Reh M.El-Shiekh , , , Mhmoud Gllh

a Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt

bCollege of Business Administration in Majmaah, Majmaah University, AL-Majmaah 11952, Saudi Arabia

c Department of Physics,College of Science at Al-Zulfi, Majmaah University, AL-Majmaah 11952, Saudi Arabia

d Geomagnetic and Geoelectric Department, National Research Institute of Astronomy and Geophysics (NRIAG), 11421 Helwan, Cairo, Egypt

ABSTRACT In this study, the generalized modified variable-coefficient KdV equation with external-force term(gvcmKdV) describing atmospheric blocking located in the mid-high latitudes over ocean is studied for integrability property by using consistent Riccati expansion solvability and the necessary integrability conditions between the function coefficients are obtained.Moreover, several new solutions have been constructed for the gvcmKdV.Additionally, the classical direct similarity reduction method is used to reduce the gvcmKdV to a nonlinear ordinary differential equation.Building on the solutions given in the previous literature for the reduced equation, many novel solitary and periodic wave solutions have been obtained for the gvcmKdV.

Keywords:Atmospheric blocking in oceans The generalized variable-coefficients modified KdV equation with external-force term Consistent Riccati expansion solvability Direct similarity reduction method Solitary wave solutions Periodic wave solutions

1.Introduction

Water waves is one of the important phenomena in Nature.The study of water waves and their several modifications is important to fluid dynamics, general, and to the dynamics of oceans in particular [1-5] .From a mathematical perspective, water waves were the reason for constructing the theory of nonlinear dispersive waves and solitons.The problem of water waves in oceans attracts the attention of many scientists like Korteweg and de Vries who investigated the KdV equation as a mathematical model for shallow water waves in channels and oceans and put the basics for the theory of solitary waves.Since that time, the KdV-type equations have become a very important model and many applications were discovered for it such as internal gravity waves in lakes of changing cross-sections, ion-acoustic waves in plasmas, interfacial waves in a two-layer liquid with varying depths,...etc.[6-16] .Therefore, The KdV-type equation is considered as a key for many investigated and modified methods in nonlinear partial differential equations like symmetry groups, B?cklund transformation, Painléve analysis,trail equation method,...etc.[17-70]

One of the imperative KdV-type equations is the generalized variable-coefficients modified KdV equation (gvcmKdV) with external-force term given by [71-73]

where fiare arbitrary functions of twith i = 1 , ..., 6 .The gvcmKdV describes the blocking events in atmospheric and oceanic dynamical systems.Moreover, Eq.(1) contains many types of KdV equation which presens other applications, for instance, internal solitary waves (ISWs) in ocean, pressure pulses in fluid-filled tubes of special value in arterial dynamics, trapped quasi-one-dimensional Bose-Einstein condensates, ion-acoustic solitary waves in plasmas and the effect of a bump on wave propagation in a fluid-filled elastic tube [15,16,46,61,70] .

Based on the aforementioned importance of the gvcmKdV, we are going to study the integrability of this equation using the consistent Riccati expansion (CRE) technique and reducing it to nonlinear ordinary differential equation of fourth order using the classical Clarkson and Kruskal (CK) direct similarity reduction method,then in both integrability and reduction procedure’s many novel solitary and periodic wave solutions will obtained.Finally, the solitary wave propagation affected with the chosen values of the variable coefficients was discussed.

2.The consistent Riccati expansion solvability

The consistent Riccati expansion solvability is a method used to discuss the integrability property as follows [69-70] :

1- Assume that a nonlinear partial differential equation is given by

2- Consider that Qhas a solution in a form

where vk( x, t ) are arbitrary functions in the independent variables x, tand R (w ) is a solution of the Riccati equation

where a, band care arbitrary constants and w = w (x, t) is an unknown function to be determined.The integer n can be obtained by the balance method, then by substituting from (3) in (2) using(4) , a polynomial in R (w ) is obtained.Additionally, by removing all coefficients of R (w ) ;a system of partial differential equations arises.

