Accepted Manuscript

Stability analysis solutions of the nonlinear modified Degasperis-Procesi water wave equation

M.A. Helal, Aly R. Seadawy, M. Zekry

PII: DOI:

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S2468-0133(17)30032-3 10.1016/j.joes.2017.07.002 JOES 48

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Journal of Ocean Engineering and Science

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28 April 2017 21 July 2017

Please cite this article as: M.A. Helal, Aly R. Seadawy, M. Zekry, Stability analysis solutions of the nonlinear modified Degasperis-Procesi water wave equation, Journal of Ocean Engineering and Science (2017), doi: 10.1016/j.joes.2017.07.002

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Stability analysis solutions of the nonlinear modified Degasperis-Procesi water wave equation

M A Helal1, Aly R. Seadawy2'3 and M Zekry3

1Mathematics Department, Faculty of Science, Cairo University, Giza, Egypt 2Mathematics Department, Faculty of science, Taibah University, Al-Ula, Saudi Arabia 3Mathematics Department, Faculty of Science, Beni-Suef University, Egypt

Corresponding Author:(Aly Seadway) E-mail:Aly742001@yahoo.com

Abstract

In the present study, the solitary wave solutions of modified Degasperis-Procesi equation are developed. Unlike the standard Degasperis-Procesi equation, where multi-peakon solutions arise, the modification caused a change in the characteristic of these peakon solutions and changed it to bell-shaped solitons. By using the extended auxiliary equation method, we deduced some new soliton solutions of the fourth-order nonlinear modified Degasperis-Procesi equation with constant coefficient. These solutions include symmetrical, non-symmetrical kink solutions, solitary pattern solutions, weiestrass elliptic function solutions and triangular function solutions. We discuss the stability analysis for these solutions.

Keywords: Modified Degasperis-Procesi water wave equation, Extended auxiliary equation method, Solitary wave solutions.

PACS Nos.: 02.30.Jr; 47.10.A; 47.11.-j; 47.35.Fg.

1 Introduction

he Degasperis-Procesi (DP) equation

At - Atxx + 4 A Ax - 3^xAxx - AAxxx = 0, (1)

was first discovered in [1] in a search for asymptotically integrable PDEs. The DP equation is a bi-Hamiltonian system and admits interesting traveling wave solutions

[2]. It arises as a model equation in the study of two-dimensional water waves propagating over a flat bed [3-5].

During the past decades, both mathematicians and physicists have devoted considerable effort to the study of this direction. With the rapid development of computer algebraic system like Maple and Mathematica, numbers of powerful methods have been developed to obtain a great many exact solutions, especially traveling wave solutions, of NLPDEs. Such as Backlund transformation method[6]; inverse scattering transformation method[7]; Method of Darboux transformation[8]; Adomi-ans decomposition method[9]; Hirotas bilinear method[10]; The method of homotopy analysis[11-12]; modified tanh-function method[13-14]; Sine-cosine method[15]; the bifurcations method of planar dynamical systems [16-17], homotopy perturbation method[18-19].

Recently, Wazwaz [20-21]; Tian and Song [22] and Liu [23] studied the modified forms of the CH and DP equations which are of the following forms:

At - Axxt + (a + 1)A^x = (b + 1)AxAxx + AAxxx (2)

case 1. a = 2, b =1, Eq.(2) is modified Camassa-Holm (mCH) shallow water equation

case 2. a = 3,b = 2 , Eq.(2) is modified Degasperis-Procesi (mDP) shallow water equation, By using tanh and sine-cosine function methods [24-25] and other direct methods [26-28] , they obtained some solitary wave solutions including peakon-type solutions of the two modified equations. By bifurcation method of planner dynamical, and Ouyang [29] and Wang and Tang [30] study mCH and mDP equations.

2 The auxiliary equation method

We have the nonlinear evolution equation in two dimensions x and t in the form

F(A, Ax, At, Axx, Att,...) = 0 (3)

here F is a polynomial function with respect to some functions and its derivatives. applying this method as

Step I: Suppose that Eq.(3) takes the following formal solution:

A(£ ) = £ (4)

and the variable p satisfying

I)2 = Co + cip(0 + C2P2(0 + C3i (£) + C4i (e ), (5)

£(x, t) = yx — Kt S (6)

where ci(i = 0,1,...., 4) are constants.

