addition, the trivial solution is a very strong attractor. This brings challenges while constructing schemes based on finite difference and finite element methods for solving the problem. To tackle the solitary-wave identification problem, we introduce an unknown coefficient and transform the original problem to a problem for finding the coefficient in similar way as in Marinov, Christov, and Marinova in [16]. Let u(ξ) 6≡ 0 be a solution of equation (2) with u(ξ) → 0 when ξ → ±∞. Obviously, the function u(−ξ) is also a solution, i.e. the solution is an even function, namely u(ξ) = u(−ξ). Therefore, the problem can be considered on the half-line. In this case, the boundary conditions at ξ = 0 are u(0) = χ,

u0 (0) = 0,

u000 (0) = 0,

(5)

where χ 6= 0 is an unknown constant. It is convenient to scale the function u(ξ) by the unknown constant χ by introducing a new unknown function w(ξ) as u(ξ) = χw(ξ). The problem for coefficient identification becomes A(w, χ) ≡ −cw + χ2

w3 + pwξξ − qwξξξξ = 0 , 3

(6)

with boundary conditions w(0) = 1,

w0 (0) = 0,

w000 (0) = 0,

w0 (ξ) → 0 when ξ → ∞.

w(ξ) → 0 when ξ → ∞,

(7) (8)

Thus, the difficulties connected with the unknown constant χ in the boundary condition (5) are circumvented. Yet, it is an identification problem of unknown coefficient χ from over-posed boundary data. Under certain conditions it is possible to find a constant χ such that the equation (6) has a solution w(ξ) and this solution also satisfies the boundary conditions (7)–(8). In such a case we say that the pair (w, χ) constitutes a solution to the problem (6)–(8). This is an inverse problem for identification of coefficient from over-posed data. 3. Solving the Coefficient Identification Problem We use the Method of Variational Imbedding (MVI) proposed by Christo I. Christov, see Christov [5], for solving the coefficient identification problem. First, we rewrite equation (6) as a system of ordinary differential equations w0 − α = 0,

α0 − β = 0,

−cw + χ2

3

β 0 − γ = 0,

w + pβ − qγ 0 = 0 . 3

4

(9) (10)

The respective boundary conditions at ξ = 0 become w(0) = 1, α(0) = γ(0) = 0.

(11)

Since w → 0 when ξ → ∞, all derivatives of the function w are approaching 0 and corresponding boundary conditions at infinity are w(∞) = α(∞) = β(∞) = γ(∞) = 0.

(12)

Following MVI, the original problem is replaced by a problem for minimizing a functional. In this case the functional is I(w, χ) =

Z∞  0

2

2

2

(w0 − α) + (α0 − β) + (β 0 − γ)

2   w3 + pβ − qγ 0 dx → min , (13) + −cw + χ2 3

where w, α, β, and γ must satisfy the conditions (11) and (12), and χ 6= 0 is an unknown constant. The functional I(w, χ) is a quadratic and homogeneous function of its arguments w and χ; hence, it attains its minimum if and only if all arguments are zero. In this sense there is one-to-one correspondence between the solution of the original problem (6)—(8) and the minimization problem (13). 3.1. Linearization The Euler-Lagrange equations for this functional possess cubic nonlinearity with respect to the function w. It means that for solving the Euler-Lagrange equations numerically, one has to linearize the said equation. Alternatively, one can linearize the integrand in (13) considering the function w as known (say, from the previous iteration) when they appear as coefficients. Following the latter approach for linearization, we introduce µ(ξ) = w3 (ξ) and consider the problem for minimization of the following functional

I(w, χ) =

Z∞ 0

2

2

2

(w0 − α) + (α0 − β) + (β 0 − γ)

2  µ + −cw + χ2 + pβ − qγ 0 dx → min . (14) 3

3.2. Imbedded Boundary-Value Problem Necessary conditions for minimization of the functional in (14) are derived from the Euler-Lagrange equations for the functions w(ξ), α(ξ), β(ξ), and γ(ξ), and for the constant χ. The equations for the functions w(ξ), α(ξ), β(ξ), and γ(ξ) are d 0 µ (w − α) + c(−cw + χ2 + pβ − qγ 0 ) = 0 , (15) dξ 3 5

d 0 d 0 (α − β) + (w0 − α) = 0 , (β − γ) + (α0 − β) = 0 , dξ dξ d µ −q (−cw + χ2 + pβ − qγ 0 ) + (β 0 − γ) = 0 , dξ 3

(16) (17)

The system (15)-(17) is of the eight order. The solution of this system may satisfy four conditions at each boundary point. There exist exact number of boundary conditions for ξ → ∞, see (12). We make use of the so-called natural boundary condition for minimization of the functional at ξ = 0, which is nothing else than the original equation (6), i.e. −cw + χ2

µ + pβ − qγ 0 = 0 . 3

(18)

The problem is coupled with the equation for χ. We rewrite the functional (13) in the form χ4 I= 9

Z∞

2χ2 w dx + 3 6

0

Z∞ 0

w3 (−cw + pβ − qγ 0 ) dx

Z∞ 2 2 2 2 + [(−cw + pβ − qγ 0 ) + (w0 − α) + (α0 − β) + (β 0 − γ) ] dx.

