COMPARISON BETWEEN THE EXACT SOLUTIONS OF THREE DISTINCT SHALLOW WATER EQUATIONS USING THE PAINLEVé APPROACH AND ITS NUMERICAL SOLUTIONS Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2020, vol. 16, no. 3, pp. 463-477. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd200305

NONLINEAR PHYSICS AND MECHANICS

MSC 2010: 35C07, 35C08, 35R11, 83C15

Comparison Between the Exact Solutions of Three Distinct Shallow Water Equations Using the Painleve Approach and Its Numerical Solutions

A . Bekir, M . S . M . Shehata, E . H . M . Zahran

In this article, we employ the Painleve approach to realize the solitary wave solution to three distinct important equations for the shallow water derived from the generalized Camassa-Holm equation with periodic boundary conditions. The first one is the Camassa-Holm equation, which is the main source for the shallow water waves without hydrostatic pressure that describes the unidirectional propagation of waves at the free surface of shallow water under the influence of gravity. While the second, the Novikov equation as a new integrable equation, possesses a bi-Hamiltonian structure and an infinite sequence of conserved quantities. Finally, the third equation is the (3 + 1)-dimensional Kadomtsev-Petviashvili (KP) equation. All the ansatz methods with their modifications, whether they satisfy the balance rule or not, fail to construct the exact and solitary solutions to the first two models. Furthermore, the numerical solutions to these three equations have been constructed using the variational iteration method.

Keywords: Camassa-Holm equation, Novikov - Veselov equation, (3 + 1)-dimensional Kadomtsev-Petviashvili (KP) equation, Painleve approach, traveling wave solutions, numerical solutions

Received August 04, 2020 Accepted August 26, 2020

Ahmet Bekir bekirahmet@gmail.com

Neighbourhood of Akcaglan, Imarli Street Number: 28/4, 26030, Eskisehir, Turkey

Maha S. M. Shehata dr.maha_32@hotmail. com

Zagazig University, Faculty of Science, Departments of Mathematics 44519, Zagazig, Egypt

Emad H. M. Zahran e_h_zahran@hotmail. com

Benha University, Faculty of Engineering, Departments of Mathematical and Physical Engineering Fareed Nada Street, 13511, Shubra, Egypt

1. Introduction

The main idea of this paper concentrates on using the Painleve approach for the first time to realize the solitary wave solution to these three important models of shallow water as well as demonstrating the corresponding numerical solutions according to the variational iteration method. Traveling wave solutions represent waves of permanent shape Z that propagate at constant speed C0. These waves are called solitary waves on the condition that the wave profile Z decays at infinity. If the solitary waves retain their shape and speed after interacting with other waves of the same type, we say that the solitary waves are solitons. There is a close connection between integrability and solitons. The solitons become peaked if they are shaped like the graph of the function Z(x) — e-lxl, and they are then called peakons. It is interesting to notice also the property of orbital stability for the solitary waves, i.e., their shape is stable under small perturbations, both for the smooth solitons and for the peakons. Independently, Fokas and Fuchssteiner [1, 2] derived the Camasss-Holm equation and studied completely the integrable generalizations of the KdV equation with bi-Hamiltonian structures. Also, in the same context Camassa and Holm [3] (see also [4]) derived it as a nonlinear wave equation in which the wavelength is larger compared with the average water depth and which models unidirectional wave propagation on shallow water, and discovered its rich mathematical structure. In this context, we employ the Camassa-Holm equation in terms of the function u(x, t) which represents the fluid velocity in the x-direction at time t. This equation can be written in the form [5]

Ut Uxxt I 2kUx I 3uux 2Ux Uxx UUxxx — 0, (l. l)

where k is related to the critical wave speed Co = y/gHo where Hq denotes the undivided depth. Specifically, we shall focus on the case k — 0 which in fact is not limited since the change of variables u — u(x, t) that represent the fluid velocity in the x-direction, u(x, t) — v(x — kt, t) + k, tends to zero at k — 0. In this case the Camassa-Holm equation has peakon solutions, i.e., solitons with a sharp peak, so with a discontinuity at the peak in the wave slope. According to [6-8], the formalism which describes the unidirectional propagation of shallow water waves over a flat bottom can be represented as

ut uxxt I 3uux 2uxuxx uuxxx — 0* (1*2)

