This paper investigates the analytical, semianalytical, and numerical solutions of the 2+1–dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation. The extended simplest equation method, the sech-tanh 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.
King Khalid UniversityR.G.P-1/151/401. 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]:(1)Qt+Uxxx−6UUx=0,where U=Ux,t 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 by(2)Qt+QxQxxQx−12QxxQx2=0,where Q=Qx,t is the unknown function. The SKdV was derived by Krichever and Novikov [10] and Weiss [11, 12].
In this paper, we study the 2+1-dimensional integrable generalization of SKdV as follows:(3)Ut+14Uxxy−UxUxy2U−UxxUy4U+Ux2Uy2U2−Ux8∫Ux2U2ydx=0.
Equation (3) was derived by Toda and Yu [13]. Using the following transformation on equation (3),(4)U=Sx,S=eF,Fx=U,Ft=R,where S=Sx,t,F=Fx,t,U=Ux,t, and R=Rx,t are the unknown functions, we obtain(5)4U2Rx−4UUxR+U2Uxxy−UUxxUy−3UUxUxy+3Ux2Uy−U4Uy=0,Ut−Rx=0.
This equation is obtained from the study by Bruzón et al. [14–16]. Using the Miura transform [17–19] on equation (5) as(6)Bx=Uxx4U−3Ux28U2−U28,By=−RU,we obtain [20, 21](7)4Bxt+Bxxxy+8BxyBx+4ByBxx=0.
If we adopt the wave transformation(8)Bx,y,t=Bℨ,ℨ=ρx+δy+ct,then we convert equation (7) into an ordinary differential equation (NLODE). The integration of the obtained NLODE with zero constant of integration leads to(9)4cB′+ρ2δB‴+6ρδB′2=0.
If we consider the substitution B′=F, then it results in(10)4cF+ρ2δF″+6ρδF2=0.
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 2+1-dimensional integrable SKdV model. We adopt the extended simplest equation method and the sech-tanh 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 n=2. Thus, the general solution of equation (10) is given by(11)Fℨ=∑i=−nnaiCℨi=a2Cℨ2+a1Cℨ+a−2Cℨ2+a−1Cℨ+a0,where aii=−2,…,2 are arbitrary constants. Additionally, Cℨ satisfies the following ordinary differential equation:(12)C′ℨ=α+λCℨ+μCℨ2,where β,α, and μ are the arbitrary constants. Substituting equations (11) and (12) into (9) and collecting all terms of Ciℨi=−4,−3,…,3,4, we get a system of algebraic equations. Solving this system, we obtain two families of solutions.
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 λ=0
For αμ>0, we obtain(15)F1x,y,t=−αμρ3−λ2ρ6−αμρcot214αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ−λραμcot14αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ,(16)F2x,y,t=−αμρ3−λ2ρ6−αμρtan214αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ−λραμtan14αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ.
For αμ<0, we obtain(17)F3x,y,t=αμρcoth214−αμρ4x−4αδμρt+4δy∓logϑ2−αμρ3,(18)F4x,y,t=αμρtanh214−αμρ4x−4αδμρt+4δy∓logϑ2−αμρ3.
When 4αμ>λ2, we obtain(19)F5x,y,t=−αμρ3−4α2μ2ρλ−4αμ−λ2tan1/84αμ−λ2ρδρtλ2−4αμ+4x+4δy+4ϑ2−λ2ρ6+2αλμρλ−4αμ−λ2tan1/84αμ−λ2ρδρtλ2−4αμ+4x+4δy+4ϑ,(20)F6x,y,t=−αμρ3−4α2μ2ρλ−4αμ−λ2cot1/84αμ−λ2ρδρtλ2−4αμ+4x+4δy+4ϑ2−λ2ρ6+2αλμρλ−4αμ−λ2cot1/84αμ−λ2ρδρtλ2−4αμ+4x+4δy+4ϑ.
According to Family 2, we have the following expressions.
2.1.2. When λ=0
For αμ>0, we obtain(21)F7x,y,t=−αμρ3−λ2ρ6−αμρtan214αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ−λραμtan14αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ,(22)F8x,y,t=−αμρ3−λ2ρ6−αμρcot214αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ−λραμcot14αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ.
For αμ<0, we obtain(23)F9x,y,t=αμρtanh214−αμρ4x−4αδμρt+4δy∓logϑ2−αμρ3,(24)F10x,y,t=αμρcoth214−αμρ4x−4αδμρt+4δy∓logϑ2−αμρ3.