(19)

0

After some algebraic manipulations, the equation for χ2 reduces to

2

χ = −3

R∞ 0

w3 (−cw + pβ − qγ 0 ) dx R∞

.

(20)

w6 dx

0

4. Finite Difference Method 4.1. Grid and Approximations The grid is chosen to be uniform allowing approximation of all operators ξ∞ with central differences. The grid spacing is defined as h = n−2 , where n is the total number of grid points and ξ∞ is a sufficient large number used to approximate infinity. Then the gird points are defined as follows: ξi = (i − 1.5)h for i = 1, . . . , n . The grid point ξ = 0 is the mid-point ξ2 12 , which is important for the numerical procedure. Let us also introduce the notation wi = w(ξi ), αi = α(ξi ), βi = β(ξi ), γi = γ(ξi ), for i = 1, . . . , n. We employ symmetric central differences for approximating the differential equations (15)–(17) and obtain wi−1 − 2wi + wi+1 −αi−1 + αi+1 −γi−1 + γi+1 − cq + pβi − c2 wi − 2 h 2h 2h µi = −cχ2 + O(h2 ), (21) 3 6

αi−1 − 2αi + αi+1 −βi−1 + βi+1 −wi−1 + wi+1 − + − αi = 0 + O(h2 ), (22) h2 2h 2h −γi−1 + γi+1 βi−1 − 2βi + βi+1 −αi−1 + αi+1 − + − βi = 0 + O(h2 ), (23) h2 2h 2h q2

−βi−1 + βi+1 γi−1 − 2γi + γi+1 −wi−1 + wi+1 + (1 − pq) + cq − γi h2 2h 2h −µi−1 + µi+1 = qχ2 + O(h2 ). (24) 6h

for i = 2, . . . , n − 1. The grid allows approximating the boundary conditions with the second order as well. Namely, at ξ = 0, the boundary conditions become w(0) = 1 =⇒ w1 + w2 = 2 + O(h2 ),

(25)

2

α(0) = 0 =⇒ α1 + α2 = 0 + O(h ), γ(0) = 0,

(26)

2

=⇒ γ1 + γ2 = 0 + O(h ).

(27)

Finally, the equation −cw + χ2 µ3 + pβ − qγ 0 = 0 is approximated as β1 + β2 γ2 − γ1 w1 + w2 χ2 p − =c − 2 h 2 3



w1 + w2 2

3

+ O(h2 ).

(28)

The boundary conditions at ξ = ξ∞ are approximated as w(ξ) → 0,

α(ξ) → 0,

β(ξ) → 0,

γ(ξ) → 0,

ξ → ∞ =⇒ wn + wn−1 = 0 + O(h2 ),

(29)

2

ξ → ∞ =⇒ αn + αn−1 = 0 + O(h ),

(30)

2

(31)

ξ → ∞ =⇒ γn + γn−1 = 0 + O(h2 ).

(32)

ξ → ∞ =⇒ βn + βn−1 = 0 + O(h ),

The linearization proposed in Subsection 3.1 allows to invert the matrix of the linear system only once, at the beginning of the iteration process, thus, save computational time. 4.2. Estimation of χ We use the ‘extended midpoint rule’ for approximating the integrals in the equation (20) for χ   n−1 P 3 −γk−1 + γk ¯ w ¯k −cw ¯k + pβk − q h χ2 = −3 k=2 + O(h2 ) , (33) n−1 P 6 w ¯k k=2

where w ¯k and β¯k are the midpoint values of the functions. This approach secures the second order approximation. 7

4.3. Iterative Algorithm Step I Solve the eight-order boundary value problem (15)-(17) for the function w with given χ and µ. Step II If the difference between two consecutive approximations of w is smaller than ε, then proceed to Step III; otherwise, µ is replaced by the newly approximated value of w3 and then go to Step I; Step III With the newly computed w, the coefficient χ is evaluated from formula (33). If the following criterion is satisfied max |wn+1 − wn | <ε h max |wn+1 |

and

max |χn+1 − χn | < ε, h max |χn+1 |

(34)

