The major concern of this work is to propose new prototypes of surface hybrid waves, in particular waves propagating without sprawl or deformation on the surface of a fluid. The model considered for this purpose is the modified KdV (Korteweg-de Vries) equation. A peculiarity of the obtained solutions is that they form packages constituted by combinations of waves be longing to the two main families of well-known bright and dark solitary waves. This putting together creates competitions between the different components of the considered packages which, following the values assigned to the parameters of the considered system and in relation to those of the wave parameters, generate hybrid or multi-form structures. The direct method of resolution which made possible the obtained results is that of Bogning-Djeumen Tchaho-Kofane extended to the new implicit Bogning functions. The existence conditions of some solutions are obtained. The numerical simulations carried out with a view to testing the observable and applicable characters of the obtained solutions revealed their stabilities over a relatively long time, and at the same time, confirmed the recommended theoretical forecasts. We are convinced that the solutions proposed as part of this work will make it possible to detect, understand and explain some physical phenomena linked to fluid molecular interactions, former or new, which constantly occur on the fluid surfaces, mainly at the shallow water surface.
Since its existence, the universe in all its extent and its diversity has made available to human beings, a multitude of physical systems for the most part complex and dynamic, whose understanding of the behavior of these different systems, is more or less beneficial for our progress and our security. These systems in our conception are presented as a domain consisting of an interior environment and an exterior environment both separated by an interface. It, therefore, appears that these three regions are susceptible to many natural or induced phenomena [
u t + a u u x + u x x x = 0 , (1)
where a is a constant, u u x the Burgers-type nonlinearity term which describes the steepening of the wave, u x x x the third-order dispersion term which accounts for the spreading or dispersion of the wave, and u t characterizes the time evolution of the wave propagating in one direction. Equation (1) is a model which accounts for both dispersion and non-linearity to the lowest order. In this context, one shows that solitons can propagate without strain or sprawl. This equation can admit infinity of solutions of the soliton types [
The aim of this work is therefore to use the Bogning-Djeumen Tchaho-Kofané method (BDKm) extended to the implicit Bogning functions (iB-functions) to construct new prototypes of hybrid solitary wave solutions of the modified Korteweg de Vries equation (mKdVE). This is how, in Section 2, we give a brief presentation of the new iB-functions on which the BDKm theory is extended. Section 3 is devoted to the construction of the analytical solutions followed by a numerical study of their stabilities while Section 4 takes care of the useful and necessary discussions for the reader’s understanding. Section 5, for its part, concludes this work while highlighting some perspectives.
The BDKm theory in its latest version is based on the use of iB-functions for the resolution of nonlinear partial differential equations. This use involves mastering the properties of the iB-function in the management and organization of calculations. The iB-functions are defined as [
J n , m ( ∑ i = 0 p α i x i ) = sinh m ( ∑ i = 0 p α i x i ) cosh n ( ∑ i = 0 p α i x i ) = sinh m ( ∑ i = 0 p α i x i ) sech n ( ∑ i = 0 p α i x i ) . (2)
The member on the left is the implicit form and the member on the right is the explicit form of the function, where α i , ( i = 0 ; 1 ; 2 ; ⋯ ; p ) are the parameters associated with the independent variables x i , ( i = 0 ; 1 ; 2 ; ⋯ ; p ) , m is the power of the numerator, n that of the denominator. Thus, these functions are generally used in nonlinear (partial) differential equations of the form
F ( ϕ , ϕ ξ , ϕ ξ ξ , ⋯ , | ϕ | 2 , ϕ | ϕ | 2 , ⋯ ) = 0 , (3)
where F is an arbitrary function which is very often polynomial, ϕ ( ξ ) the unknown function to be determined and ξ = ∑ k = 0 p η ( x k − ν t ) the independent variable. The principle consists in building the solution in the form
ϕ ( ξ ) = ∑ i j λ i j J i , j ( η ξ ) , (4)
where λ i j are constant coefficients to be determined and η a wave parameter. Then, taking into account Equation (4) in Equation (3) landed to the main equation of gamuts
∑ i j n P n ( λ i j , η , ν ) J n , 0 ( η ξ ) + ∑ i j m Q m ( λ i j , η , ν ) J m , 1 ( η ξ ) + ∑ i j k R k ( λ i j , η , ν ) J − k , 0 ( η ξ ) + ∑ i j l S l ( λ i j , η , ν ) J − l , 1 ( η ξ ) + ∑ i j Y ( λ i j , η , ν ) J 0 , 0 ( η ξ ) = 0 , (5)