When α=0: For λ>0, we get(25)F11x,y,t=−λ2μ2ρexp1/2λρδλ2ρt+4x+4δy+4ϑ6μe1/4δλ3ρ2t+λρx+δλy+λϑ−12−2λ2μρe1/4λρδλ2ρt+4x+δλy+λϑ3μe1/4δλ3ρ2t+λρx+δλy+λϑ−12−λ2ρ6μe1/4δλ3ρ2t+λρx+δλy+λϑ−12.
For λ<0, we obtain(26)F12x,y,t=−μ4ρexp1/2λρδλ2ρt+4x+4δy+4ϑμe1/4δλ3ρ2t+λρx+δλy+λϑ+12−λ2ρ6μe1/4δλ3ρ2t+λρx+δλy+λϑ+12−λ2μ2ρexp1/2λρδλ2ρt+4x+4δy+4ϑ6μe1/4δλ3ρ2t+λρx+δλy+λϑ+12+λμ2ρe1/4λρδλ2ρt+4x+δλy+λϑμe1/4δλ3ρ2t+λρx+δλy+λϑ+12−λ2μρe1/4λρδλ2ρt+4x+δλy+λϑ3μe1/4δλ3ρ2t+λρx+δλy+λϑ+12+λμ3ρexp1/2λρδλ2ρt+4x+4δy+4ϑμe1/4δλ3ρ2t+λρx+δλy+λϑ+12.
When 4αμ>λ2, we obtain(27)F13x,y,t=2αμρ3−λ2ρ6+14λ2ρsec2184αμ−λ2δρ2tλ2−4αμ+4ρx+4δy+4ϑ−αμρsec2184αμ−λ2δρ2tλ2−4αμ+4ρx+4δy+4ϑ,(28)F14x,y,t=2αμρ3−λ2ρ6+14λ2ρcsc2184αμ−λ2δρ2tλ2−4αμ+4ρx+4δy+4ϑ−αμρcsc2184αμ−λ2δρ2tλ2−4αμ+4ρx+4δy+4ϑ.
2.2. Sech-Tanh Method
The general solution of equation (10) according to the sech-tanh method and calculated value of balance is given by(29)Fℨ=∑i=1nsechi−1ℨaisechℨ+bitanhℨ+a0=sechℨa2sechℨ+b2tanhℨ+a1sechℨ+a0+b1tanhℨ,where a0,a1,a2,b1, and b2 are the arbitrary constants. Substituting equation (29) into (10) and collecting all terms of sechℨ,sech2ℨ,sech3ℨ,tanhℨ,tanhℨsech2ℨ, and tanhξsechℨ, we obtain a system of algebraic equations. Solving this system, we obtain(30)a0⟶2c3δ,a1⟶0,a2⟶−cδ,b1⟶0,b2⟶0,ρ⟶−cδ,where c>0,δ>0.
Consequently, the explicit solution of equation (7) is given by(31)F15x,y,t=c2−3sech2ct+ρx+δy3δ.
3. Stability Investigation
We now examine the stability property for 2+1-dimensional integrable SKdV model with the Miura transformation by means of an Hamiltonian system. The momentum ℌ in the Hamiltonian system is given by(32)ℌ=12∫−JJB2ℨdℨ,where Bℨ is the solution of the model. Consequently, the condition for stability of the solutions can be formulated as(33)∂ℌ∂c>0,where c is the wave velocity. The momentum in the Hamiltonian system for equations (18) and (31) are given, respectively, by(34)ℌ=1c3200c−sech210c+26−sech210c+34+sech226−10c+sech234−10c−4loge52−1×sinh10c+1+e52cosh10c−4loge68−1sinh10c+1+e68cosh10c+4log1+e52cosh10c−e52−1sinh10c+4log1+e68cosh10c−e68−1sinh10c,(35)ℌ=1724100c+loge7−5c+e5c+loge23−5c+e5c−loge−5c+e5c+7−loge−5c+e5c+23−sech25c+72−sech25c+232+sech272−5c+sech2232−5c.And thus,(36)∂ℌ∂cc=72=2.37146×10−298>0,(37)∂ℌ∂cc=9=5.55556>0.
