 Research
 Open Access
 Published:
New dualmode Kadomtsev–Petviashvili model with strong–weak surface tension: analysis and application
Advances in Difference Equations volume 2018, Article number: 433 (2018)
Abstract
Dualmode \((2+1)\)dimensional Kadomtsev–Petviashvili (DMKP) equation is a new model which represents the spread of two simultaneously directional waves due to the involved term “\(u_{tt}(x,y,t)\)” in its equation. We present the construction of DMKP and search for possible solutions. The innovative tanhexpansion method and Kudryashov technique will be utilized to find the necessary constraint conditions which guarantee the existence of soliton solutions to DMKP. Supportive 3D plots will be provided to validate our findings.
Introduction
Dualmode type is a new family of nonlinear partial differential equations which fall in the following form: [1, 2]
where \(N(y,y_{x},\ldots)\) and \(L(y_{kx}) : k\geq2\) are the nonlinear and linear terms involved in the equation. \(y(x,t)\) is the unknown fieldfunction, \(s>0\) is the phase velocity, \(\vert \beta \vert \leq1\), \(\vert \alpha \vert \leq1\), β is the dispersion parameter, and α is the parameter of nonlinearity. With \(s=0\) and integrating with respect to t, the dualmode problem is reduced to a partial differential equation of the first order in time t.
A few dualmode models have been established and studied. In [3,4,5,6,7,8], authors extracted abundant soliton solutions for the secondorder KdV. [9, 10] established the dualmode Burgers and fourthorder Burgers and obtained multiple soliton solutions by means of the simplified Hirota technique. In [11,12,13], the tanh technique and Hirota method were implemented to seek possible solutions of the twomode coupled Burgers equation, coupled mKdV, and coupled KdV. Finally, the dualmode perturbed Burgers, Ostrovsky, and Schrodinger equations were established in [14, 15].
Structure of dualmode \((2+1)\)dimensional Kadomtsev–Petviashvili
The Kadomtsev–Petviashvili (KP) equation reads [16, 17]
where \(W=W(x,y,t)\). It models three connected aspects: weakly dispersive, longer wave length compared with its wave amplitude, and slower variation in ycoordinate compared with its propagation in xcoordinate. σ gives the strength of the surface tension, strong with \(\sigma>0\) and weak if \(\sigma<0\).
To derive the dualmode version of KP, we apply the operator \(N=(\frac{\partial}{\partial t}\alpha_{1} s \frac{\partial }{\partial x}\alpha_{2} s \frac{\partial}{\partial y})\) to the nonlinear terms of (1.2) and the operator \(L=(\frac{\partial}{\partial t}\beta_{1} s \frac{\partial}{\partial x}\beta_{2} s \frac{\partial}{\partial y})\) to the linear terms, i.e.,
\(\alpha_{1}\), \(\alpha_{2}\) are the nonlinearity parameters, \(\beta_{1}\), \(\beta_{2}\) are the dispersive parameters, and s is the phase velocity.
We aim in this work to seek possible soliton solutions for (1.3) and study graphically the effects of the aforementioned parameters on the propagations of the obtained dual waves such model possesses.
Solutions of DMKP by tanhexpansion technique
First, we use the new variable \(z=a x+ b y c t\) to convert (1.3) into the following reduced differential equation:
where \(W=W(z)\). Balancing \(W^{2}\) with \(W''\) in tanh technique sense [18, 19], the solution of (2.1) is
To determine the values of \(A_{1}\), \(A_{2}\), \(A_{3}\), a, b, and c, we substitute (2.2) in (2.1) and apply the identity \(\operatorname {sech}^{2}(z)=1\tanh^{2}(z)\) to get a polynomial of tanh function. Setting the coefficient of the same power of tanh to zero, we obtain the following nonalgebraic system:
We study the solution of the above system via compatible constraint relations:
For \(\alpha_{1}=\alpha_{2}=\beta_{1}=\beta_{2}=\gamma\) with \(\gamma<1\), we get two solution sets.
The first solution set
Therefore, the resulting dualwave solution of DMKP (1.3) is
where \(\frac{1}{\sqrt{2}}<  \gamma<1\). Figure 1 shows the proximity of the two waves with increasing the phase velocity s and fixing γ. Figure 2 shows the extent of convergence and spacing of the two waves by the sign of γ and fixing s.
The second solution set
Therefore, the resulting dualwave solution of DMKP (1.3) is
where \(\frac{1}{\sqrt{2}}<  \gamma <1\). Figure 3 shows the shape of the obtained solution described in (2.7).
Solutions of DMKP by Kudryashov expansion technique
It is to be noted that the tanh solution obtained in the preceding section is σindependent and, therefore, using another method to study the solution of DMKP is needed. We use here the Kudryashov technique where the solution takes the following form [20, 21]:
The variable Y satisfies the differential equation
Solving (3.2) gives
The index n to be determined by applying orderbalance procedure between \(W^{2}\) and \(W''\) appears in (2.1). Thus, \(n=2\) and accordingly we write (3.1) as
Differentiating both (3.2) and (3.4) implicitly leads to
and
Now, we insert (3.2) through (3.6) in (2.1) to get a polynomial in Y. By setting each coefficient of \(Y^{i}\) to zero, a nonlinear algebraic system is obtained. Seeking a solution to this system, we get
Therefore, a new solution of DMKP (1.3) is
Figures 4 and 5 provide 3D plots of the Kudryashov solutions when \(\sigma> 0\) and \(\sigma<0\), respectively.
Conclusion
A new dualmode Kadomtsev–Petviashvili (DMKP) equation is introduced. This model describes the spread of two simultaneously directional waves. We have studied possible solutions for DMKP and obtained the following findings:

σindependent tanh soliton solution is obtained for the DMKP.