Step II: By balancing the highest order nonlinear term with the highest order derivative term of Eq.(3), we determined the parameters n in (4).

Step III: By substituting from the Eq.(4); Eq.(5) and Eq.(6) into the Eq.(3) and collecting coefficients of <pkwe get a set of over-determined equations for ai; 7, k and ci. we deduce these parameters by solving the above system.

Step IV: By substituting obtained in step III and the value of ai; 7, k, ci and into equation (4) can be find the solutions of equation (3)

3 Modified Degasperis-Procesi equation

_y applying the method mention in section II to the modified Degasperis-Procesi equatio

equatio

At - Axxt + 4A Ax = 3AxAxx + AAxxx (7)

Equivalently

At - Axxt + 3(A3)x - ((Ax)2)x - (AAxx)x = 0

Assuming the variable £(x, t) = yx — Kt, the equation (7) transform to the following ordinary differential equation

—ka + yW + 4 ya3 — yV)2 — Y 3AA" = 0 (8)

considering homogeneous balance between 4A2Ax and AAxxx in equation (7), according to the above method in section 2, we obtain n = 2. By supposing the solution of equation (7) in form

A = ao + ai<p(£) + ^ (£) (9)

By substituting from equation(5); (6) and (9) into equation (8) and collecting coefficients of k = 0,1, 2..., 6, l = 0, and suppose that each coefficients equals

zero, we find a set of over-determined equations:

With mathematics software, we obtain the parameters of ai; 7, k and ci via symbolic computation.

Case I. When c0 = c1 = c3 = 0, equation (5) possesses a bell-shaped solitary wave solution

—sech(^c2C) c2 > 0 c4 < 0 (10)

= C3 = 0, eq

^2 = 2sec(^-c2C) C2 < 0 C2 > 0 The parameters ■i and ci can be derived

15 2 1 5y

■0 = 0, ai = 0, a2 = — C47 , C2 = - 2, K = —

2 4Y2 2

■0 = Ô(_9 ^v/15i), ai = 0, a2 = — C472, 82

(-25 ±vT5i), k = ^(117 T (12)

80y2V " ' " 8

We obtain the following solutions of Eq.(7)

Figure(l — b)

Figure(l - <?)

Velocity potential in the fluid solution (13) with various different shapes is plotted: Dark solitary waves in (1a) and contour plot in (1b).

Ala = g(-9 ±

-15 l2/ 1 /

Al = -3-sech (^~(Yx - Kt)) 2Y

--(-25 ± v^5i))sech2(^(yx - Kt))

Figures (1) shown that the dark solitary wave solutions and contour plot of Eq.(13);

in the interval [-5, 5] and [0, 5]

Stability analysis: By applying the stability conditions for the solutions, we obtain the Hamiltonian for the solutions equation (13) in the form

^ = 211A2dx

nhe sufficient condition to discuss the stability of solution as > 0

225e (—1 + e ) I 2e10 + (1 + e20) cosh(—) sinh(—) < 0

Case 2. When c3 = ci = 0, c0 = —, Eq.(5) possesses Kink-shaped solitary wave

solution ___

c2 -c2

= a/_2c4 tanh(y— i) C2 < 0 C4 > 0

Figure(2 — /')

Figure(2 - a)

Velocity potential in the fluid solution (2o) with various different shapes is plotted: Kink-shaped solitary waves in (2a) and contour plot in (2b).

Eq.(5)also possesses a tringular solutic

^4 = V2C4 tan^YC) C2 > 0 C4 > 0

The parameters ai and ci can be derived.

-15 ^ 15 2 5y

■2 = Y C4Y , K = Y,

= 3^(39 ^v/15i), ai = 0, a2 = yC4Y2,

C2 = ^ (18)

(25 t v^5i),

k = t(11Y T 3VÎ5Yi) 8

ain the following solutions of Eq.(7)

-15 15 2 1 A3 = ~8— ytanh (yx - Kt))

A4 = 32(39 ± VI5i) + 32(25 T V^i) tan2^(7x - Kt))

1 / ^ X 3

— (39 ± v15i) +--,

32v ; 32v 1 7 VV 2

Figures (2) shown that the dark solitary wave solutions of Eq.(20); in the interval -20,20] and [0,2]

FigHre(3 — h)

Figure^ - a)

Velocity potential in the fluid solution (27) with various different shapes is plotted: Bell-shaped solitary waves in (3a) and contour plot in (3b).