We conclude that this solution is stable on the interval x∈−5,5,t∈−5,5. 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 2+1-dimensional integrable SKdV model. The Adomian decomposition method and cubic b-spline 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:(38)∑j=0∞Fjℨ=F0+F′0ℨ−4cρ2δL−1∑j=0∞Fjℨ−6ρL−1∑j=0∞Aj.
Thus, with respect to equation (18) and the following conditions α=−1,a0=4,a−1=0,a−2=−3,c=72,δ=2,λ=0,μ=4, and ρ=3, we obtain(39)F0=−2,F1=12ℨ2,F2=−8ℨ4,F3=8ℨ4−16ℨ63.
Consequently, the semianalytical solution of equation (10) is written in the following form:(40)F=−16ℨ63+12ℨ2−2+⋯.
However, with respect to equation (31) and the following conditions a0=1,a1=0,a2=−3/2,b1=0,b2=0,c=9,δ=4, and ρ=−3/2, we obtain(41)F0=−12,F1=3ℨ22,F2=−ℨ4,F3=13ℨ630−ℨ42.
Consequently, the semianalytical solution of equation (10) is written in the following form:(42)F=13ℨ630−3ℨ42+3ℨ22−12+⋯.
4.2. Cubic B-Spline Scheme
Employing the cubic B-spline 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 Fx,y,t to highlight the properties of the 2+1-dimensional integrable SKdV model with Miura transformation. In the follow-up, we fix the value of y to characterize these solutions and the interpretation is based on three different types of representations (three- and two-dimensional charts and contour plot). In the following steps, the physical interpretation of the represented figures is discussed:
Figure 1 shows the bright solitary for (18) in the three-dimensional plot (a) to illustrate the perspective view of the solution, the two-dimensional plot (b) to present the wave propagation pattern of the wave along x- axis, and the contour plot (c) to explain the overhead view of the solution when α=−1,a0=4,a−1=0,a−2=−3,c=72,δ=2,λ=0,μ=4, and ρ=3
Figure 2 shows the dark solitary for (31) in the three-dimensional plot (a) to illustrate the perspective view of the solution, the two-dimensional plot (b) to present the wave propagation pattern of the wave along the x-axis, and the contour plot (c) to explain the overhead view of the solution when a0=1,a1=0,a2=−3/2,b1=0,b2=0,c=9,δ=4, and ρ=−3/2
Figure 3 illustrates the exact and semianalytical obtained solutions, respectively, by the extended simplest equation method and Adomian decomposition method
Figure 4 illustrates the exact and semianalytical obtained solutions, respectively, by the sech-tanh expansion method and Adomian decomposition method
Figure 5 illustrates the exact and numerical obtained solutions, respectively, via the sech-tanh expansion method and cubic B-spline scheme
Figure 6 illustrates the exact and numerical obtained solutions, respectively, via the sech-tanh expansion method and cubic B-spline scheme
Now, we shall show the accuracy of our obtained solution and explain the comparison between the two employed analytical schemes:
Tables 1 and 2 show calculated values of the exact, semianalytical, and numerical solutions with different values of ℨ. These values show the accuracy of the obtained analytical solutions via the sech-tanh expansion method over the obtained analytical solutions via the extended simplest equation method where the absolute values of error in the sech-tanh method is smaller that those obtained by the extended simplest equation method. Figure 7 explains the absolute value of error in 1 and 2.
Solitary wave in three different forms of equation (18).
Solitary wave in three different forms of equation (31).
Exact and numerical solutions of equation (10) according to the obtained analytical solution via the extended simplest equation method.
Exact and numerical solutions of equation (10) according to the obtained analytical solution via the sech-tanh expansion method.
Exact and numerical solutions of equation (10) according to the obtained analytical solution via the extended simplest equation method.
Exact and numerical solutions of equation (10) according to the obtained analytical solution via the sech-tanh expansion method.
Absolute value of error via the Adomian decomposition method (a) and cubic B-spline scheme (b) based on the shown value in Tables 1 and 2.
Exact, semianalytical, and absolute values of error with different values of ℨ with respect to the obtained solution via extended simplest equation method (18) and sech-tanh method (31) via the Adomian decomposition method.