then the iterations are terminated. Otherwise the index of iteration is incremented n := n + 1 and the algorithm is returned to step Step I. The chosen tolerance in (34) is ε ≤ 10−10 . 5. Results and Discussion 5.1. Verification of the Scheme In order to verify the performance of the described numerical scheme we started the numerical experiments with an analytical solution (see [18])  √p  4p2 3p 2 √ √ ξ , where c = sech . (35) uan (ξ) = − 25q 10q 2 5q Choosing p = 5 and q = 4 in the equation (35), produces the following value √ 3 10 for the coefficient: χan = u(0) = 4 ≈ 2.371708245126285 . . . . 5.1.1. The mesh-size h In the test for verification of the scheme, the value of ξ∞ is chosen to be ξ∞ = 30. The discretization error term is O(h2 ), and the total error of approximation is expected to be O(h2 ). The results from the numerical calculations clearly demonstrate these error orders. Figure 1 shows the pointwise numerical errors for the function u calculated with for different spacings: h = 0.1; 0.05; 0.025; 0.0125. It can be seen that if the spacing decreases twice, the errors decrease approximately four times. This shows that the rate of convergence is 2. The rates of convergence, calculated as rateu = log2

||u2h − uexact || , ||uh − uexact ||

rateχ = log2

are given in Table 1.

8

||χ2h − χexact || , ||χh − χexact ||

(36)

Numerical error for u(ξ)

−3

x 10

h=1/10 h=1/20 h=1/40 h=1/80

3.5 3

unum−uexact

2.5 2 1.5 1 0.5 0 0

5

10 ξ

15

20

Figure 1: The difference between numerical and exact values of u(ξ).

Table 1: Number of iterations, l2 norm of u − uexact , the estimated values of the coefficients χ and the rates of convergence.

h 0.1 0.05 0.025 0.0125

Iterations 56 55 50 44

||u − uexact ||l2 0.00886880 0.00220981 0.00055174 0.00013789

rateu — 2.0048 2.0019 2.0005

χ 2.373540 2.372175 2.371826 2.371738

|χ − χexact | 0.001832 0.000467 0.000117 0.000029

rateχ — 1.9719 1.9969 2.0124

The l2 norm of the difference between the numerical and analytical solutions with four different spacings h = 0.1; 0.05; 0.025; 0.0125 are given in Table 1 along with the rates of convergence. The estimated values of the coefficients χ and the rates of convergence using the same four grid spacings are also given in Table 1. The numerical experiments clearly confirm the quadratic convergence of the numerical solution to the exact solution. 5.1.2. The approximation of infinity (ξ∞ ) It is well known fact that the value of ξ∞ , approximating infinity, is critical for the numerical calculations. If the value of ξ∞ is too large, the computational time would increase. On the other hand, if the value of ξ∞ is very small, this may cause loss of accuracy. The shapes of the logarithm of |uexact −unum | for four values of the numerical 9

Numerical error for u(ξ) −2 −3

log|unum−uexact|

−4 −5 −6 −7

ξ∞=15 ξ =20 ∞

−8

ξ =25 ∞

ξ∞=30 −9 0

5

10

15 ξ

20

25

30

Figure 2: Difference log10 |uh − uanalytic | for four different values of ξ∞ (h = 1/40).

infinity (ξ∞ = 15; 20; 25; 30) are shown in Figure 2. The corresponding values of the l1 , l2 , and l∞ norms for ξ∞ = 15; 20; 25; 30; 35 are given in Table 2. These numerical results show that ξ∞ = 30 is a reasonable value for ξ∞ in this experiment. 5.2. Evolution of the Solitary Waves The evolution of the solitary waves with monotone shapes is examined through a series of calculations by keeping p = 5, q = 4, while varying the value of c in the interval 1.0 ≤ c ≤ 1.5. The obtained results are shown in Figure 3. We want to emphasize that in our numerical calculations the condi-

Table 2: Norms of the difference between numerical and exact solution for five values of ξ∞ .

ξ∞ 15 20 25 30 35

||u − uexact ||l1 0.0160 0.0028 0.0018 0.0017 0.0017

||u − uexact ||l2 0.0069 7.8978e-004 5.5370e-004 5.5174e-004 5.5172e-004

10

||u − uexact ||l∞ 0.0053 4.3335e-004 2.2921e-004 2.2920e-004 2.2920e-004

|χ − χexact | 0.000103 0.000117 0.000117 0.000117 0.000117

−c u + u3/3 + 5u’’− 4u’’’’ =0 3 c=1.0 c=1.1 c=1.2 c=1.3 c=1.4 c=1.5

2.5

u

2

1.5

1

0.5

0 0

2

4

6

8 ξ

10

12

14

16

Figure 3: Solitary waves with monotone shapes for p = 5, q = 4.