Definition 1.If the partial differential system obtained by finishing all powers of the polynomial R ( w ) is consistent, or, not overdetermined, we call that the expansion (3) is a CRE and Eq.(2) is integrable according to CRE solvable.

3.Integrability and solutions for the gvcmKdV

In this section, we have applied the CRE solvability steps given in the previous section on the gvcmKdV Eq.(1) .From the balance method, we get n = 1 , therefore, we can assume that the solution of equation ( 1 ) takes the form

where v0(x, t) , v1(x, t) are arbitrary functions.Using Maple software, and by back substitution from (5) into Eq.(1) using (4) ,a fourth-order polynomial in R is constructed.Furthermore, by equating the coefficients of R with zero, a partial differential system of only five equations is obtained.We have found that the obtained system for gvcmKdV equation is consistent, or, not overdetermined with the following solutions for it (the detailed calculations are neglected because its too long):

where C1is an integration constant.With integrability conditions

where ′ means differentiation with respect to time t.Additionally,by using the known solutions for the Riccati Eq.(4) , see [74] , we can obtain many new solutions for the gvcmKdV equation as follows:

Type I: If b2-4 ac ＞ 0 and bc0 (or ac0) , parabolic type solutions for the gvcmKdV are obtained

where A, B are two non-zero real constants.

where dis an arbitra ry constant.

Type III: If b2-4 ac ＜ 0 and bc0 (or ac0) , several new periodic wave solutions are obtained for equation ( 1 ) as follows:

with vi= vi( x, t ) , i = 1 , ..., 14 and v 0 = v0(x, t) given by (6).

4.CK similarity reduction method

The classical Clarkson and Kruskal (CK) direct similarity reduction method was first introduced by Clarkson and Kruskal [75] , and later on, it was enlarged and modified by many authors [15,16,61-68] .In this paper, we have used the classic CK method because the other modified CK methods, specially the connected CK with homogeneous balance method, couldn’t be used to reduce Eq.(1) .

In the following, the main steps of the classic CK method show below:

1) If ?is a partial differential equation given by

where v = v (x, t) , fi(t) , i = 1 , ..., 6 are arbitrary functions in t.

2) Assume that

where ? = ?(x, t) is a similarity variable, σ(x, t) , and γ(x, t) are arbitrary functions in (x, t) and will be determined later.Collecting all coefficients of V (?) and equating it with the coefficient of the most linear term multiplied with an arbitrary function Θj(?) , j =1 , 2 , ...k.Then, a partial differential system in σ, γand ?will be obtained, to solve the determined system we can use the following rules.

a ) If σ(x, t) = γ(x, t) Θ(?) + σ0(x, t) , then we can assume that Θ(?) = 0 ,

b) If γ(x, t) = γ0(x, t)Θ(? ) , then Θ(? ) can be assumed as constant.

c) If Θ(?) = ?0(x, t) , then we can take Θ(?) = ?.

At last, Eq.(23) is reduced to a nonlinear ordinary differential equation with constant coefficients.

5.Reduction and solutions for gvcmKdV

The target of this section is to transform the gvcmKdV equation into ODE using the classical CK method, so by substituting from( 24 ) into ( 1 ) , the following partial differential equation is given

In order to convert equation ( 25 ) to an ordinary differential equation in V, we need to make the coefficients of equation (26) constants or functions on ?, therefore, we take the coefficient of V ′′′as a normalized coefficients and equate other coefficients in ( 25 )with f1γ( ? ) , i = 1 , ..., 8 as follows:

By using the assumptions (a -c) in section (3) , we get

where σ, γand ?take the form

and integrability conditions

By backing substitution from (28) into (25), the similarity solution of gvcmKdV takes the form:

with the following similarity variable

where Cis an integration constant.Then equation ( 25 ) becomes

Integrate equation ( 32 ) twice with respect to ?, we obtain

where A1and A2are integration constants.Equation (33) is a Riccati equation with many solutions [63,64] , where we concentrate our attention on solutions contain all variable coefficients only, it means that b 20 and b 30 , therefore, we get the following solutions for Eq.(33)

where A1= 0 and A2= 0 , then back substitution from (34-35) into(30) the following new solutions for the gvcmKdV are arising

with ?given by Eq.(31) under integrability conditions (29).If b 3 =0 in equation ( 33 ) , then many new Jacobi elliptic wave solutions could be obtained for the gvcmKdV.