Secondly, we will solve the Novikov - Veselov equation, which is a natural (2 + 1)-dimensional analogue of the Korteweg-de Vries (KdV) equation [9-11]. This equation was named in 2009 after S.P.Novikov and A.P. Veselov [12], who derived a new integrable Camassa-Holm type equation with cubic nonlinearity which possesses a bi-Hamiltonian structure and an infinite sequence of conserved quantities and admits exact peaked solutions in the form

ut uxxt "I 4u ux 3uuxuxx u u^xx — 0* (1*3)

Some results on the existence and uniqueness of global weak solutions to the Novikov equation have been realized by Wu and Yin [13]. The periodic Novikov equation is locally well-posed and was proved by Tiglay [14]. In [15] the range of regularity index of local well-posedness was extended; furthermore, it is clear that there is incompatible continuity from any bounded subset between the maps solution to both the periodic boundary-value problem and the Cauchy problem of the Novikov equation. The character of solutions to the Novikov - Veselov equation depends basically on the regularity of the scattering data for this solution. If the scattering data have no singularity, then the solution disappears uniformly with time. If the scattering data

are not regular, then the solution may develop solitons. Furthermore, at positive energy the solitons of the Novikov - Veselov equation are transparent potentials as in the one-dimensional case (in which solitons have reflection less than potentials). Also, the well-known exponentially decaying solitons are realized when the similarity of one-dimensional case is not satisfied.

Finally, we will solve the (3 + 1)-dimensional Kadomtsev-Petviashvili (KP) equation [16]

Uxt + 6ux + 6uuxx uxxxx uyy uzz 0, (1.4)

which describes the dynamics of weakly dispersive, nonlinear waves whose wavelength is large compared to its amplitude and whose variations in the main direction of propagation (rescaled X) are fast compared to their variations in the second space dimension (rescaled Y). Examples of these phenomena include: (1) the surface waves in shallow water, in which u is a rescaled wave amplitude and a rescaled velocity; the wavelength is long compared to the depth of the water h, which is large compared to the wave amplitude; (2) magneto-elastic waves in antiferromagnetic materials, in which u is a rescaled strain tensor and a rescaled velocity.

2. Description of the Painleve approach

To propose the general formalism of the nonlinear evolution equation, let us introduce R as a function of (x, t) and its partial derivatives as

,Pt ,Vxx,Vtt ,...) = 0, (2.1)

which involves the highest-order derivatives and nonlinear terms. With the aid of the transformation y(x,t) = ), Z = x — CQt Eq. (2.1) can be reduced to the following ODE:

5 (^W" ,...) = 0, (2.2)

where S is a function in ) and its total derivatives.

According to the Painleve approach [17], the exact solution to the nonlinear ordinary differential equation can be written in the following form:

<P(Z) = Aq + S(X) e-NZ, X = R(Z) (2.3)

or

<p(Z) = Aq + AiS(X) e-NZ + A2S2(X) e-2NZ, X = R(Z), (2.4)

— NZ

where X = R(Z) = Ci — e and S(X) in Eq. (2.3) satisfies the Riccati-equation in the form SX — AS2 = 0. One can solve this equation to get

S(X) = sxTT,- (2"5)

Consequently,

= —N e-NZS (X) + Rz e-NZSx, (2.6)

= N2 e-NZS(X) — 2NRCe-NZSx + R((e-NCSx + R\e-NZSxx, (2.7) = —N3 S(X)e-NZ + 3N2RcSxe-NZ — 3NRccSxe-NZ — — 3NR2 Sxxe-NZ + 3RC RccSxx + RzzzSx e-NZ + R\Sxxxe-NC.

(2.8)

3. Application

In this section, we will apply the Painleve approach as a new technique to realize the exact solutions to the Camassa-Holm model, the Novikov- Veselov equations in terms of some variables. Hence, we can easily obtain the traveling wave solutions when these variables take specific values.