Ext. Sim Eq. method
Sech-tanh method
Absolute error
Value of 3
Exact
Approximate
Exact
Approximate
First scheme
Second scheme
0.001
1.99998
1.99999
0.499999
0.499999
1.19999E−05
4.99711×10−13
0.002
1.9999
1.99995
0.499994
0.499994
0.000047999
7.99988×10−12
0.003
1.99978
1.99989
0.499987
0.499987
0.000107995
4.05003×10−11
0.004
1.99962
1.99981
0.499976
0.499976
0.000191984
1.28001×10−10
0.005
1.9994
1.9997
0.499963
0.499963
0.00029996
3.12502×10−10
0.006
1.99914
1.99957
0.499946
0.499946
0.000431917
6.48006×10−10
0.007
1.99882
1.99941
0.499927
0.499927
0.000587846
1.20052×10−9
0.008
1.99846
1.99923
0.499904
0.499904
0.000767738
2.04804×10−9
0.009
1.99806
1.99903
0.499879
0.499879
0.00097158
3.28057×10−9
0.01
1.9976
1.9988
0.49985
0.49985
0.00119936
5.00013×10−9
Exact, numerical, and absolute value of error with different value of ℨ with respect to the obtained solution via the extended simplest equation method (18) and sech-tanh method (31) via the cubic B-spline scheme.
Value of ℨ
Ext. Sim Eq. Method
Sech-tanh method
Absolute error
Exact
Numerical
Exact
Numerical
First scheme
Second scheme
0
−2
−2
−0.5
−0.5
0
0
0.001
−1.99999
−1.99987
−0.499999
−0.499998
0.00012596
8.99958×10−12
0.002
−1.99998
−1.99971
−0.499994
−0.499994
0.000263919
1.59986×10−11
0.003
−1.99995
−1.99953
−0.499987
−0.499987
0.000413876
2.09974×10−11
0.004
−1.9999
−1.99933
−0.499976
−0.499976
0.000575829
2.39971×10−11
0.005
−1.99985
−1.9991
−0.499963
−0.499963
0.000749778
2.49963×10−11
0.006
−1.99978
−1.99885
−0.499946
−0.499946
0.000935721
2.39959×10−11
0.007
−1.99971
−1.99857
−0.499927
−0.499927
0.00113366
2.09953×10−11
0.008
−1.99962
−1.99827
−0.499904
−0.499904
0.00134358
1.59955×10−11
0.009
−1.99951
−1.99795
−0.499879
−0.499879
0.0015655
8.99764×10−12
0.01
−1.9994
−1.9976
−0.49985
−0.49985
0.0017994
0
6. Conclusion
In this paper, the extended simplest equation and sech-tanh expansion methods have been successfully implemented on the 2+1-dimensional integrable SKdV model with Miura transform. Moreover, the stability properties of the solutions have also been tackled. The Adomian decomposition method and cubic B-spline 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 sech-tanh 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 real-world 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.P-1/151/40.
SeadawyA.Stability analysis of traveling wave solutions for generalized coupled nonlinear KdV equations201610120921410.18576/amis/1001202-s2.0-84959154710OsmanM. S.WazwazA.-M.An efficient algorithm to construct multi-soliton rational solutions of the (2+1)-dimensional KdV equation with variable coefficients201832128228910.1016/j.amc.2017.10.0422-s2.0-85034111941WazwazA.-M.El-TantawyS. A.A new integrable ($$3+1$$ 3+1)-dimensional KdV-like model with its multiple-soliton solutions20168331529153410.1007/s11071-015-2427-02-s2.0-84955205894JacksonC. B.ØsterlundC.MugarG.HassmanK. D.CrowstonK.Motivations for sustained participation in crowdsourcing: case studies of citizen science on the role of talkProceedings of the 2015 48th Hawaii International Conference on System SciencesJanuary 2015Kauai, HI, USAIEEE1624163410.1109/HICSS.2015.1962-s2.0-84944245831BhrawyA. H.DohaE. H.Ezz-EldienS. S.AbdelkawyM. A.A numerical technique based on the shifted legendre polynomials for solving the time-fractional coupled KdV equations201653111710.1007/s10092-014-0132-x2-s2.0-84959146510SeadawyA. R.The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schrödinger equation and its solutions2017139314310.1016/j.ijleo.2017.03.0862-s2.0-85016519768El-AjouA.ArqubO. A.MomaniS.Approximate analytical solution of the nonlinear fractional KdV-Burgers equation: a new iterative algorithm2015293819510.1016/j.jcp.2014.08.0042-s2.0-84937057969LouS.HuangF.Alice-Bob physics: coherent solutions of nonlocal KdV systems20177186910.1038/s41598-017-00844-y2-s2.0-85018164012HuW.