Traveling Wave Solutions of the Extended Calogero-Bogoyavlenskii-Schiff Equation

Traveling Wave Solutions of the Extended Calogero-Bogoyavlenskii-Schiff Equation

S. M. Mabrouk

Assistant Professor, Department of physics and Engineering Mathematics Faculty of Engineering, Zagazig University, Egypt.

Abstract Traveling wave solutions of the extended

+

+ 1

+ 1

+ 1 1

+

= 0 (2)

CalogeroBogoyavlenskiiSchiff equation (ECBS) is

2

4

4

investigated. The ()-expansion method is applied to find the explicit solutions. New traveling wave solutions of the ECBS equation are obtained. The solitary wave solutions show one-soliton, N-solitons, kink and periodic waves. Some of the resulting solutions are plotted.

Keywords Extended Calogero-Bogoyavlenskii-Schiff equation; ()-expansion method; Traveling wave solutions.

1. INTRODUCTION

   Nonlinear integrable equations are playing a substantial role in modeling most of the scientific and engineering applications [1-5], such as propagation of shallow water waves, optical fibers, condensed matter, electromagnetic, plasma and fluid mechanics. The study of higher dimensional nonlinear integrable equations is one of the most important recent studies [6-8]. Large assortments of mathematical methods of solutions are demoralized in studying these equations, some of these methods are the ()-expansion method [9-11], exponential function method [12, 13], Lax pairs [14, 15], Extended homoclinic

   test [16, 17], Hirota bilinear method [18, 19], Darboux transformation [20, 21], sinecosine and tanhcoth methods [22].

   Bogoyavlenskii and Schiff described the interaction of a Riemann waves along two spatial dimensions, by the nonlinear integrable equation CalogeroBogoyavlenskii Schiff (GBS) equation [23, 24]. Riemann waves dynamics

   He in [18], showed that the extension term does not disturb the integrability of the Songs equation (1), and found its N- soliton solutions using the simplified Hirotas method.

   The purpose of this paper is to find the traveling wave solutions of the extended CalogeroBogoyavlenskiiSchiff equation (2), using the () expansion method. The paper is organized as follows: Section II, is devoted to summarize the () expansion method. In section III, the () expansion method is applied to the extended CalogeroBogoyavlenskiiSchiff equation. Section IV, is devoted to discuss the results. The paper ends with conclusions in Section V.

2. MATHEMATICAL METHOD

   ()-expansion method is excessively used in finding the traveling wave solutions of nonlinear evolution equations [29-32]. It can be summarized as; for any partial differential equation (PDE);

   P(u, ux, uy, ut, uxx, uxy,) = 0

   (3)

   Where p is a polynomial in u and its partial derivatives. Suppose the solution of the partial differential equation (2.1) is in the form;

   u(x, y, t) = u() , = + ± . (4) The constant c is the velocity of the travelling wave. The PDE (3) is reduced to a nonlinear ODE; which can be integrated many times with setting the constants of integration equal to zero for simplicity. Consider the traveling wave solution of the final ODE in the form;

   is one of the most important applications of physics and

   engineering; such as tsunami and tidal in rivers, magento-

   () =

   ()

   (5)

   =0

   =0

   sound waves in plasmas, internal waves in oceans and optical tsunami in fibers.

   Song et al. [25], introduced an integrable (2 + 1)-

   Where G = G() satisfies the second order linear ODE;

   2

   2

   () + () + () = 0 (6)

   Where, = , = , a , and µ are real constants

   dimensional equation. This equation describes the interaction between Riemann wave promulgated along the

   to be determined.

   2 i

   y-axis and long wave promulgated along the x-axis as; The positive integer m is determined through balancing

   +

   + 1

   + 1

   + 1 1

   = 0 (1)

   the highest order linear and nonlinear terms' derivatives

   2

   4

   4

   appearing in the ODE. Substitute (5) and (6) into the final

   Where u(x, y, t) is a function of space variables x, y and temporal variable t. Equation (1) is also obtained by unifying two directional generalization of the potential KdV equation and Calogero-Bogoyavlenskii-Shiff equation [26- 28]. The explicit N-soliton solutions of equation (1) is obtained in [25]. Wazwaz [18], extended equation (1) by adding the planar flux term , introducing the ECBS equation;

   ODE, then collect all terms with the same order of () and set each coefficient to zero yield a set of algebraic equations for ai, c, µ and .

3. TRAVELING WAVE SOLUTION OF THE ECBS EQUATION

   = 2 + tanp ( ) (17)

   2

   2

   2

   2

   2 2 2

   This section presents, the application of ()-

   2) Equation (12) for < 0, is;

   2 2

   2 2

   expansion method to find the explicit solutions of the extended CalogeroBogoyavlenskiiSchiff equation (2).

   1 sin( )+2( )

   = 2 2 [ + ( )]

   2 2 1()+2 sin()

   Differentiate equation (2) with respect to x, yields;

   2 2

   2

   4

   + 6

   + 4

   + 2

   +

   +

   2

   2

   [

   +

   (1 sin( 2 )+2( 2 ))]

   (18)

   + 4

   = 0 (7)

   2 2 1()+2 sin()

   2

   2

   2 2

   We now utilize the wave transformation equation (4) in reducing (7) to the nonlinear ODE;

   (4 4 + 1) + 6()2 + 6 + (5) = 0 (8) Where dashes refer to the derivatives with respect to. Let (4 4 + 1) = (9)

   Then integrate (8) twice with respect to yields;

   Which can be simplified for C1 = 0 and C2 = 1, to;

   = 2 + cot2 ( ) (19)

   3

   3

   2 2 2

   2

   2

   And for C1 = 1 and C2 = 0, to;

   = 2 + + tan2 ( ) (20)