3.1. The Camassa —Holm equation

Firstly, we will apply the constructed approach to Eq. (1.2) mentioned above:

Ut ^^xxt I 3UUx 2Ux*Uxx u^u^xxx — 0. (3.1) By admitting the transformation u(x,t) — ), ( — x — Cot, we get

-Co^c + Co^cCC + 3^c — 2^c PCC — WCCC — (3.2) According to the proposed method, the suggested solution is

<p(Q =A0 + e~N<S(X), X(() = R(0 = Ci - —. (3.3)

Thus, we can easily obtain

PC — —Ne-NCS — Ae-2NCS2, (3.4)

m — N2 e-NCS + 3ANe-2NCS2 + 2A2e-3NCS3, (3.5)

^ccc — —n3 e-NCS — 7AN2e-2NCS2 — 12A2 Ne-3NCS3 — 6A3 e-4NCS3. (3.6)

Substituting , PCC and pzcz into Eq. (3.2) and equating the coefficients of different powers of S(()e-NC to zero implies the following two cases:

(i) when

25

N = 1, and A = (3.7) 16 Co

the solution is

16Co e-C

^) = 25X + 16CoXo; (3"8)

(ii) when

N = -1, and A = (3.9) 2Co

the solution is

*K> = rak- (3'10)

3.2. The Novikov — Veselov equation

Secondly, we will apply the constructed approach to Eq. (1.3) mentioned above: Ut Uxxt I 4u Ux 3uUxUxx U Uxxx — 0. By admitting the transformation U(x,t) — ), ( — x — C0t, we get

—CoPz + CoPzCC + — 2PPC PCC — P2PCCC — 0. (3.11)

Using the transformation (3.1), substituting pz, PCC and pzcz into Eq. (3.11) and equating the coefficients of different powers of ) to zero, we obtain the following two cases:

1.5

l.Oy

0.5

-10 -5 -0.5 -1.0 5 10

Fig. 1. The plot of Eq. (3.8) in 2D and 3D with values: TV = 1, A = C0 = 1, X0 = 1, Cx = 1.

16Cq

0.20 1

0.15 \

0.10

1 , . 0.05 ..........

-10 -5 -o.os -0.10 5 10

Fig. 2. The

(i) when

the solution is

(ii) when the solution is

plot of Eq. (3.10) in 2D and 3D with values: TV = -1, A = C0 = 1, X0 = 1, Cx = 1.

2Cq

N = 1, and A =

p(Z ) =

-1 - 2CQ ± v7! + 1174CQ

9Co

9Co e-z

(-1 - 2C0 ± v/TTll74Cf) X + 9C0X0' 2 - 3C0 ± \/9Cg ~ 102Co + 4

N = -1, and A =

p(Z ) =

6Co

6C0 ez

(2 - 3 c0 ± v^c^ - 102C0 + 4) x + 6C0x0

(3.12)

(3.13)

(3.14)

(3.15)

-10

-5

0.4 0.2

-0.2 ^8.4

Fig. 3. The plot of Eq. (3.13) in 2D and 3D with values: N = 1, A

Cq = 1, Xq = 1, Ci = 1.

10

- 1 - 2Co + Vl + U74Cg

9Co

0.4

0.2

-10 -5 -0.2 5 10

4

0.B

Fig. 4. The plot of Eq. (3.13) in 2D and 3D with values: N = 1, A =

Cq = 1, Xq = 1, Ci = 1.

- 1 - 2C0 - Vl + 1174Cl

9Cq

10

-10 -5 5 10

-5

-10 r

-15

-20 \

Fig. 5. The plot of Eq. (3.15) in 2D and 3D with values: N = -1, A =

Cq = 1, Xq = 1, Ci = 1.

2 - 3C0 + V9Co - 102Q, + 4 6C~n :

2.0

1.5

1.0!

№ ..........

-10

-5

10

Fig. 6. The plot of Eq. (3.15) in 2D and 3D with values: N = -1, A =

Co = 1, Xq = 1, Ci = 1.

