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New dual-mode Kadomtsev–Petviashvili model with strong–weak surface tension: analysis and application

  • 1,
  • 1Email author,
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  • 2, 3
Advances in Difference Equations20182018:433

https://doi.org/10.1186/s13662-018-1893-3

  • Received: 26 September 2018
  • Accepted: 19 November 2018
  • Published:

Abstract

Dual-mode \((2+1)\)-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 tanh-expansion 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

  • Dual-mode
  • \((2+1)\)-dimensional Kadomtsev–Petviashvili
  • tanh-expansion method
  • Kudryashov expansion method

MSC

  • 35C08
  • 74J35

1 Introduction

Dual-mode type is a new family of nonlinear partial differential equations which fall in the following form: [1, 2]
$$ y_{tt}-s^{2}y_{xx}+\biggl( \frac{\partial}{\partial t}-\alpha s \frac{\partial}{\partial x}\biggr)N(y,y_{x},\ldots)+ \biggl(\frac{\partial}{\partial t}-\beta s \frac{\partial}{\partial x}\biggr)L(y_{kx})=0, $$
(1.1)
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 field-function, \(s>0\) 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 dual-mode problem is reduced to a partial differential equation of the first order in time t.

A few dual-mode models have been established and studied. In [38], authors extracted abundant soliton solutions for the second-order KdV. [9, 10] established the dual-mode Burgers and fourth-order Burgers and obtained multiple soliton solutions by means of the simplified Hirota technique. In [1113], the tanh technique and Hirota method were implemented to seek possible solutions of the two-mode coupled Burgers equation, coupled m-KdV, and coupled KdV. Finally, the dual-mode perturbed Burgers, Ostrovsky, and Schrodinger equations were established in [14, 15].

1.1 Structure of dual-mode \((2+1)\)-dimensional Kadomtsev–Petviashvili

The Kadomtsev–Petviashvili (KP) equation reads [16, 17]
$$ \frac{\partial}{\partial x} (W_{t}-6 W W_{x}+W_{xxx} )+3\sigma W_{yy}=0, $$
(1.2)
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 y-coordinate compared with its propagation in x-coordinate. σ gives the strength of the surface tension, strong with \(\sigma>0\) and weak if \(\sigma<0\).
To derive the dual-mode 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.,
$$ \begin{aligned}[b] 0={}&\frac{\partial}{\partial x} \biggl(W_{tt}-s^{2} W_{xx}-s^{2} W_{yy}-6 \biggl(\frac{\partial}{\partial t}- \alpha_{1} s \frac{\partial}{\partial x}-\alpha_{2} s \frac{\partial}{\partial y} \biggr) (W W_{x})\\ &+\biggl(\frac{\partial}{\partial t}-\beta_{1} s \frac{\partial }{\partial x}-\beta_{2} s \frac{\partial}{\partial y}\biggr) (W_{xxx}) \biggr) \\ &+3\sigma\biggl(\frac{\partial}{\partial t}-\beta_{1} s \frac{\partial}{\partial x}- \beta_{2} s \frac{\partial}{\partial y}\biggr) (W_{yy}),\end{aligned} $$
(1.3)
\(\alpha_{1}\), \(\alpha_{2}\) 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.

