Research Article  Open Access
Raghda A. M. Attia, S. H. Alfalqi, J. F. Alzaidi, Mostafa M. A. Khater, Dianchen Lu, "Computational and Numerical Solutions for Dimensional Integrable Schwarz–Korteweg–de Vries Equation with Miura Transform", Complexity, vol. 2020, Article ID 2394030, 13 pages, 2020. https://doi.org/10.1155/2020/2394030
Computational and Numerical Solutions for Dimensional Integrable Schwarz–Korteweg–de Vries Equation with Miura Transform
Abstract
This paper investigates the analytical, semianalytical, and numerical solutions of the –dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation. The extended simplest equation method, the sechtanh method, the Adomian decomposition method, and cubic spline scheme are employed to obtain distinct formulas of solitary waves that are employed to calculate the initial and boundary conditions. Consequently, the numerical solutions of this model can be investigated. Moreover, their stability properties are also analyzed. The solutions obtained by means of these techniques are compared to unravel relations between them and their characteristics illustrated under the suitable choice of the parameter values.
1. Introduction
The Korteweg–de Vries (KdV) equation is a seminal model in fluid mechanics. This model was introduced by Boussinesq in 1877 and reintroduced by Diederik Korteweg and Gustav de Vries in 1895. The KdV has the following formula [1–9]:where characterizes the weakly nonlinear shallow water waves. Equation (1) can be written in many distinct forms and combined with other models. One of them is the Schwarz–Korteweg–de Vries (SKdV) equation given bywhere is the unknown function. The SKdV was derived by Krichever and Novikov [10] and Weiss [11, 12].
In this paper, we study the dimensional integrable generalization of SKdV as follows:
Equation (3) was derived by Toda and Yu [13]. Using the following transformation on equation (3),where , and are the unknown functions, we obtain
This equation is obtained from the study by Bruzón et al. [14–16]. Using the Miura transform [17–19] on equation (5) aswe obtain [20, 21]
If we adopt the wave transformationthen we convert equation (7) into an ordinary differential equation (NLODE). The integration of the obtained NLODE with zero constant of integration leads to
If we consider the substitution , then it results in
Having these ideas in mind, this paper is organized as follows: Section 2 presents the two methods and derives the solutions of the SKdV equation. Section 5 represents the solutions for several numerical values of the parameters. Additionally, their stability and properties are also discussed. Finally, Section 6 summarises the main conclusions.
2. Explicit Solutions
In this section, we apply two analytical techniques for deriving the solutions of the dimensional integrable SKdV model. We adopt the extended simplest equation method and the sechtanh method to obtain various distinct formulas of solitary wave solutions of equation (3). For further details about the two methods, see [22–26].
2.1. Extended Simplest Equation
According to the homogeneous balance rule between the highest derivative and the nonlinear term in equation (9), we obtain . Thus, the general solution of equation (10) is given bywhere are arbitrary constants. Additionally, satisfies the following ordinary differential equation:where are the arbitrary constants. Substituting equations (11) and (12) into (9) and collecting all terms of , we get a system of algebraic equations. Solving this system, we obtain two families of solutions.
Family 1.
Family 2. From these two families, the solitary wave solutions of equation (7) can be obtained.
According to Family 1, we have the following expressions.
2.1.1. When
For , we obtain
For , we obtain
When , we obtain
According to Family 2, we have the following expressions.
2.1.2. When
For , we obtain
For , we obtain
When : For , we get
For , we obtain
When , we obtain
2.2. SechTanh Method
The general solution of equation (10) according to the sechtanh method and calculated value of balance is given bywhere are the arbitrary constants. Substituting equation (29) into (10) and collecting all terms of , we obtain a system of algebraic equations. Solving this system, we obtain
Consequently, the explicit solution of equation (7) is given by
3. Stability Investigation
We now examine the stability property for dimensional integrable SKdV model with the Miura transformation by means of an Hamiltonian system. The momentum in the Hamiltonian system is given bywhere is the solution of the model. Consequently, the condition for stability of the solutions can be formulated aswhere is the wave velocity. The momentum in the Hamiltonian system for equations (18) and (31) are given, respectively, byAnd thus,
We conclude that this solution is stable on the interval . This result shows the ability of the solutions for their application. Using the same steps, we can check the stability property of all other obtained solutions.