2 - 3C0 - V9Co - 102Co + 4

6Co

3.3. The (3 + 1)-dimensional Kadomtsev — Petviashvili equation

Finally, we apply the constructed approach to Eq. (1.4) mentioned above:

uxt + 6ux + 6uuxx uxxxx uyy UZZ 0,

Using the transformation u(x, t) = p(Z), Z = x + y + z — Cot, we get

— (Co + 2)pcc + 6(pc )2 + 6ppcc — pccc = 0. (3.16) Integrating twice, we obtain

— (Co + 2)p + 6p2 — pcc = 0. (3.17)

Again, using the transformation (3.1), substituting into Eq. (3.17) and equating the coefficients of different powers of cp(() to zero gives A = N = ±\/Cq — 2:

(i) when

the solution is

A = N = Co - 2,

p(Z ) =

(>/Cb^2 - e-v^C) + (Co - 2)X0 '

(3.18)

(ii) when

the solution is

1

A = -, N = -y/Co-2,

p(Z ) =

(2 - C0

+ ev/^c) + (2 - Co)Xo

(3.19)

Fig. 7. The plot of Eq. (3.18) in 2D and 3D with values: TV = 2, A = TV = y/C0 - 2, C0 = 6, X0 = 1, C1 = 1.

4. The variational iteration method

Consider a differential equation with inhomogeneous term f (Z). The linear and the nonlinear operators R and S become

LH + NH = f (Z). (4.1)

The VIM proposes the following correction functional for Eq. (4.1):

C

Hm+1 (Z) = Hm(Z) + J X(t)(LHm(t) + NHm(t) - g(t)) dt, (4.2)

0

where A is a general Lagrange multiplier, which can be optimally identified via the variational theory, and Hm is a restricted variation which means 5Hm. The Lagrange multiplier A is crucial and critical in the method, and it can be a constant or a function. With A determined, an iteration formula should be used for determination of the successive approximations Hm+1(Z);

n ^ 0 of the solution H(Z). The zeros approximation H0 can be any selective function. However, using the initial values, H(0) and H'(0) are preferably used for the selective zeros approximation u0, as will be seen later. Consequently, the solution is given by H(Z) = lim Hm(Z). It is

C ^^

interesting to point out that we formally derived the distinct forms of the Lagrange multipliers A in [18], hence we skip details. We only set a summary of the obtained results.

It is important to give the significant forms of Eq. (4.2) briefly according to the Lagrange multipliers in these results.

For the first-order ODE of the form

H' + q(Z )H = p(Z), H (0)= p, (4.3)

we find that A = -1, and the correction function gives the iteration formula

C

Hm+1 (Z) = Hm(Z) -J (H'm (t) + q(t)Hm(t) - p(t)) dt. (4.4)

0

For the second-order ODE of the form

H ''(Z) + cH' (Z) + dh(Z ) = g(Z), H (0)= p, H'(0)= n,

we find that A = t - x, and the correction function gives the iteration formula

C

Hm+1 (Z) = Hm(Z) + j(t - x)(H'm(t) + cH'm(t) + dHm - g(t)) dt. (4.6)

0

For the third-order ODE of the form

H'''(Z) + cH''(Z) + dH'(Z) + eH (Z) = g(Z), H (0)= p, H'(0)= n, H''(0)= a,

we find that A = — (t — x)2, and the correction function gives the iteration formula

C

1

(4.5)

(4.7)

Hm+1(0 = Hm(C) x)2(H'^(t) + cH^t) + dH'm(t) + eHm - g(t)) dt. (4.8)

0

(4.9)

Consequently, for the general form of ODE

H(m) + f (H', H", H'" H(m-1)) = g(Z), H (0) = po, H '(0) = pl, H ''(0) = p2 ..., Hm-1 (0) = pm-1.

The Lagrange multiplier A takes the general form A = ---(t — x)m~l, while the general

(m — 1)!

form of iteration rule becomes

m Z

Hm+1(0 = Hm(C) I J{t-x)m~l [H^ + f(H',H",H"',...,H^)-g(t))dt. (4.10)

0

Furthermore, the zero approximation Ho (Z) can be perfectly selected to be

H0(O = ffo(o) + H'{ 0)C + 0)C2 + 0)C3... +

m—1 /-m— 1

2!

where m is the order of the ODE.

3!

(m — 1)!

