 Research
 Open Access
New dualmode Kadomtsev–Petviashvili model with strong–weak surface tension: analysis and application
 Issam Abu Irwaq^{1},
 Marwan Alquran^{1}Email author,
 Imad Jaradat^{1} and
 Dumitru Baleanu^{2, 3}
https://doi.org/10.1186/s1366201818933
© The Author(s) 2018
 Received: 26 September 2018
 Accepted: 19 November 2018
 Published: 23 November 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.
Keywords
 Dualmode
 \((2+1)\)dimensional Kadomtsev–Petviashvili
 tanhexpansion method
 Kudryashov expansion method
MSC
 35C08
 74J35
1 Introduction
A few dualmode models have been established and studied. In [3–8], 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 [11–13], 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].
1.1 Structure of dualmode \((2+1)\)dimensional Kadomtsev–Petviashvili
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.
2 Solutions of DMKP by tanhexpansion technique
For \(\alpha_{1}=\alpha_{2}=\beta_{1}=\beta_{2}=\gamma\) with \(\gamma<1\), we get two solution sets.
2.1 The first solution set
2.2 The second solution set
3 Solutions of DMKP by Kudryashov expansion technique
4 Conclusion

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 \(\frac{1}{\sqrt{2}} < \gamma< 1\).
Declarations
Funding
Not applicable.
Authors’ contributions
The authors declare that this study was accomplished in collaboration with the same responsibility. All authors read and approved the final manuscript.
Competing interests
The authors declare that there is no conflict of interests regarding the publication of the paper.
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.
Authors’ Affiliations
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