New diverse soliton solutions for the coupled Konno-Oono equations

The main aim of this article is to establish new impressive diverse soliton solutions to the nonlinear Coupled Konno-Oono Model (NCKOM) that represents current-field string interact with an external magnetic field. The achieved soliton solutions will give stretch study for this model and all related phenomena’s. Three different schemes have been called for this purpose. The first one is the extended direct algebraic method (EDAM), while the second is the Paul-Painlevé approach method (PPAM) and the third one is the Riccati-Bernoulli Sub-ODE method (RBSODM). Brief comparisons between our results and that achieved previously have been listed.


Introduction
In this paper, we will establish various types of the impressive distinct soliton solutions to NCKOM through the extended direct algebraic method (Seadawy 2016). In the same vein, the PPAM (Bekir et al. 2020a;Zahran 2020, 2021a) has been applied perfectly to propose other new vision of these types of solutions. In related subject other new perceptions of these types have been established using the RBSODM (Shehata et al. 2019;Bekir and Zahran 2021b, 2021c, 2021d). The suggested model which plays a vital rule in the magnetic field profile was independently introduced by (Konno and Oono 1994;Konno and Kakuhata 1996;Souleymanou et al. 2012) as application for current-field string interacting with an external magnetic field system.
The mathematical model which represents this model as the coupled integrable dispersionless system can be written in the form where 1 , 1 and 3 are constants, this model is the famous one which was constructed as applications for current-field string interacting with an external magnetic field (Konno and Oono 1994;Konno and Kakuhata 1996;Souleymanou et al. 2012).
A special case of a system, considered to be transformed into a new Konno-Oono equation system which is a coupled integrable dispersionless equation, is given in the form (Alam and Belgacem 2016;Mirhosseini-Alizamani et al. 2020) as follow Let us consider U(x, t) = G( ), V(x, t) = S( ), = (x − wt) , hence the system (2) becomes By integrating the second equation of the system (3) w.r.t. we get where is constancy of integration, by inserting Eq. (4) into the first part of Eq. (3) we get Some trials through few set of authors have been constructed to establish different types of the soliton solutions for this model, sea for example, (Alam and Belgacem 2016) who constructed the exact solutions for this model using (G′/G)-expansion method, (Yel et al. 2017) who investigated the analytical solution of this model using the Sin-Gorden expansion method, (Mirhosseini-Alizamani et al. 2020) who applied the new extended direct algebraic method to construct the exact solution for this model and (Mirhosseini-Alizamani et al. 2020) who applied the Jacobi-Elliptic functions to implement the closed form solution for this model. In the same connection, some recent studies on obtaining the traveling wave solutions for many nonlinear models arising in various branches of science have been listed in Shehata et al. (2019);Taghizadeh et al. 2012;Bekir and Zahran 2021e;Shehata et al. 2022;Hosseini et al. 2020;Bekir et al. 2021a;Younis et al. 2020;Bekir et al. 2021b;Bekir et al. 2020b;Zahran et al. 2022a;Zahran et al. 2021a;Zahran and Bekir 2021;Zahran et al. 2021b;Zahran and Bekir 2022;Bekir et al. 2021c;Bekir and Zahran 2021f;Bekir et al. 2022a;Zahran et al. 2022b;Bekir et al. 2022b).
Moreover, there are recent studies have been documented to investigate the soliton solutions by using the integrable equations see for example (Ma 2022a) who provided a brief overview of soliton solutions obtained through the Hirota direct method, discussed the bilinear formulation of soliton solutions in both (1 + 1)-dimensions and (2 + 1)-dimensions together with applications to various integrable equations and are analyzed the Hirota conditions for N-soliton solutions, (Kuo and Ma 2022) who verified, demonstrated the (1) (4) S = 1 (G 2 + ).
(5) w 2 2 G �� + 2G 3 + 2 G = 0. existence of resonant multi-soliton solutions for three time derivative terms and the three space dissipative terms NLPDE respectively that arising from the B-type Kadomtsev-Petviashvili equation, proved the accuracy of the extracted resonant multi-soliton solutions at the same time, (Ma, W-X 2021a) who discussed haw to construct and classify nonlocal PTsymmetric integrable equations via nonlocal group reductions of matrix spectral problems. The nonlocalities considered are reverse-space, reverse-time and reverse-space-time that involve either the transpose or the Hermitian transpose as well as derived the Soliton solutions generated from specific Riemann-Hilbert problems with the identity jump matrix for the nonlocal PT-symmetric matrix nonlinear Schrödinger and modified Korteweg-de Vries equations, (Ma, W-X 2021b) who presented the soliton solutions for nonlocal reversetime nonlinear Schrödinger (NLS) hierarchies associated with higher-order matrix spectral problems with the aid of the Riemann-Hilbert problems and used the Sokhotski-Plemelj formula to transform the Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations. He proposed new formulation of solutions to special Riemann-Hilbert problems with the identity jump matrix, corresponding to the reflection less inverse scattering transforms and constructed soliton solutions to each system in the considered nonlocal reverse-time NLS hierarchies and (Ma 2022b) who conducted two group reductions of the Ablowitz-Kaup-Newell-Segur matrix spectral problems, presented a class of novel reduced nonlocal reverse-spacetime integrable modified Korteweg-de Vries equations and generate soliton solutions from the reflection less inverse Riemann-Hilbert problems.
The main idea of this work is to achieve new visions for the different types of exact solutions of the NCKOM in terms of some variables through the manners mentioned above, whenever these variables take specific values, the solitary wave solutions could be achieved.