H m-1 (0)Z

4.1. The Camassa —Holm equation

For the third-order differential equation (3.2) mentioned above we have

—Co pz + Copzzz + 3ppz — 2pz pzz — PP((( = 0. Case 1: Under the initial condition

25

A = —, N = 1, p(0) = 16, y>'(0) = -400, 16

(4.11)

(4.12)

according to the variational iteration method, the first and second iterations at p(0) = 16 and p(0) = —400 for each of the following cases can be introduced as

po(Z) = p(0) + Zp'(0), po(Z) = 16 — 400Z,

z

pi(Z) = po(Z) — i ( —po(t) + Po (t) + 3po(t)p'o(t) — 2po(tp(t) — po(t)p/o//(t)) dt,

z

pi = i6 — 400c— J i400 — 1200(16 — 400t dt = 16 + 48400Z — 240000Z 2

z

(4.13)

P2 (Z) = P1 (Z) — J ( —p1(t) + pi(t) + 3pi (t)pi'(t) — 2pi (t)p1 (t) — pi (t)p1 (t)) dt, o

p2 (Z ) = 16 — 884336Z — 519763976Z2 + 13558729530Z3 — 86544060000Z4. (4.14)

1 , , , -10 .......... 10

-2 1014 ; \

-4 1014 : \

-6 1014 : \

-8 1014 : \

Fig. 9. The plot of Eq. (4.14) in 2D and 3D with values: N=1,A= C0 = 1, X0 = 1, Cx = 1.

16 Co

1

Case 2: Under the initial condition A = N = -1, (/7(0) = -1, <¿/(0) = 0.5,

Vq (Z ) = v(0) + Zv'(0), VQ (Z ) = 1 + 0.5Z,

c

V1 (Z ) = vq (Z ) -J ( -Vo (t) + vQ" (t) + 3vq (t)vQ (t) - 2vQ (t)vQ' (t) - vq (t)v'S (t)) dt,

Q

c

^ = -1 + 0.5C - J[-0.5 + 3(-l + 0.5i] dt = -1 + 4C -

(4.15)

c

V2 (Z ) = V1 (Z ) -J ( -vi(t) + vi" (t) + 3^1 (t)v/ (t) - 2^1 (t)vï (t) - (t)vi' (t)) dt,

Q

V2(Z) = -1 + 18.5Z - 32.6Z2 + 7.7Z3 - 1.2Z4. (4.16)

-10 10

/ -5000 : \

-10000

-15000

Fig. 10. The plot of Eq. (4.16) in 2D and 3D with values: N = -1, A = -0.5.

4.2. The Novikov — Veselov equation

For the third-order differential equation (3.11) mentioned above we have

-CQVc + CQVCCC + 4vvc - 2VVCVCC - VVCCC = 0.

Case 1: Under the initial condition

A = 3.5, N = 1, v(0) = 1, V (0) = -4.5,

(4.17)

according to the variational iteration method, the first and second iterations at p(0) = 1, p'(0) = -4.5 for each of the following cases can be introduced as

VQ(Z) = v(0) + ZV(0), VQ(Z) = 1 - 4.5Z,

c

V1 (Z ) = VQ (Z ) -/(-vQ (t) + vQ" (t) + 4vq (t)vQ (t) - 2v(t)vQ (t)vg (t) - vQ (t)(t)vii/ (t) dt,

Q

1.5-107 /

. 1.0 -107

\ 5.0-106 J

-10 -5

10

Fig. 11. The plot of Eq. (4.19) in 2D and 3D with values: N = 1, A = 3.5, Co = 1, Xo = 1, Ci = 1.

z

pi = 1 — 4X — J l4.5 + 4(1 — 4.5t)]dt = 1 — 13Z + 9Z 2,

(4.18)

z

p2(Z) = pi(Z) p1 (t) + pi"(t) + 4pi(t)pi(t) — 2p(t)pi(t)p'i(t) — pi(t)(t)pi//(t)) dt, o

p2 (Z) = 1 — 442Z + 3042.4Z2 — 3744Z3 + 1296Z4. (4.19)