2 Solutions of DMKP by tanh-expansion technique

First, we use the new variable \(z=a x+ b y -c t\) to convert (1.3) into the following reduced differential equation:
$$ \begin{aligned}[b] 0={}& \bigl(a\bigl(c^{2}-\bigl(a^{2}+b^{2} \bigr)s^{2}\bigr) -3\sigma b^{2}(c+\beta_{1} a s+ \beta_{2} b s) \bigr)W+3a^{2} (c+\alpha_{1} a s+ \alpha_{2} b s) W^{2} \\ &-a^{4} (c+\beta_{1} a s+\beta_{2} b s) W'',\end{aligned} $$
(2.1)
where \(W=W(z)\). Balancing \(W^{2}\) with \(W''\) in tanh technique sense [18, 19], the solution of (2.1) is
$$ W(z)=A_{1}+A_{2} \tanh(z)+A_{3} \tanh^{2}(z). $$
(2.2)
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 non-algebraic system:
$$\begin{aligned}& \begin{aligned}0={}&3 a^{2} A_{1}^{2} c - 2 a^{4} A_{3} c + a A_{1} c^{2} - a^{3} A_{1} s^{2} - a A_{1} b^{2} s^{2} + 3 a^{3} A_{1}^{2} s \alpha_{1} + 3 a^{2} A_{1}^{2} b s \alpha_{2} \\ &- 2 a^{5} A_{3} s \beta_{1}- 2 a^{4} A_{3} b s \beta_{2} - 3 A_{1} b^{2} c \sigma- 3 a A_{1} b^{2} s \beta_{1} \sigma- 3 A_{1} b^{3} s \beta_{2} \sigma,\end{aligned} \\& \begin{aligned}0={}& 2 a^{4} A_{2} c + 6 a^{2} A_{1} A_{2} c + a A_{2} c^{2} - a^{3} A_{2} s^{2} - a A_{2} b^{2} s^{2} + 6 a^{3} A_{1} A_{2} s \alpha_{1} + 6 a^{2} A_{1} A_{2} b s \alpha_{2} \\ &+ 2 a^{5} A_{2} s \beta_{1}+ 2 a^{4} A_{2} b s \beta_{2} - 3 A_{2} b^{2} c \sigma- 3 a A_{2} b^{2} s \beta_{1} \sigma- 3 A_{2} b^{3} s \beta_{2} \sigma,\end{aligned} \\& 0= 3 a^{2} A_{2}^{2} c + 8 a^{4} A_{3} c + 6 a^{2} A_{1} A_{3} c + a A_{3} c^{2} - a^{3} A_{3} s^{2} - a A_{3} b^{2} s^{2} + 3 a^{3} A_{2}^{2} s \alpha_{1} \\& \phantom{0=}{} + 6 a^{3} A_{1} A_{3} s \alpha_{1}+ 3 a^{2} A_{2}^{2} b s \alpha_{2} + 6 a^{2} A_{1} A_{3} b s \alpha_{2} + 8 a^{5} A_{3} s \beta_{1} + 8 a^{4} A_{3} b s \beta_{2} \\& \phantom{0=}{} - 3 A_{3} b^{2} c \sigma- 3 a A_{3} b^{2} s \beta_{1} \sigma- 3 A_{3} b^{3} s \beta_{2} \sigma, \\& 0= -2 a^{4} A_{2} c + 6 a^{2} A_{2} A_{3} c + 6 a^{3} A_{2} A_{3} s \alpha_{1} + 6 a^{2} A_{2} A_{3} b s \alpha_{2} - 2 a^{5} A_{2} s \beta_{1} - 2 a^{4} A_{2} b s \beta_{2}, \\& 0=-6 a^{4} A_{3} c + 3 a^{2} A_{3}^{2} c + 3 a^{3} A_{3}^{2} s \alpha_{1} + 3 a^{2} A_{3}^{2} b s \alpha_{2} - 6 a^{5} A_{3} s \beta_{1} - 6 a^{4} A_{3} b s \beta_{2}. \end{aligned}$$
(2.3)
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.

2.1 The first solution set

$$ \begin{gathered} a= \text{free constant},\qquad A_{1}= \text{free constant},\qquad A_{2}= 0, \\ A_{3}=2 a^{2}, \\ c=\frac{s \gamma(a\pm\sqrt{a^{2}(2\gamma^{2}-1)})}{\gamma^{2}-1},\qquad b= \frac{ a \gamma^{2} \pm\sqrt{a^{2}(2\gamma^{2}-1)}}{1-\gamma^{2}}.\end{gathered} $$
(2.4)
Therefore, the resulting dual-wave solution of DMKP (1.3) is
$$ W(x,y,t)=A_{1}+3 a^{2} \tanh^{2} \biggl(a x +\frac{ a \gamma^{2} \pm \sqrt{a^{2}(2\gamma^{2}-1)}}{1-\gamma^{2}} y-\frac{s \gamma(a\pm \sqrt{a^{2}(2\gamma^{2}-1)})}{\gamma^{2}-1} t \biggr), $$
(2.5)
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.
Figure 1
Figure 1

3D plots of the dual waves \(W(x,y,t)\) and the impact of phase velocity s on its propagating

Figure 2
Figure 2

3D plots of the dual waves \(W(x,y,t)\) and the impact of linearity-dispersive parameter γ on its propagating

2.2 The second solution set

$$ \begin{gathered}a= \text{free constant},\qquad A_{1}= 0,\qquad A_{2}= \text{free constant}, \\ A_{3}=2 a^{2}, \\ c=-(a+b)s \gamma,\qquad b= \frac{ a \gamma^{2} \pm\sqrt{a^{2}(2\gamma^{2}-1)}}{1-\gamma^{2}}.\end{gathered} $$
(2.6)
Therefore, the resulting dual-wave solution of DMKP (1.3) is
$$ \begin{aligned}[b] W(x,y,t)={}&A_{2} \tanh \biggl(a x +\frac{ a \gamma^{2} \pm \sqrt{a^{2}(2\gamma^{2}-1)}}{1-\gamma^{2}} y\\&- \biggl(a+\frac{ (a \gamma^{2} \pm\sqrt{a^{2}(2\gamma^{2}-1)}}{1-\gamma^{2}} \biggr)s \gamma t \biggr) \\ &+2 a^{2} \tanh^{2} \biggl(a x +\frac{ a \gamma^{2} \pm \sqrt{a^{2}(2\gamma^{2}-1)}}{1-\gamma^{2}} y\\ &- \biggl(a+\frac{ (a \gamma^{2} \pm\sqrt{a^{2}(2\gamma^{2}-1)}}{1-\gamma^{2}} \biggr)s \gamma t \biggr),\end{aligned} $$
(2.7)
where \(\frac{1}{\sqrt{2}}< | \gamma| <1\). Figure 3 shows the shape of the obtained solution described in (2.7).
Figure 3
Figure 3