The soliton solutions according to the EDAM
Let us firstly propose the formalism of any NLPDE by introducing the function Υ in terms of R(x, t) and its partial derivatives as That involve the highest order derivatives and nonlinear terms. When Eq. (6) surrenders to the transformation h( where S H is in terms of h( ) G( ) and its total derivatives and so on. Here I omit to list them one by one.
The exact solution of Eq. (7) in the framework of the EDAM can be written in the form Let us now apply this technique to the suggested model Eq. (5) mentioned above which is (9) w 2 2 G �� + 2G 3 + 2 G = 0. The homogeneous balance between G ′′ , G 3 implies M = 1 hence the solution is Via substituting about G ′′ , G, G 3 into Eq. (9) mentioned above, equating the coefficients of various powers i to zero leads to system of equations and by solving it we get these results The above results will generate four different solutions, for simplicity and similarity we will implement the solution corresponding to the third result and plot it This result can be simplified to be The solution according to the suggested method is By the same manner we can implement the solutions of the other results.

The soliton solutions according to the PPAM
The PPAM states the exact solution for Eq. (7) in the form Or (10) G( ) = a 0 + a 1 .
, and R(X) in Eqs. (17) and (18) satisfy the Riccati-equation in the form R X − AR 2 = 0 and its solution is Consequently, By substituting about G, G , G mentioned in the relations (17-21) into the suggested model Eq. (5) mentioned above we get By equating the coefficients of various powers of e −N R N by zero the following system will be emerged By solving this system, we get According to these results there are 4-achieved solutions, for simplicity and similarity we will implement only two from them which are.

When
.Then the solution is   (29) G � = aG 2−n + bG + cG n (30) G �� = ab(3 − n)G 2−n + a 2 (2 − n)G 3−2n +nc 2 G 2n−1 + bc(n + 1)G n + (2ac + b 2 )G where the constant C that appears through the relations (32-38) represents the integral consistency. Now, by inserting the derivatives of u into Eq. (5) yields an algebraic equation of u . Noticing the symmetry of the right-hand item of Eq. (5) and setting the highest power exponents of G to be equivalence in Eq. (5), m can be determined. Comparing the coefficients of G i yields a set of algebraic equations for a, b, c, and C. Solving the set of algebraic equations and substituting m, a, b, c, C, = (x + y − 2wt) into any one of the Eqs. (31)-(38), we can get the traveling wave solutions of Eq. (34).
According to RBSODM, substitute about G ′′ Eq. (30) into the suggested model mentioned above we obtain, By equating the coefficients of different power of G i to zero after choose suitable value for n we get the following system a 2 2 w 2 + 1 = 0 2ab 2 w 2 = 0 2 w 2 (2ac + b 2 ) + 2 = 0 The system (39) lead to these two results These two results imply only one case og the suggested method in which b 2 − 4ac > 0, hence the solution is (39) 2 w 2 bc = 0.

Conclusion
In this study, three various schemes have been implemented to construct new diverse types of the traveling wave solutions to the NCKOM. The first schema is the EDAM which has successfully applied to realize new types of soliton solutions to this model Fig. 1. In the same vein and parallel the PPAM has been applied to achieve other new visions of the traveling wave solutions for this model namely Figs. 2-5. In same connection, the RBSOM has been used to realize other new distinct traveling wave solutions namely Figs. 6 and 7. The obtained results via these three techniques denote to these three methods are reliable, effective and can used to solve any other evolution equation. Our obtained solutions are more impressive, general instead of that achieved by Alam and Belgacem (2016); Yel et al. 2017;Mirhosseini-Alizamani et al. 2020;Abdelrahman et al. 2020;Wang 2021) and will give stretch study not only for this model but also for all related phenomena which widely used in magnetic field.