Case 2: Under the initial condition

A = —4, N = 1, p(0) = 1, p' (0) = —3,

(4.20)

according to the variational iteration method, the first and second iterations at p(0) = 1, p(0) = —3 are

po (Z ) = p(0) + Zp'(0), po (Z ) = 1 — 3Z,

z

pi(Z) = po(Z) —J {—p'o(t) + pf(t) +4po(t)po(t) — 2p(t)po(t)po'(t) — po(t)(t)po''(t)) dt,

o

z

2 (4.21)

pi = 1 — 3c — J [-l + 4(l - ^ = 1 — 6c + 6Z 2,

o

z

p2(Z) = pi(Z) —J(—pi(t) + pi"(t)+4pi(t)pi(t) — 2p(t)pi(t)pi(t) — pi(t)(t)pi//(t)) dt, o

p2 (Z) = 1 — 125Z + 433Z2 — 718Z3 + 360Z4. (4.22)

4.3. The (3 + 1)-dimensional Kadomtsev — Petviashvili equation

For the second-order differential equation (3.17) mentioned above we have

—8p + 6p — pzz = 0.

\ 4-106

\ 3-106

\ 2-106 J

\ 1-106

-10 -5

10

Fig. 12. The plot of Eq. (4.22) in 2D and 3D with values: N = —1, A = —4, Co = 1, Xq = 1, Ci = 1.

Case 1: Under the initial condition

A = 0.5, N = 2, p(0) = 1, p' (0) = —3,

(4.23)

according to the variational iteration method, the first and second iterations at p(0) = 1, p(0) = —3 are

po (Z ) = p(0) + Zp'(0), po (Z ) = 1 — 3Z,

z

pi (Z) = po (Z) — J (—8po (t) + 6p2 (t) — po'(t)) dt, o

z

pi = 1 - 3c — J—8(1 — 3c ) + 6(1 - 3Z )2idt = 1 - z + 6Z 2 - 18c3,

o

z

p2 (Z) = pi (Z) — J ( — 8pi (t) + 6p2 (t) — pi (t)) dt,

o

p2(Z ) = —1 + 16Z — 106Z2 — 3Z3 + 36Z4 — 48.4Z5 + 216Z6.

(4.24)

(4.25)

2.0-108

1.5-108

1.0-108

5.0-107 J

10

Fig. 13. The plot of Eq. (4.25) in 2D and 3D with values: N = 2, A = 0.5, Co =6, Xq = 1, Ci = 1.

Case 2: Under the initial condition

A = -0.5, N = -2, p(0) = 1, p'(0) = -1,

(4.26)

according to the variational iteration method, the first and second iterations at p(0) = 1, p' (0) = 1 are

P0 (Z ) = p(0) + Zp (0), po (Z ) = 1 + Z,

C

P1 (Z) = P0 (Z) -J -8p0 (t) + 6p2 (t) - p0 (t)) dt,

0

C

P1 = 1 + Z -J [-8(1 + Z) + 6(1 + Z )2] dt = 1 + 3Z - 2Z2 - 2Z3, 0

C

P2 (Z) = P1 (Z) -J -8p1 (t) + 6p2 (t) - p'1 (t)) dt, 0

P2(Z) = 1 + 9Z - 2Z2 - 17.3Z3 + 20Z4 + 9.6Z5 - 8Z6 - 3.4Z7.

(4.27)

(4.28)

, 6-106 \ 4-106 \ 2-106

-10 -5 10

-2-106 \

-4-106 \

-6-106 \

Fig. 14. The plot of Eq. (4.28) in 2D and 3D with values: N = -2, A = -0.5, Co =6, Xo = 1, Ci = 1.

And for all previous cases the higher iteration can be demonstrated according to the following steps:

C

P3 (Z ) = P2 (Z ) - y(-P2(t) + P2' (t) + 3p2 (t)p2 (t) - 2p2 (t)p2' (t) - P2 (t)p'2 (t)) dt, 0

£

PN+1 (Z ) = PN (Z ) -J-'n (t) + pN (t)+3pN (t)p'N (t) - 2p'N (t)p'N (t) - PN (t)p'N (t)) dt, 0

(4.29)

using the fact that the exact solution is obtained by using p(Z) = lim pN(Z).

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