   2

   2

   4

   4

   2 2 2

   + 32 + = 0 (10)

   B. Case 2; = 2 , a = -2, a

   =-2 and = 2 4

   Let = , yields

   + + 32 = 0 (11)

   The balance between and 2, leads to m = 2, and the solution of equation (11) is written as;

   0 3 3 1 2

   1. For > 0, equation (12) becomes;

      2

      2

      +

      +

      [

      [

      = 2 2

      3 3 2

      () = 0 + 1 (

      ) + 2 (

      )2

      (12)

      1 sinh()+2()

      ( 2 2 )] 2

      +

      2

      [ 2

      Where G=G() satisfies equation (6) and ai, and µ, are real

      1( 2 )+2 sinh( 2 )

      2 2

      2 2

      constants to be determined. Substitute from (12) using (6)

      2

      into (11) yields;

      1 sinh( )+2( )

      ( )]

      (21)

      4 3

      2 1()+2 sinh()

      (6

      + 32) ()

      + (10 + 6

      + 2 ) ( ) + 2 2

      2 2

      2 1 2

      1

      at C1 = 0 and C2 = 1, we have;

      2 2 2

      = 2 cotp ( ) (22)

      2

      2

      (31 + 42 + 82 + 2 + 31 + 602) ( ) +

      5 6 3 2 2

      ( 2 + 2 + 6 + + 6 ) ( ) + ( +

      2

      2

      And at C1 = 1 and C2 = 0, we have;

      0

      0

      1 1 2

      1 0 1 1

      = 2 tanp ( ) (23)

      222 + 0 + 32) = 0 (13)

      6 6 3 2 2

   2. For < 0, equation (12) becomes;

   Collecting all terms with the same order of (), and

   2 2

   setting each coefficient to zero yielding; a set of algebraic equations for ai, , µ and ,

   = 2 [ +

   3 3 2

   62 + 32 = 0

   2

   2

   1 sin()+2()

   ( 2 2 )] 2

   +

   2

   [ 2

   102 + 612 + 21 = 0 1( 2 )+2 sin( 2 )

   31 + 422 + 82 + 2 + 32 + 602 = 0

   (14)

   2

   1

   12 + 21 + 62 + 1 + 601 = 0

   (1 sin( 2 )+2( 2 ))]

   (24)

   2 2 2 1()+2 sin()

   { 1 + 22 + 0 + 30 = 0 2 2

   2

   2

   C1 = 0 and C2 = 1, yields;

   Solutions of this system of equations result in two cases;

   = 2 + cot2 ( ) (25)

   7 6 3 2 2

   C1 = 1 and C2 = 0, yields;

   1. Case 1; a0

      = -2µ, a1

      = -2, a2

      = -2 and = 4 2

      8 =

      2

      6

      2 +

      3

      tan2 2

      (

      2

      ) (26)

      The function G() is found through the solution of

      equation (6) by setting, 2 4 =

      1) The solution (12) For > 0 is;

      1 sinh()+2()

4. RESULTS AND DISCUSSION

   This section is motivated to identify and plot the traveling wave solutions of ECBS equation (2), for the two cases;

   = 2 2

   + ( 2

   2 )]

   [ 2 2

   1( 2 )+2 sinh( 2 ) A. Traveling wave solutions for case 1

   2

   Integrating equation (16) with respect to yields, a one-

   1 sinh( 2 )+2( 2 )

   2 [ + ( )]

   (15)

   soliton solution

   of the ECBS equation;

   2 2 ()+ sinh() 1

   1 2 2 2

   2

   2

   Which can be simplified for C1 = 0 and C2 = 1, to;

   = 2 + cotp ( ) (16)

   1

   1

   2 2 2

   1 = coth ( 2 ) = coth ( 2 ( + + )) (27) This solution is plotted in Fig.1 for = 2, t = 2 and = 1.

   And for C1 = 1 and C2 = 0;

   Fig.1. One soliton solution of EBCS equation for =2, t =2 and = -1

   Integrating (17) with respect to yields, a kink-soliton solution u2 of the ECBS equation, as shown in Fig.2, for

   = 2, t = 0.5 and = -2.

   Fig.3. Periodic solution of ECBS equation for = 2, t =1 and = 1.

   Integrating (20) with respect to yields, an N-soliton

   solution u4 of the ECBS equation, as presented in Fig.4, for

   2 = tanh ( 2 ) = tanh ( 2 ( + + ))

   (28)

   = 2, t = 0.2 and = 1.

   = tan ( ) = tan ( ( + + )) (30)

   4 2 2

   Fig.2. Kink solution of ECBS equation for =2, t = 0.5 and =-2.

   Fig.4. N-soliton solution of ECBS equation for = 2, t = 0.2 and = 1.

   Integrating (19) with respect to yields, a periodic solution

   u3 of the ECBS equation, as presented in Fig.3, for = 2, t

   = 1 and = 1.

   1. Traveling wave solutions for case 2

   Integrating (22) with respect to yields, a one-soliton solution u5 of the ECBS equation;

   = cot ( ) = cot ( ( + + )) = + coth ( ) (31)

   3 2 2 5 3 2

   1. Integrating (23) with respect to yields;

      = + tanh ( ) (32)

      6 3 2

      Integrating (25) with respect to yields,

      = + cot ( ) (33)

      7 3 2

      Integrating (26) with respect to yields,

      = tan ( ) (34)

      8 3 2

5. CONCLUSIONS

In this paper, the ()-expansion method is effectively employed to reveal many explicit solutions for the extended Calogero-Bogoyavlenskii-Schiff equation. The presented solutions include a variety of one-soliton, kink, periodic and N-soliton solutions. These aggregations of traveling wave solutions are applicable for different conditions. The results help in studying the wave behaviors in several applications as, deep oceans and nonlinear optics.

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