Table 3: Number of iterations and the estimated values of the coefficients χ for different values of the parameter q.

q Iterations χ

4 56 2.372750

8 330 2.346457

16 849 2.320944

32 2032 2.299291

64 5073 2.284701

128 12818 2.281947

4p2 is not satisfied for c > 1. Therefore, the obtained here numerical 25q solutions are not members of the family of the exact solutions (35).

tion c =

5.3. Transition from Solitary Waves with Monotone Shapes to Non-monotone Shapes As we already mentioned in subsection 1.2, for certain values of the coefficients we could expect solitary-wave solutions with non-monotone shapes or oscillatory tails. To show the transition from solitary waves with monotone shapes to solitary waves with non-monotone shapes we provide the following test: starting from the set of parameters of the exact solution p = 5, q = 4, and c = 1, used for verification of the numerical algorithm, we calculate numerically the solution for q = 4; 8; 16; 32; 64; 128. For these calculations we use ξ∞ = 60. The shapes of the solution for 13 ≤ ξ ≤ 40 are given in Figure 4. The obtained values 11

− u + u3/3 + 5u’’− qu’’’’ =0 0.03 q=4 q=16 q=32 q=64 q=128

0.02

0.01

u

0

−0.01

−0.02

−0.03

15

20

25

ξ

30

35

40

Figure 4: Transition from solitary waves with monotone shapes (q = 4) to solitary waves with non-monotone shapes (q = 16; 32; 64; 128) for c = 1, p = 5.

Table 4: Number of iterations and the estimated values of the coefficients χ for different values of the parameter c.

c It. χ

11 12383 6.81419

13 6756 5.93167

15 6101 5.54281

17 5836 5.30174

19 5696 5.14281

21 5616 5.03776

23 5570 4.97102

of χ and the number of iterations are show in Table 3 and in Figure 5. The non-monotone behavior is more and more visible as q increases. 5.4. Evolution of the Solitary Waves with Non-monotone Shapes on c In this subsection we keep the coefficients p = 0.05 and q = 25 and change the values of the velocity c to c = 11; 13; 15; 17; 19. The shape of the obtained solution is given in Figure 6. For this calculations we use ξ∞ = 60. The nonmonotone behavior is more and more visible as c increases. The obtained values of χ and the number of iterations are shown in Table 4 and in Figure 7. 6. Conclusions We have proposed a new approach for finding traveling wave solutions of the modified Kawahara equation numerically. Results have been presented to demonstrate the efficiency of the numerical method. The results show that 12

2.4

14000

2.38

12000 10000 number of iterations

2.34

2.32

8000 6000 4000

2.3

0

2000

20

40

60

80

100

120

0 0

140

20

40

60

q

80

100

120

q

(a)

(b)

Figure 5: Results with c = 1, p = 5 for: (a) χ; (b) the number of iterations.

−c u + u3/3 + 0.05u’’− 25u’’’’ =0 1.5 c=11 c=13 c=15 c=17 c=19

1

0.5

u

χ

2.36

0

−0.5

−1

−1.5 2

4

6

8

ξ

10

12

14

16

Figure 6: Solitary wave solutions with non-montone shapes for p = 0.05, q = 25.

13

140

7

13000 12000

6.5

number of iterations

11000

χ

6

5.5

10000 9000 8000 7000

5

6000

4.5 10

12

14

16

18

20

22

5000 10

24

12

14

16

c

18

20

22

24

c

(a)

(b)

Figure 7: Solitary wave solutions with non-monotone shapes for p = 0.05, q = 25; (a) the behavior of χ; (b) the number of iterations.

the method is second order accurate and proved to be efficient for solving the modified Kawahara equation. Solitary wave solutions with monotone and nonmonotone shapes for different sets of the problem parameters are presented. Some of the solutions are entirely new. The method based on coefficient identification is reliable and original approach to establishing new traveling wave solutions numerically. [1] Araruna, F., Capistrano-Filho, R., Doronin, G.: Energy decay for the modified Kawahara equation posed in a bounded domain. J.Math.Anal.Appl. 385, 743-756 (2012) ¨ Bilgil, H.: Optical soliton solutions for the variable [2] Bekir, A., G¨ uner, O., coefficient modified Kawahara equation. Optik 126, 2518-2522 (2015) [3] Bellomo, N., Preziosi, L.: Modelling Mathematical Methods and Scientific Computation. CRC Press Inc. (1995) [4] Biswas, A.: Solitary wave solution for the generalized Kawahara equation. Applied Mathematics Letters 22, 208–210 (2009) [5] Christov C. I.: A method for identifying homoclinic trajectories. Proceedings of the 14th Spring Conference, Union of Bulgarian Mathematicians 571–577 (1985) [6] Christov, C. I., Marinov, T. T., Marinova, R. S.: Identification of SolitaryWave Solutions as an Inverse Problem: Application to Shapes with Oscillatory Tails. Mathematics and Computers in Simulation Journal 80 (1) 56-65 (2009) [7] Engl, H. W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)

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