4. Semianalytical and Numerical Solutions
This section applies semianalytical and numerical schemes for deriving the solutions of the dimensional integrable SKdV model. The Adomian decomposition method and cubic bspline schemes are employed to the method to investigate the accuracy of the obtained analytical solutions. Also, this study aims to give a comparison between both used analytical schemes. For further details about the two methods, see [27–30].
4.1. Adomian Decomposition Method
Applying this scheme gives equation (10) in the following form:
Thus, with respect to equation (18) and the following conditions , we obtain
Consequently, the semianalytical solution of equation (10) is written in the following form:
However, with respect to equation (31) and the following conditions , we obtain
Consequently, the semianalytical solution of equation (10) is written in the following form:
4.2. Cubic BSpline Scheme
Employing the cubic Bspline scheme to evaluate the numerical solutions of equation (10). Using same initial and boundary conditions with respect to the obtained solutions (18) and (31), yields
5. Discussion
This section illustrates several of the results for to highlight the properties of the dimensional integrable SKdV model with Miura transformation. In the followup, we fix the value of to characterize these solutions and the interpretation is based on three different types of representations (three and twodimensional charts and contour plot). In the following steps, the physical interpretation of the represented figures is discussed:(i)Figure 1 shows the bright solitary for (18) in the threedimensional plot (a) to illustrate the perspective view of the solution, the twodimensional plot (b) to present the wave propagation pattern of the wave along  axis, and the contour plot (c) to explain the overhead view of the solution when (ii)Figure 2 shows the dark solitary for (31) in the threedimensional plot (a) to illustrate the perspective view of the solution, the twodimensional plot (b) to present the wave propagation pattern of the wave along the axis, and the contour plot (c) to explain the overhead view of the solution when (iii)Figure 3 illustrates the exact and semianalytical obtained solutions, respectively, by the extended simplest equation method and Adomian decomposition method(iv)Figure 4 illustrates the exact and semianalytical obtained solutions, respectively, by the sechtanh expansion method and Adomian decomposition method(v)Figure 5 illustrates the exact and numerical obtained solutions, respectively, via the sechtanh expansion method and cubic Bspline scheme(vi)Figure 6 illustrates the exact and numerical obtained solutions, respectively, via the sechtanh expansion method and cubic Bspline scheme Now, we shall show the accuracy of our obtained solution and explain the comparison between the two employed analytical schemes:(vii)Tables 1 and 2 show calculated values of the exact, semianalytical, and numerical solutions with different values of . These values show the accuracy of the obtained analytical solutions via the sechtanh expansion method over the obtained analytical solutions via the extended simplest equation method where the absolute values of error in the sechtanh method is smaller that those obtained by the extended simplest equation method. Figure 7 explains the absolute value of error in 1 and 2.
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)


6. Conclusion
In this paper, the extended simplest equation and sechtanh expansion methods have been successfully implemented on the dimensional integrable SKdV model with Miura transform. Moreover, the stability properties of the solutions have also been tackled. The Adomian decomposition method and cubic Bspline scheme have also employed to investigate the semianalytical and numerical solutions, and the two show the accuracy of the obtained analytical solutions. The rigor of the obtained solutions that have been obtained by the sechtanh expansion method has been discussed. The solutions were represented by allowing a physical interpretation and better interpretation of their properties. In summary, this paper studied the SKdV and found relevant solutions that provide new interpretations of the realworld phenomena.
Data Availability
Data sharing is not applicable for this article as no datasets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Authors’ Contributions
All authors conceived the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha, KSA, for funding this work through the research group under grant number R.G.P1/151/40.