where i , j , k , l are positive natural integers and n , m the real numbers [
In this section, we first construct the surface hybrid analytical solitary waves, solutions of the modified KdV equation using the BDKm [
In 1895, the Dutch D. Korteweg and G. de Vries had proposed an equation translating nonlinearity and the dispersion phenomenon. This equation, presented in its simplest form by Equation (1) came to explain the existence of the solitary wave such as observed in 1834 by Russel. The KdV equation is very relevant in the shallow water theory. We consider the KdV equation in a modified form and which is written [
ω t = ω x x x + γ ω ω x + β ω 2 ω x (6)
where γ and β are constant coefficients of the quadratic (the weak nonlinearity) and higher-order cubic nonlinear terms; ω x x x the dispersion effect term which makes the wave form spread. t and x are both independent temporal and spatial coordinates respectively. ω ( x , t ) represents the water’s free surface in non-dimensional variables. The competition among the dispersion effect, quadratic and cubic nonlinearities constitute the main interest [
ν ϕ ξ + ϕ ξ ξ ξ + γ ϕ ϕ ξ + β ϕ 2 ϕ ξ = 0 , (7)
Since we are looking for hybrid surface solitary wave solutions, these can be chosen after an extension to the new iB-functions as the functions ϕ ( ξ ) of the form:
ϕ ( ξ ) = a J 2 , 0 ( η ξ ) + b J 1 , 1 ( η ξ ) + c J 3 , 0 ( η ξ ) + d J 2 , 2 ( η ξ ) , (8)
where a , b , c and d are real constants to be determined later, η , the inverse of the width at half-height of each component of the ansatz given by Equation (8). It is important to point out here that, Equation (8) is a kind of package [
∑ n P n ( a , b , c , d , γ , β , η , ν ) J n ,0 ( η ξ ) + ∑ m Q m ( a , b , c , d , γ , β , η , ν ) J m ,1 ( η ξ ) = 0, (9)
where n ∈ { 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 } ; m ∈ { 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 } . So, Equation (9) has delivered in its formulation, two ranges of equations in the terms of J n ,0 ( η ξ ) , and J m ,1 ( η ξ ) , thus constituting the most important ranges according to the BDKm theory. At the same time, this theory also suggests that, in the solving of the series of algebraic equations come from these two ranges, priority is given to the ranges of equations from the coefficients of the terms in J n ,0 ( η ξ ) which, in most cases and according to their degree of enrichment in coherent information, suffices to obtain the expressions of the coefficients a , b , c and d as function of parameters γ , β , η and ν . Further, the investigations carried out in the present case show that the series of equations outcome from the second term of Equation (9) provided only the irrelevant (trivial) results, this in comparison with the results provided by equations of the first series (given by the first term of Equation (9). In this context, We should limit ourselves to solving only the equations outcome from terms in J n ,0 ( η ξ ) . So, the identification at zero, of the coefficients of the terms in J n ,0 ( η ξ ) , n ∈ { 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 } results in the series of algebraic equations, with unknowns a , b , c and d as follows
The term in J 8,0 ( η ξ ) ,
7 η β b c 2 = 0 , (10)
The term in J 7,0 ( η ξ ) ,
12 η β a b c = 0 , (11)
The term in J 6,0 ( η ξ ) ,
( 5 η β a 2 − 2 η β a d − 3 η β d 2 − 6 η β c d ) b = 0 , (12)
The term in J 5,0 ( η ξ ) ,
( 4 η γ − 4 η β a ) b c = 0 , (13)
The term in J 4,0 ( η ξ ) ,
( 6 η β a d + 3 η γ a − 2 η β b 2 − 2 η β d 2 − 3 η γ d − 4 η β a 2 − 6 η 3 ) b = 0, (14)
The term in J 3,0 ( η ξ ) ,
( − 3 η γ − 6 η β d ) b c = 0, (15)
The term in J 2,0 ( η ξ ) ,
( η ν + 4 η 3 + 3 η γ d − 2 η γ a + η β b 2 + 5 η β d 2 − 4 η β a d ) b = 0, (16)
We shall not be able to pursue our analyzes without however emphasize the fact that, this series of equations going from Equations (10) to (16) constitutes the key points from which all future analyzes will be born. Thus, on observing the structure of this first series of equations (Equation (10) to Equation (16)), it becomes obvious that the second term of Equation (8) is the most disruptive element of the constituted package [
b = 0 or c = 0 (17)
and
a = 0 or b = 0 or c = 0. (18)
Equations (17) and (18) are those which will guide the choices of the different families of solutions to be constructed. We note in this context that when we set a condition, the major term changes. For example, if c = 0 , Equations (10), (11), (13) and (15) are verify, only Equations (12), (14) and (16) are those to be considered. In the concern of obtaining non-trivial solutions, we shall retain only five families of solutions represented by the following five conditions: a ≠ 0 , b ≠ 0 , c ≠ 0 ; a ≠ 0 , b ≠ 0 , c = 0 ; a = 0 , b ≠ 0 , c = 0 ; a ≠ 0 , b = 0 , c = 0 and a ≠ 0 , b = 0 , c ≠ 0 .