6.Physical and engineering applications

The gvcmKdV describes the atmospheric blocking weather phenomena happens at mid-high latitudes, it often causes extraordinary flood, extreme drought or extreme cold, and other abnormal phenomena [14,71-73] .Also, gvcmKdV equation can describe internal solitary waves in ocean, pressure pulses in fluid-filled tubes of special value in arterial dynamics, trapped quasi-one-dimensional Bose-Einstein condensates, ion-acoustic solitary waves in plasmas and the effect of a bump on wave propagation in a fluid-filled elastic tube [15,16,46,61,70] .In this section, we are going to present some figure plots for solution v1( x, t ) with different values for the variable coefficients to show how it could change the solitary wave shape

Fig.1.Propagation of solitary wave solution v 1 with f 1 (t) = f 3 ( t ) = tand f 2 ( t ) =f 4 ( t ) = -t.

Fig.2.The solitary wave solution v 1 when f 1 ( t ) = f 3 ( t ) = t , f 2 ( t ) = -t , and f 4 ( t ) =sin ( t ) .

In Figs.( 1-4 ), the propagation of the solitary wave solution v1for the gvcmKdV equation is determined by using the different selected values of the variable coefficients f 1 ( t ) , f 2 ( t ) , f 3 ( t ) , and f 4 ( t ) .We can observe in Figs.( 1-4 ), that its shape changed from one figure to other because of the change in the variable values.With fixed parameters a = c = C 1 = 1 , b = 3 , and k 1 = 0 .5 .

Fig.3.Shape of the solitary wave solution v 1 with f 1 ( t ) = -t , f 3 ( t ) = f 2 ( t ) = t , and f 4 ( t ) = sin ( t ) .

Fig.4.The shock wave solution v 1 when f 1 ( t ) = f 3 ( t ) = t , f 2 ( t ) = -t , and f 4 ( t ) =exp ( t ) .

7.Conclusion

In this paper, we have studied the gvcmKdV which describes atmospheric blocking located in the mid-high latitudes over ocean.Two different techniques, the CRE solvability and the CK direct reduction method were adopted.Finally the following main important results have been obtained: 1) The nonlinear partial differential system (PDEs) obtained from the CRE solvability method is consistent with solutions given by equations (6-7) under the integrability conditions (8) , therefore, the gvcmKdV is considered as an integrable equation.2) The PDEs system obtained from the CRE is solvable; if we take v0(x, t) = σ(x, t) , v1(x, t) = γ(x, t)and w = ?, with integrability conditions (29), in this case, Riccati Eq.(4) yields if a = -then, solutions in equations (9-23) are obtained by the CK direct similarity method as well.3) The reason why we have used the classic CK technique, is because of the external force term f 6 ( t ) ,which makes other modifications made for the CK method in literatures [15,16,61-68] couldn’t be applied.4) The analytical and numerical solutions obtained before in literatures [7,8,14,71-73] can be considered as a special case of the obtained solutions.5) The solitary wave propagation is affected by the different choices of the variable coefficients, so, the atmospheric blocking weather phenomena have occured at mid-high latitude.It often causes flood,extreme drought, or extreme cold, and other abnormal phenomena that can be controlled by choosing the suitable variable coefficients values.

Finally, as we discussed in the introduction, the importance of gvcmKdV as a model describes many phenomena.We hope that the investigation of novel solitary and periodic wave solutions shed light on new types of applications of this equation in physics and ocean engineering, and describe realistic behavior of the atmospheric blocking in the mid-high latitudes over ocean.

Declaration of Competing Interest

There is no conflict of interest for the author.

Acknowledgement

The author would like to thank the Deanship of Scientific Research, Majmaah University, Saudi Arabia, for funding this work under project No.R-2021-222.