σdependent Kudryashov soliton solution is obtained for the DMKP.

The above two solutions exist when \(\alpha_{1}=\alpha_{2}=\beta_{1}=\beta_{2}=\gamma\) with \(\frac{1}{\sqrt{2}} < \gamma< 1\).
Also, a graphical analysis is provided to show the impact of both linearitydispersive parameter γ and the phase velocity s on the spread of the obtained dual waves for DMKP.
References
 1.
Korsunsky, S.V.: Soliton solutions for a secondorder KdV equation. Phys. Lett. A 185, 174–176 (1994)
 2.
Wazwaz, A.M.: Multiple soliton solutions and other exact solutions for a twomode KdV equation. Math. Methods Appl. Sci. 40(6), 1277–1283 (2017)
 3.
Xiao, Z.J., Tian, B., Zhen, H.L., Chai, J., Wu, X.Y.: Multisoliton solutions and Bäcklund transformation for a twomode KdV equation in a fluid. Waves Random Complex Media 27(1), 1–14 (2017)
 4.
Lee, C.T., Liu, J.L.: A Hamiltonian model and soliton phenomenon for a twomode KdV equation. Rocky Mt. J. Math. 41(4), 1273–1289 (2011)
 5.
Lee, C.C., Lee, C.T., Liu, J.L., Huang, W.Y.: Quasisolitons of the twomode Korteweg–de Vries equation. Eur. Phys. J. Appl. Phys. 52, Article ID 11301 (2010)
 6.
Lee, C.T., Lee, C.C.: On wave solutions of a weakly nonlinear and weakly dispersive twomode wave system. Waves Random Complex Media 23, 56–76 (2013)
 7.
Hong, W.P., Jung, Y.D.: New nontraveling solitary wave solutions for a secondorder Korteweg–de Vries equation. Z. Naturforsch. 54 a, 375–378 (1999)
 8.
Alquran, M., Jarrah, A.: Jacobi elliptic function solutions for a twomode KdV equation. J. King Saud Univ., Sci. (2017). https://doi.org/10.1016/j.jksus.2017.06.010
 9.
Wazwaz, A.M.: A twomode Burgers equation of weak shock waves in a fluid: multiple kink solutions and other exact solutions. Int. J. Appl. Comput. Math. (2016). https://doi.org/10.1007/s4081901603024
 10.
Wazwaz, A.M.: Twomode Sharma–Tasso–Olver equation and twomode fourthorder Burgers equation: multiple kink solutions. Alex. Eng. J. (2017). https://doi.org/10.1016/j.aej.2017.04.003
 11.
Syam, M., Jaradat, H.M., Alquran, M.: A study on the twomode coupled modified Korteweg–de Vries using the simplified bilinear and the trigonometricfunction methods. Nonlinear Dyn. 90(2), 1363–1371 (2017)
 12.
Jaradat, H.M., Syam, M., Alquran, M.: A twomode coupled Korteweg–de Vries: multiplesoliton solutions and other exact solutions. Nonlinear Dyn. 90(1), 371–377 (2017)
 13.
Alquran, M., Jaradat, H.M., Syam, M.: A modified approach for a reliable study of new nonlinear equation: twomode Korteweg–de Vries–Burgers equation. Nonlinear Dyn. 91(3), 1619–1626 (2018)
 14.
Jaradat, I., Alquran, M., Ali, M.: A numerical study on weakdissipative twomode perturbed Burgers and Ostrovsky models: right–left moving waves. Eur. Phys. J. Plus 133, Article ID 164 (2018)
 15.
Jaradat, I., Alquran, M., Momani, S., Biswas, A.: Dark and singular optical solutions with dualmode nonlinear Schrodinger’s equation and Kerrlaw nonlinearity. Optik 172, 822–825 (2018)
 16.
Fokou, M., Kofane, T.C., Mohamadou, A., Yomba, E.: Twodimensional third and fifthorder nonlinear evolution equations for shallow water waves with surface tension. Nonlinear Dyn. 91(2), 1177–1189 (2018)
 17.
Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15, 539–541 (1970)
 18.
Alquran, M., AlKhaled, K.: Sinc and solitary wave solutions to the generalized Benjamin–Bona–Mahony–Burgers equations. Phys. Scr. 83, Article ID 065010 (2011)
 19.
Alquran, M., AlKhaled, K.: The tanh and sine–cosine methods for higher order equations of Korteweg–de Vries type. Phys. Scr. 84, Article ID 025010 (2011)
 20.
Kudryashov, N.A.: One method for finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 17(6), 2248–2253 (2012)
 21.
Wang, L., Shen, W., Meng, Y., Chen, X.: Construction of new exact solutions to timefractional twocomponent evolutionary system of order 2 via different methods. Opt. Quantum Electron. 50, Article ID 297 (2018)
Funding
Not applicable.
Author information
Affiliations
Contributions
The authors declare that this study was accomplished in collaboration with the same responsibility. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that there is no conflict of interests regarding the publication of the paper.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Abu Irwaq, I., Alquran, M., Jaradat, I. et al. New dualmode Kadomtsev–Petviashvili model with strong–weak surface tension: analysis and application. Adv Differ Equ 2018, 433 (2018). https://doi.org/10.1186/s1366201818933
Received:
Accepted:
Published:
MSC
 35C08
 74J35
Keywords
 Dualmode
 \((2+1)\)dimensional Kadomtsev–Petviashvili
 tanhexpansion method
 Kudryashov expansion method