For a ≠ 0, b ≠ 0, c ≠ 0 ; one obtains from Equations (13) and (15), respectively:
a = γ β (19)
and
d = − γ 2 β . (20)
By substituting Equations (19) and (20) in Equation (12), One obtains:
c = − 7 γ 4 β . (21)
We continue our investigations by taking into account Equations (19) and (20) into Equation (14), and we reach the results:
b 2 = − 3 γ 2 + 6 β η 2 2 β 2 , (22)
with the first constraint
β ≺ − γ 2 2 η 2 . (23)
The constraint given by Equation (23) shows that, this first family of hybrid solitary waves exists if and only if β ≺ 0 (due to even powers of parameters γ and β ). So, it is the appearance of the cubic nonlinearity in the propagation medium that is at the origin of the existence of this first new prototype of hybrid solitary waves of the modified KdV equation. One more time, taking into account Equations (19) and (20) in Equation (16), leads respectively to:
b 2 = − γ 2 − 16 β η 2 − 4 β ν 4 β 2 , (24)
and the second constraint:
β ≺ γ 2 16 η 2 + 4 ν . (25)
Taking into account Equation (23) in Equation (25) imposes choosing the wave speed according to the third constraint below:
ν ≺ − 4 η 2 . (26)
It follows from Equation (26) that, the wave speed must remain a lower bound of the singleton { − 4 η 2 } . So, we can control the speed of the wave from the opposite of the square of the inverse of the width at half-height of each component of the ansatz given by Equation (8), and vice versa. Besides, Equations (23) and (24) must be identical, thus giving rise to the equality: b 2 = b 2 . Therefore, we derive from this equality, the constraint giving the expression of the parameter β depending on the parameters γ , η and ν as being:
β = 7 γ 2 4 η 2 + 4 ν , (27)
with the aid of Equation (23): ν ≺ − η 2 . In the end, the combination of the constraints given by Equations (23), (25), (26) and (27) imposes the choice of the wave speed ν such that:
ν ∈ ] − 9 2 η 2 ; − 4 η 2 [ . (28)
Equation (22) or Equation (24), taking into account Equation (27) gives, successively:
b 2 = − 108 η 4 + 132 η 2 ν + 24 ν 2 49 γ 2 (29)
and
b = ± − 108 η 4 + 132 η 2 ν + 24 ν 2 49 γ 2 . (30)
Thus, from Equation (8), we obtain the first family of solutions in the form:
ϕ ( ξ ) = γ β J 2 , 0 ( η ξ ) ± − 108 η 4 + 132 η 2 ν + 24 ν 2 49 γ 2 J 1 , 1 ( η ξ ) − 7 γ 4 β J 3 , 0 ( η ξ ) − γ 2 β J 2 , 2 ( η ξ ) , (31)
with the constraints given by Equations (23), (25), (26), (27) and (28), respectively. It is sound to point out here that, Equation (31) is the main new prototype of the hybrid solitary wave solutions of Equation (6) which is proposed in this work. This new prototype is a package [
are directly proportional to the ratio X = γ β , which, in their turns means that
the taller these wave components the greater the coupled effects of the quadratic and cubic nonlinearity.