References
 A. Seadawy, “Stability analysis of traveling wave solutions for generalized coupled nonlinear KdV equations,” Applied Mathematics & Information Sciences, vol. 10, no. 1, pp. 209–214, 2016. View at: Publisher Site  Google Scholar
 M. S. Osman and A.M. Wazwaz, “An efficient algorithm to construct multisoliton rational solutions of the (2+1)dimensional KdV equation with variable coefficients,” Applied Mathematics and Computation, vol. 321, pp. 282–289, 2018. View at: Publisher Site  Google Scholar
 A.M. Wazwaz and S. A. ElTantawy, “A new integrable ($$3+1$$ 3+1)dimensional KdVlike model with its multiplesoliton solutions,” Nonlinear Dynamics, vol. 83, no. 3, pp. 1529–1534, 2016. View at: Publisher Site  Google Scholar
 C. B. Jackson, C. Østerlund, G. Mugar, K. D. Hassman, and K. Crowston, “Motivations for sustained participation in crowdsourcing: case studies of citizen science on the role of talk,” in Proceedings of the 2015 48th Hawaii International Conference on System Sciences, pp. 1624–1634, IEEE, Kauai, HI, USA, January 2015. View at: Publisher Site  Google Scholar
 A. H. Bhrawy, E. H. Doha, S. S. EzzEldien, and M. A. Abdelkawy, “A numerical technique based on the shifted legendre polynomials for solving the timefractional coupled KdV equations,” Calcolo, vol. 53, no. 1, pp. 1–17, 2016. View at: Publisher Site  Google Scholar
 A. R. Seadawy, “The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schrödinger equation and its solutions,” Optik, vol. 139, pp. 31–43, 2017. View at: Publisher Site  Google Scholar
 A. ElAjou, O. A. Arqub, and S. Momani, “Approximate analytical solution of the nonlinear fractional KdVBurgers equation: a new iterative algorithm,” Journal of Computational Physics, vol. 293, pp. 81–95, 2015. View at: Publisher Site  Google Scholar
 S. Lou and F. Huang, “AliceBob physics: coherent solutions of nonlocal KdV systems,” Scientific Reports, vol. 7, no. 1, p. 869, 2017. View at: Publisher Site  Google Scholar
 W.Q. Hu and S.L. Jia, “General propagation lattice Boltzmann model for variablecoefficient nonisospectral KdV equation,” Applied Mathematics Letters, vol. 91, pp. 61–67, 2019. View at: Publisher Site  Google Scholar
 I. M. Krichever and S. P. Novikov, “Holomorphic bundles over algebraic curves and nonlinear equations,” Russian Mathematical Surveys, vol. 35, no. 6, pp. 53–79, 1980. View at: Publisher Site  Google Scholar
 J. Weiss, “The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative,” Journal of Mathematical Physics, vol. 24, no. 6, pp. 1405–1413, 1983. View at: Publisher Site  Google Scholar
 J. Weiss, “The Painlevé property and Bäcklund transformations for the sequence of Boussinesq equations,” Journal of Mathematical Physics, vol. 26, no. 2, pp. 258–269, 1985. View at: Publisher Site  Google Scholar
 K. Toda and S.J. Yu, “The investigation into the SchwarzKortewegde Vries equation and the Schwarz derivative in (2+1) dimensions,” Journal of Mathematical Physics, vol. 41, no. 7, pp. 4747–4751, 2000. View at: Publisher Site  Google Scholar
 M. S. Bruzón, M. L. Gandarias, C. Muriel, J. Ramírez, and F. R. Romero, “Travelingwave solutions of the Schwarz–Korteweg–de Vries equation in 2+1 dimensions and the Ablowitz–Kaup–Newell–Segur equation through symmetry reductions,” Theoretical and Mathematical Physics, vol. 137, no. 1, pp. 1378–1389, 2003. View at: Publisher Site  Google Scholar
 J. Ramírez, J. L. Romero, M. S. Bruzón, and M. L. Gandarias, “Multiple solutions for the Schwarzian Kortewegde Vries equation in (2+1) dimensions,” Chaos, Solitons & Fractals, vol. 32, no. 2, pp. 682–693, 2007. View at: Publisher Site  Google Scholar
