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In this work, the HB method is extended to search for similarity reduction of nonlinear partial differential equations. This method is generalized and will apply for a (2 + 1)-dimensional higher order Broer-Kaup System. Some new exact solutions of Broer-Kaup System are found.

In the past few decades, there has been the noticeable progress in the construction of the exact solutions for nonlinear partial differential equations, which has long been a major concern for both mathematicians and physicists. The effort in finding exact solutions to nonlinear differential equation, when they exist, is very important for the understanding of most nonlinear physical phenomena. For instances, the nonlinear wave phenomena observed in fluid dynamics, plasma and optical fibers are often modelled by the bell shaped sech solutions and the kink shaped tanh solutions.

We consider the following a (2 + 1)-dimensional higher order Broer-Kaup system:

which is obtained from the Kadomtsev-Petviashvili (KP) equation by the symmetry constraint [

The systems (1)-(4) were given by Li et al [

The homogenous balance (HB) method is a powerful tool to find solitary wave solutions of nonlinear partial differential equations. Fan et al. [

In this work, the HB method is extended to search for similarity reduction of nonlinear partial differential equations. So, more solutions can be obtained by the improved HB method. This method is generalized and can be applied to other nonlinear partial differential equations [9-15].

Similarity Reduction of Nonlinear Partial Differential EquationsWe describe the main steps of our method. For a given PDE, say in three variables, say

we seek its similarity reductions in the form

where

Substituting from Equation (6) into Equation (5) and collecting all terms of

To explain this method, we will apply for a (2 + 1)-dimensional higher order Broer-Kaup system (1)-(4), we suppose their similarity solutions are of the form

where

are determined functions. Balancing the highest order of linear term with the nonlinear terms in every equations (1)-(4) to determining

the Equation (64) take the following form

Substituting Equations (9)-(12) into the original system (1)-(4) and collecting all terms of

To make Equations (13)-(16) be an ordinary differential equations of

where

(86)

There are freedoms in the determination of

If

If

If

If

If

From Equation (17), we get

integrating Equation (87) with respect to

integrating Equation (89) with respect to

By using the rule (e) into Equation (90), we obtain a function

substituting from Equation (90) into Equations (17), (20), (24), (25), (26), (53), (54), (55), (56), (63), (64), (71), (72), (73), (76), (77), (80), (83)and (84) we obtain

where

By using Equation (91) into Equation (18), we get

where

By apply the rule (a) on Equation (93), we obtain

(94)

substituting from Equations (91), (94) into Equation (20), we obtain

using Equations (91)and (94) into Equation (19), we get

where

By using the rule (c) into the above equation (96) to become in the form

then Equation (96) take the form

substituting from Eqs.(

from Equation (91) into Equation (57), we get

where

By using the rule (b) into the above equation (99)

(100)

substituting from Equations.(91), (94) and (97), into Equations.(61), (62), (65), (68), (75), (78), (81)and (82) then,we obtain

Substituting from Equations (91), (94), (97) and (101) into Equations (60), (69), (70), we obtain

where

and

Using this notation, the equation (97) take the following form

substituting Equations (97) and (100) into Equation (74), we obtain

using any equation which we need into Equation (28), we obtain

where

By using the rule (d) after differential with respect to

(111)

Substituting into Equations (9)-(12), we obtain the similarity solutions of the Broer-Kaup system Equations (1)-(4) in the form

(112)

where

with

Substituting from Equation (112) to obtain an ordinary differential equations from the origin system (1)-(4), we get

where

The general solution for the variable

(118)

where

There some subcases for the constants

(119)

where

the solutions for equations (120)-(123) are

(124)

To obtain the solutions for the original system (1)-(4), we substituting from the equations (119), (124) into Equations (112), we get

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