When One sets c = 0 , it follows that Equations (10), (11), (13) and (15) are satisfied, while, Equations (12), (14) and (16) are reduced, successively to:
5 a 2 − 2 a d − 3 d 2 = 0 , (32)
6 β a d + 3 γ a − 2 β b 2 − 2 β d 2 − 3 γ d − 4 β a 2 − 6 η 2 = 0 , (33)
ν + 4 η 2 + 3 γ d − 2 γ a + β b 2 + 5 β d 2 − 4 β a d = 0. (34)
From these three equations, it clearly emerges that Equation (32) is verified if the following condition is satisfied: a = d . As a consequence, Equation (33) gives respectively:
b 2 = − 3 η 2 β (35)
and
b = ± η − 3 β , (36)
with the constraint:
β ≺ 0. (37)
Moreover, Equation (34) taking into account Equation (35), leads to the quadratic equation in unknown coefficient a below:
β a 2 + γ a + ν + η 2 = 0. (38)
Equation (38) has for discriminant:
Δ = γ 2 − 4 β ( ν + η 2 ) . (39)
For Δ ≥ 0 , Equation (38) admits as solution
a = − γ ± γ 2 − 4 β ( ν + η 2 ) 2 β , (40)
with ν ≤ − η 2 . It is necessary to point out here that, the case Δ ≺ 0 is not necessary because it leads to the complex values of the coefficient a, since we are looking for the real solutions. Thus, the second family of solutions of Equation (6), given that: cosh 2 ( η ξ ) − sinh 2 ( η ξ ) = 1 , are the functions ϕ ( ξ ) of the form:
ϕ ( ξ ) = − γ ± γ 2 − 4 β ( ν + η 2 ) 2 β ± η − 3 β J 1,1 ( η ξ ) , (41)
with the constraints given by Equation (37) and ν ≤ − η 2 . Equation (41) is a prototype that, in its constitution within Equation (8), appeared a priori to be a hybrid structure. But, during the resolution it is formally established that, for this prototype to be a solution of the mKdV equation considered here, in addition to the fact that c = 0 , it is also necessary that a = d . As a consequence, we obtain the solution given by Equation (41) which is a sum of two terms, the first of which is a constant and the second a kink. It becomes easy to note that this second family of solutions can mainly generate only kink and anti-kink structures according to the values taken by the different parameters present in this expression.
Under this condition, Equations (10), (11), (13) and (15) are verified. Furthermore, Equation (12) leads to β = 0 , while, Equation (14) delivers d in the form
d = − 2 η 2 γ . (42)
The combination of Equations (16) and (42) permits to obtain de following constraint:
ν = 2 η 2 , (43)
for all values of b such as b ∈ ℜ * . So, the third family of the solutions of Equation (6) is written:
ϕ ( ξ ) = b J 1 , 1 ( η ξ ) − 2 η 2 γ J 2 , 2 ( η ξ ) , (44)
with Equation (43) as constraint, for β = 0 and b ∈ ℜ * . Equation (44) is a sub-package derived from the large package given by Equation (8). It is a hybrid prototype that highlights two solitons well-known in the literature, namely the kink represented here by the first and the dark represented by the second term of Equation (44). These two solitons in their interactions give rise to hybrid structures which can be either with strong kink or anti-kink tendencies, or with strong dark or bright tendencies depending on the values taken by the coefficients and parameters b , d , ν and η of the wave, as well as those taken by the parameters γ of the system whose dynamics are governed by Equation (6). Equation (44) also highlights the opposition of signs between the amplitude d and the quadratic nonlinearity parameter γ as well as the inversely proportional character of d and γ for a given value of η of the second term of this equation. This means that, the smaller γ , the larger d and vice versa. This observation also reflects the antagonistic effect between the two: the taller the wave formed by the second term of Equation (44), the weaker the effect of quadratic nonlinearity and vice versa. This compared to the values taken by b justifies the different hybrid structures formation which appears for β = 0 . Thus, this third family of solutions reflects behavior of the considered system and is described by Equation (6) with respect to the phenomena induced by the cubic nonlinearity.