3D plots of the first wave, second wave, and dual waves for \(W(x,y,t)\) given in (2.7)

3 Solutions of DMKP by Kudryashov expansion technique

It is to be noted that the tanh solution obtained in the preceding section is σ-independent 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]:
$$ W(x,y,t)=W(z)=\sum_{i=0}^{n} \lambda_{i} Y^{i},\quad Y=Y(z), z=a x+b y-c t. $$
(3.1)
The variable Y satisfies the differential equation
$$ Y'=Y(Y-1)\ln(k). $$
(3.2)
Solving (3.2) gives
$$ Y(z)=\frac{1}{1+d k^{z}}. $$
(3.3)
The index n to be determined by applying order-balance procedure between \(W^{2}\) and \(W''\) appears in (2.1). Thus, \(n=2\) and accordingly we write (3.1) as
$$ W(z)= \lambda_{0} +\lambda_{1} Y + \lambda_{2} Y^{2}. $$
(3.4)
Differentiating both (3.2) and (3.4) implicitly leads to
$$ Y''=Y(Y-1) (2Y-1)\ln^{2}(k) $$
(3.5)
and
$$ \begin{gathered} W'(z)=\lambda_{1} Y' +2 \lambda_{2} Y Y', \\ W''(z)=\lambda_{1} Y'' +2 \lambda_{2} \bigl(Y Y''+ \bigl(Y'\bigr)^{2}\bigr).\end{gathered} $$
(3.6)
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
$$ \begin{gathered}\alpha_{1} =\alpha_{2} = \beta_{1}= \beta_{2}= \gamma, \qquad \vert \gamma \vert < 1, \\ \lambda_{0}=0, \\ \lambda_{1}=-2 a^{2} \ln^{2}(k) , \\ \lambda_{2}=-\lambda_{1}, \\ \begin{aligned} c={}&\bigl(3 b^{2} \sigma+ a^{4} \ln^{2}(k)\\& \pm\sqrt{(3 b^{2} \sigma+ a^{4}\ln^{2}(k))^{2} +4 a s (a^{3} s + 3 b^{3} \gamma\sigma+ a b^{2} (s + 3 \gamma\sigma) + a^{4} (a + b) \gamma\ln^{2}(k))}\bigr)\\&/(2 a).\end{aligned}\end{gathered} $$
(3.7)
Therefore, a new solution of DMKP (1.3) is
$$ W(x,y,t)= -\frac{2 a^{2} d \ln^{2}(k) k^{c t + a x + b y}(2 + \ln(k))}{ (k^{c t} + d k^{a x + b y})^{2}}. $$
(3.8)
Figures 4 and 5 provide 3D plots of the Kudryashov solutions when \(\sigma> 0\) and \(\sigma<0\), respectively.
Figure 4
Figure 4

3D soliton solution of \(W(x,y,t)\) given in (3.8) when \(\sigma=2\) and respectively \(t=1\) and \(t=2\). The other assigned values are \(a=b=d=1\), \(k=e\), \(\gamma=\frac{1}{2}\)

Figure 5
Figure 5

3D soliton solution of \(W(x,y,t)\) given in (3.8) when \(\sigma=-2\) and respectively \(t=1\) and \(t=2\). The other assigned values are \(a=b=d=1\), \(k=e\), \(\gamma=\frac{1}{2}\)

4 Conclusion

A new dual-mode 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:
  • σ-independent tanh soliton solution is obtained for the DMKP.

  • σ-dependent 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 linearity-dispersive parameter γ and the phase velocity s on the spread of the obtained dual waves for DMKP.

