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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.
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Correspondence to Marwan Alquran.
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MSC
 35C08
 74J35
Keywords
 Dualmode
 \((2+1)\)dimensional Kadomtsev–Petviashvili
 tanhexpansion method
 Kudryashov expansion method