 N. A. Kudryashov and A. Pickering, “Rational solutions for Schwarzian integrable hierarchies,” Journal of Physics A: Mathematical and General, vol. 31, no. 47, pp. 9505–9518, 1998. View at: Publisher Site  Google Scholar
 A. Nakamura, “The Miura transform and the existence of an infinite number of conservation laws of the cylindrical KdV equation,” Physics Letters A, vol. 82, no. 3, pp. 111112, 1981. View at: Publisher Site  Google Scholar
 C. E. Kenig and Y. Martel, “Global wellposedness in the energy space for a modified KP II equation via the Miura transform,” Transactions of the American Mathematical Society, vol. 358, no. 6, pp. 2447–2488, 2006. View at: Publisher Site  Google Scholar
 J. Mas and E. Ramos, “The constrained KP hierarchy and the generalised Miura transformation,” Physics Letters B, vol. 351, no. 13, pp. 194–199, 1995. View at: Publisher Site  Google Scholar
 M. Senthil Velan and M. Lakshmanan, “Lie symmetries, KacMoodyVirasoro algebras and integrability of certain (2+1)dimensional nonlinear evolution equations,” Journal of Nonlinear Mathematical Physics, vol. 5, no. 2, pp. 190–211, 1998. View at: Publisher Site  Google Scholar
 H. Ma and Y. Bai, “New solutions of the SchwarzKortewegde Vries equation in 2+1 dimensions with the Gauge transformation,” International Journal of Nonlinear Science, vol. 17, no. 1, pp. 41–46, 2014. View at: Google Scholar
 M. Khater, R. Attia, and D. Lu, “Modified auxiliary equation method versus three nonlinear fractional biological models in present explicit wave solutions,” Mathematical and Computational Applications, vol. 24, no. 1, 2019. View at: Google Scholar
 M. M. Khater, D. Lu, and R. A. Attia, “Dispersive long wave of nonlinear fractional WuZhang system via a modified auxiliary equation method,” AIP Advances, vol. 9, no. 2, p. 25003, 2019. View at: Publisher Site  Google Scholar
 R. Attia, D. Lu, and M. Khater, “Chaos and relativistic energymomentum of the nonlinear time fractional Duffing equation,” Mathematical and Computational Applications, vol. 24, no. 1, p. 10, 2019. View at: Publisher Site  Google Scholar
 H. M. Baskonus, D. A. Koç, and H. Bulut, “Dark and new travelling wave solutions to the nonlinear evolution equation,” Optik, vol. 127, no. 19, pp. 8043–8055, 2016. View at: Publisher Site  Google Scholar
 F. Dusunceli, “New exponential and complex traveling wave solutions to the KonopelchenkoDubrovsky model,” Advances in Mathematical Physics, vol. 2019, Article ID 7801247, 9 pages, 2019. View at: Publisher Site  Google Scholar
 Z. Odibat, “An optimized decomposition method for nonlinear ordinary and partial differential equations,” Physica A: Statistical Mechanics and Its Applications, vol. 541, p. 123323, 2020. View at: Publisher Site  Google Scholar
 H. O. Bakodah, A. A. Al Qarni, M. A. Banaja, Q. Zhou, S. P. Moshokoa, and A. Biswas, “Bright and dark Thirring optical solitons with improved Adomian decomposition method,” Optik, vol. 130, pp. 1115–1123, 2017. View at: Publisher Site  Google Scholar
 A. T. Ali, M. M. A. Khater, R. A. M. Attia, A.H. AbdelAty, and D. Lu, “Abundant numerical and analytical solutions of the generalized formula of HirotaSatsuma coupled KdV system,” Chaos, Solitons & Fractals, vol. 131, p. 109473, 2020. View at: Publisher Site  Google Scholar
 M. M. Khater, C. Park, A.H. AbdelAty, R. A. Attia, and D. Lu, “On new computational and numerical solutions of the modified Zakharov–Kuznetsov equation arising in electrical engineering,” Alexandria Engineering Journal, vol. 59, no. 3, pp. 1099–1105, 2020. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2020 Raghda A. M. Attia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.