-Q.JiaS.-L.General propagation lattice Boltzmann model for variable-coefficient non-isospectral KdV equation201991616710.1016/j.aml.2018.12.0022-s2.0-85058374382KricheverI. M.NovikovS. P.Holomorphic bundles over algebraic curves and non-linear equations1980356537910.1070/rm1980v035n06abeh0019742-s2.0-84956211082WeissJ.The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative19832461405141310.1063/1.525875WeissJ.The Painlevé property and Bäcklund transformations for the sequence of Boussinesq equations198526225826910.1063/1.5266552-s2.0-0009231934TodaK.YuS.-J.The investigation into the Schwarz-Korteweg-de Vries equation and the Schwarz derivative in (2+1) dimensions20004174747475110.1063/1.5333742-s2.0-0034371970BruzónM. S.GandariasM. L.MurielC.RamírezJ.RomeroF. R.Traveling-wave solutions of the Schwarz–Korteweg–de Vries equation in 2+1 dimensions and the Ablowitz–Kaup–Newell–Segur equation through symmetry reductions200313711378138910.1023/a:10260923040472-s2.0-0347753282RamírezJ.RomeroJ. L.BruzónM. S.GandariasM. L.Multiple solutions for the Schwarzian Korteweg-de Vries equation in (2+1) dimensions200732268269310.1016/j.chaos.2005.11.0192-s2.0-33749833507KudryashovN. A.PickeringA.Rational solutions for Schwarzian integrable hierarchies199831479505951810.1088/0305-4470/31/47/0112-s2.0-0032573320NakamuraA.The Miura transform and the existence of an infinite number of conservation laws of the cylindrical KdV equation198182311111210.1016/0375-9601(81)90924-52-s2.0-0002255956KenigC. E.MartelY.Global well-posedness in the energy space for a modified KP II equation via the Miura transform200635862447248810.1090/s0002-9947-06-04072-42-s2.0-33744732698MasJ.RamosE.The constrained KP hierarchy and the generalised Miura transformation19953511-319419910.1016/0370-2693(95)00357-q2-s2.0-0000910308Senthil VelanM.LakshmananM.Lie symmetries, Kac-Moody-Virasoro algebras and integrability of certain (2+1)-dimensional nonlinear evolution equations19985219021110.2991/jnmp.1998.5.2.102-s2.0-0008071969MaH.BaiY.New solutions of the Schwarz-Korteweg-de Vries equation in 2+1 dimensions with the Gauge transformation20141714146KhaterM.AttiaR.LuD.Modified auxiliary equation method versus three nonlinear fractional biological models in present explicit wave solutions2019241KhaterM. M.LuD.AttiaR. A.Dispersive long wave of nonlinear fractional Wu-Zhang system via a modified auxiliary equation method2019922500310.1063/1.50876472-s2.0-85061263807AttiaR.LuD.KhaterM.Chaos and relativistic energy-momentum of the nonlinear time fractional Duffing equation20192411010.3390/mca24010010BaskonusH. M.KoçD. A.BulutH.Dark and new travelling wave solutions to the nonlinear evolution equation2016127198043805510.1016/j.ijleo.2016.05.1322-s2.0-84991442209DusunceliF.New exponential and complex traveling wave solutions to the Konopelchenko-Dubrovsky model201920199780124710.1155/2019/78012472-s2.0-85062353381OdibatZ.An optimized decomposition method for nonlinear ordinary and partial differential equations202054112332310.1016/j.physa.2019.123323BakodahH. O.Al QarniA. A.BanajaM. A.ZhouQ.MoshokoaS. P.BiswasA.Bright and dark Thirring optical solitons with improved Adomian decomposition method20171301115112310.1016/j.ijleo.2016.11.1232-s2.0-85006386579AliA. T.KhaterM. M. A.AttiaR. A. M.Abdel-AtyA.-H.LuD.Abundant numerical and analytical solutions of the generalized formula of Hirota-Satsuma coupled KdV system202013110947310.1016/j.chaos.2019.109473KhaterM. M.ParkC.Abdel-AtyA.-H.AttiaR. A.LuD.On new computational and numerical solutions of the modified Zakharov–Kuznetsov equation arising in electrical engineering20205931099110510.1016/j.aej.2019.12.043