For a ≠ 0 , b = 0 and c = 0 , all equations of the first range are verified (from Equation (10) to Equation (16)). The second range leads to a trivial solution which is not important. Thus, according to the implementation of the BDKm, the equations of the first range are the best enriched, thereupon. This makes it possible to extract and retain the approximate solution resulting from this first range for all the values of a and d belonging to the set of non-zero real numbers. We, therefore, obtain the fourth family of solutions of Equation (6) under the form:
ϕ ( ξ ) = a J 2 , 0 ( η ξ ) + d J 2 , 2 ( η ξ ) , (45)
where a ∈ ℜ * and d ∈ ℜ * . Let us note that the sub-package formed by Equation (45) is a hybrid prototype at least under its mathematical form. The structures likely to form oscillate between the bright and the dark solitons depending on the values taken by the various characteristic parameters of the wave and those of parameters that characterize the system whose dynamics are described by Equation (6).
In this context, all equations from (10) to (16) are verified, while, the second range of equations leads to a trivial solution which is not important. So, according to the implementation of the BDKm, the equations of the first range are the best enriched on this subject. This permits to unearth and retain the approximate solution resulting from this first range for all the values of a, c and d belonging to the set of non-zero real numbers. We therefore display the fifth family of solutions of Equation (6) as being:
ϕ ( ξ ) = a J 2 , 0 ( η ξ ) + c J 3 , 0 ( η ξ ) + d J 2 , 2 ( η ξ ) , (46)
where a ∈ ℜ * , c ∈ ℜ * and d ∈ ℜ * . This fifth family of solutions is a sub-package of which the first two terms are bright solitons and the third term is a dark soliton. Equation (46) is also a hybrid prototype in its mathematical form. Depending on the values taken by parameters of the wave and those of the system whose dynamics is described by Equation (6) mainly generates either bright structures or dark soliton structures.
This subsection is devoted to numerical simulations aiming to reassure oneself of the observable and applicable characters of certain new hybrid solitary wave prototypes obtained in this work. This should allow laboratories specializing in propagation tests to have the matter to exercise in order to reveal, understand and explain certain new phenomena or behaviors developed by systems whose dynamics are governed by Equation (6). To achieve this, we used certain digital tools, in particular, the MATLAB toolbox pdepe [
In order to allow a better understand how the above profiles (from Figures 1-3) are obtained, one can illustrate this through an example. Thus, in the case of
Equations (38); (39) and (40), respectively. For more details on the proceedings for obtaining the profiles displayed in the various figures, one can refer to [
In this part of manuscript, one engages some detailed discussions which help the reader to better appropriate the new ideas offered by the new prototypes of solitary waves obtained and which are the solutions of Equation (6).
A thorough observation of these four profiles reveals at least as regards of their natures, that,
to the ( x , t ) plane and vice versa.
It is the same between
Moreover,
After all these discussions, it is clear that numerical simulations revealed many hybrid or multi-form characters of certain obtained solutions, thus, corroborating the theoretical forecasts made on this subject. These results thus confirm the hybrid and novel character, in all its forms (package, sub-packages, and so on.), of the chosen ansatz and represented by Equation (8). Compared to the results obtained in [
We come to propose at the end of this study, some hybrid prototypes of solitary waves (as well as their existence conditions) of the modified KdV equation depicted by Equation (6). This is made possible thanks to the BDKm extended to the iB-functions, which is very relevant in the resolution of certain types of differential equations which involve dispersive terms and nonlinear terms. The surmises issued on the basic mathematical form (see Equation (8)) from which all the families of the obtained solutions come were validated by numerical simulations carried out with good accuracy [
The authors’ thanks go to both ministries of higher education of Cameroon and Gabon for their assistance through their respective research support programs. These thanks also go to Madame Nchong Glory Arrey who, despite her many occupations, has always been present whenever she has been asked to read this manuscript in order to improve the quality of the English used.
The authors declare no conflicts of interest regarding the publication of this paper.
Tchaho, C.T.D., Ngantcha, J.P., Ekogo, T.B., Um, B.R.M., Omanda, H.M. and Bogning, J.R. (2022) Construction of New Surface Wave Solutions of the Modified KdV Equation. Open Journal of Applied Sciences, 12, 196-215. https://doi.org/10.4236/ojapps.2022.122014