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

(1)
Department of Mathematics and Statistics, Jordan University of Science & Technology, Irbid, Jordan
(2)
Department of Mathematics, Cankaya University, Ankara, Turkey
(3)
Institute of Space Sciences, Bucharest, Romania

References

  1. Korsunsky, S.V.: Soliton solutions for a second-order KdV equation. Phys. Lett. A 185, 174–176 (1994) MathSciNetView ArticleGoogle Scholar
  2. Wazwaz, A.M.: Multiple soliton solutions and other exact solutions for a two-mode KdV equation. Math. Methods Appl. Sci. 40(6), 1277–1283 (2017) MathSciNetGoogle Scholar
  3. Xiao, Z.J., Tian, B., Zhen, H.L., Chai, J., Wu, X.Y.: Multi-soliton solutions and Bäcklund transformation for a two-mode KdV equation in a fluid. Waves Random Complex Media 27(1), 1–14 (2017) MathSciNetView ArticleGoogle Scholar
  4. Lee, C.T., Liu, J.L.: A Hamiltonian model and soliton phenomenon for a two-mode KdV equation. Rocky Mt. J. Math. 41(4), 1273–1289 (2011) MathSciNetView ArticleGoogle Scholar
  5. Lee, C.C., Lee, C.T., Liu, J.L., Huang, W.Y.: Quasi-solitons of the two-mode Korteweg–de Vries equation. Eur. Phys. J. Appl. Phys. 52, Article ID 11301 (2010) View ArticleGoogle Scholar
  6. Lee, C.T., Lee, C.C.: On wave solutions of a weakly nonlinear and weakly dispersive two-mode wave system. Waves Random Complex Media 23, 56–76 (2013) MathSciNetView ArticleGoogle Scholar
  7. Hong, W.P., Jung, Y.D.: New non-traveling solitary wave solutions for a second-order Korteweg–de Vries equation. Z. Naturforsch. 54 a, 375–378 (1999) Google Scholar
  8. Alquran, M., Jarrah, A.: Jacobi elliptic function solutions for a two-mode KdV equation. J. King Saud Univ., Sci. (2017). https://doi.org/10.1016/j.jksus.2017.06.010 View ArticleGoogle Scholar
  9. Wazwaz, A.M.: A two-mode 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/s40819-016-0302-4 View ArticleGoogle Scholar
  10. Wazwaz, A.M.: Two-mode Sharma–Tasso–Olver equation and two-mode fourth-order Burgers equation: multiple kink solutions. Alex. Eng. J. (2017). https://doi.org/10.1016/j.aej.2017.04.003 View ArticleGoogle Scholar
  11. Syam, M., Jaradat, H.M., Alquran, M.: A study on the two-mode coupled modified Korteweg–de Vries using the simplified bilinear and the trigonometric-function methods. Nonlinear Dyn. 90(2), 1363–1371 (2017) MathSciNetView ArticleGoogle Scholar
  12. Jaradat, H.M., Syam, M., Alquran, M.: A two-mode coupled Korteweg–de Vries: multiple-soliton solutions and other exact solutions. Nonlinear Dyn. 90(1), 371–377 (2017) MathSciNetView ArticleGoogle Scholar
  13. Alquran, M., Jaradat, H.M., Syam, M.: A modified approach for a reliable study of new nonlinear equation: two-mode Korteweg–de Vries–Burgers equation. Nonlinear Dyn. 91(3), 1619–1626 (2018) View ArticleGoogle Scholar
  14. Jaradat, I., Alquran, M., Ali, M.: A numerical study on weak-dissipative two-mode perturbed Burgers and Ostrovsky models: right–left moving waves. Eur. Phys. J. Plus 133, Article ID 164 (2018) View ArticleGoogle Scholar
  15. Jaradat, I., Alquran, M., Momani, S., Biswas, A.: Dark and singular optical solutions with dual-mode nonlinear Schrodinger’s equation and Kerr-law nonlinearity. Optik 172, 822–825 (2018) View ArticleGoogle Scholar
  16. Fokou, M., Kofane, T.C., Mohamadou, A., Yomba, E.: Two-dimensional third- and fifth-order nonlinear evolution equations for shallow water waves with surface tension. Nonlinear Dyn. 91(2), 1177–1189 (2018) View ArticleGoogle Scholar
  17. Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15, 539–541 (1970) MATHGoogle Scholar
  18. Alquran, M., Al-Khaled, K.: Sinc and solitary wave solutions to the generalized Benjamin–Bona–Mahony–Burgers equations. Phys. Scr. 83, Article ID 065010 (2011) View ArticleGoogle Scholar
  19. Alquran, M., Al-Khaled, K.: The tanh and sine–cosine methods for higher order equations of Korteweg–de Vries type. Phys. Scr. 84, Article ID 025010 (2011) View ArticleGoogle Scholar
  20. Kudryashov, N.A.: One method for finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 17(6), 2248–2253 (2012) MathSciNetView ArticleGoogle Scholar
  21. Wang, L., Shen, W., Meng, Y., Chen, X.: Construction of new exact solutions to time-fractional two-component evolutionary system of order 2 via different methods. Opt. Quantum Electron. 50, Article ID 297 (2018) View ArticleGoogle Scholar

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