Stability analysis solutions: T

can be given as

^ = 2 11 A2dx The Sufficient conditio:

iltonian for the solutions equation (20)

Case 3. Wher

solution solution

n for d

he Hamilt iscuss the stability of solution dK > 0

) 2e20 + (1 + e40) cosh(—) ) sinh(—) < 0 V Y J Y

c1 = c0 = 0, Eq.(5) possesses a bell-shaped solitary wave

c2 ^/V^x =--sech i)

c2 2(\f~c2 Pa =--sec (

e parameters a and c can be derived.

a0 = 0,

ai = yc3Y >

«2 = 0,

c2 > 0 c2 < 0

c2 = ""J>

k = f (25)

ao = -(-9 ±</L5i),

a2 = 0,

Figure(4 — b)

Figure(4 - a)

Velocity potential in the fluid solution (36) wit arious different shapes is plotted: Bright solitary waves in (4a) and contour plot in (4b).

Velocity potential in the fluid solution (37) with various different shapes is plotted: Dark solitary waves in (5a) and contour plot in (5b).

C2 = ^(-25 ± Vl5z}, k =1(ll7 T 3^l57i) (26)

20y 2 8

We obtain the following solutions of Eq.(7)

A 4 = - y sech2(—(YX - Kt}} ^ (27)

A = ^(-9 i^i) - ¿(-25 ±-/l5¿)sec2(^ Kt)) (28)

Figures (3) shown that the dark solitary wave solutions o Eq.(27); in the interval [-10,10] and [0, 2]

Stability analysis solution: The Hamiltonian solutions equation (27)

can be derived as ^ = 2 11 A2dx

The Sufficient condition to demonstre ; stability of solution dp > 0

225e20(-1 + e20) 10 + (1 + e20) cosh(—) ) sinh(—) < 0 (29)

Case 4. When c4 = c2 = 0 > 0, Eq.(5) possesses a Weierstrass elliptic function solution

= tf(^T C,g2,g3) (30)

Where g2 = - 4 nd g3 = - 4^ are called invariants of Weierstrass elliptic function. The parameters and Ci can be derived.

-K 15y2c3

a0 = -¡—, ai = —Ô—, a2 = 0, 4y 8

16k2(k - 2y) = 4k(k - 4y) ( )

675y9c3 , Ci = 45y6c3 (3 )

We obtain the following solutions of Eq.(7)

A7 = ^ + ^tf (^(YX - Kt), g2, g3) (32)

Case 5. When c0 = ci = 0, Eq.(5) possesses solitary wave solutions

=_4c2e^_

1 - 2c3e^? + eW(c3 - 4C2C4) (33)

=_4c2e^_

eW - 2c3eVc2? + c2 - 4C2C4 (34)

The parameters a and c can be derived.

af 5ai 25

C4 9k2 ' C3 3k2 ' C2 4k2

ao = 0' a2 = M ' Y = ^ (35)

We obtain the following solutions of Eq.(7)

12K2c2eVC2(YX-KÍ) (c3 - 2e^c2(YX-Ki)(c3 - 4c2c4) + c3e2^(YX-Ki) (c2 - 4c2c4))

C3e2VC2(Tx

5(1 - 2c3e^(YX-Kí) + (c| - 4c2c4))2

12K2c2e^(YX-Kí)(c3e2^(YX-Kí) - 2e^(YX-Ki)(c3 - 4c2c4) + c3 - 4c2c3c4)

„„ = 2_y ^___v 3 ^^^ i ^^^^ (37)

- 2c3^x/C2(7x-Kt) + c2 - 4c2c4)2 V ;

Figures (4a,4b) shown that the bright and dark solitary wave solutions of Eq.(36), Eq.(37) with C3 = 1, -5, k = 0.9; in the interval [-50, 50] and [0, 2] References

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