| Citation: |
Zhou-Zheng Kang, Tie-Cheng Xia. American Institute of Mathematical Sciences[J]. Journal of Applied Analysis & Computation, 2020, 10(2): 729-739. doi: 10.11948/20190128
|
American Institute of Mathematical Sciences
-
Abstract
This work aims to study the negative-order KdV equation in (3+1)-dimensions which is developed via using the recursion operator of the KdV equation by employing the three-wave methods. As a consequence, a variety of novel multiwave solutions with several arbitrary parameters to the considered equation are presented. Moreover, selecting particular values for the parameters, some graphs are plotted to show the spatial structures and dynamics of the resulting solutions. These results enrich the variety of the dynamics in the field of nonlinear waves.-
Keywords:
- Negative-order KdV equation /
- three-wave methods /
- multiwave solutions
-
-
References
[1] H. Chen, Y. C. Lee and C. Liu, Integrability of nonlinear Hamiltonian systems by inverse scattering method, Phys. Scr., 1979, 20, 490-492. doi: 10.1088/0031-8949/20/3-4/026 [2] J. Chai, B. Tian, X. Xie and Y. Sun, Conservation laws, bilinear Bäcklund transformations and solitons for a nonautonomous nonlinear Schrödinger equation with external potentials, Commun. Nonlinear Sci. Numer. Simulat., 2016, 39, 472-480. doi: 10.1016/j.cnsns.2016.02.024 [3] C. He, Y. Tang, W. Ma and J. Ma, Interaction phenomena between a lump and other multi-solitons for the (2+1)-dimensional BLMP and Ito equations, Nonlinear Dynam., 2019, 95(1), 29-42. doi: 10.1007/s11071-018-4548-8 [4] Z. Kang, T. Xia and W. Ma, Abundant multiwave solutions to the (3+1)-dimensional Sharma-Tasso-Olver-like equation, Proc. Rom. Acad. Ser. A, 2019, 20(2), 115-122. [5] S. Lou and J. Lu, Special solutions from the variable separation approach: the Davey-Stewartson equation, J. Phys. A: Math. Gen., 1996, 29, 4209-4215. doi: 10.1088/0305-4470/29/14/038 [6] J. Liu, L. Zhou and Y. He, Multiple soliton solutions for the new (2+1)-dimensional Korteweg-de Vries equation by multiple exp-function method, Appl. Math. Lett., 2018, 80, 71-78. doi: 10.1016/j.aml.2018.01.010 [7] Z. Lan, Y. Gao, J. Yang, C. Su, C. Zhao and Z. Gao, Solitons and Bäcklund transformation for a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation in fluid dynamics, Appl. Math. Lett., 2016, 60, 96-100. doi: 10.1016/j.aml.2016.03.021 [8] J. Liu and Y. He, Abundant lump and lump-kink solutions for the new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation, Nonlinear Dynam., 2018, 92(3), 1103-1108. doi: 10.1007/s11071-018-4111-7 [9] W. Ma and Z. Zhu, Solving the (3+1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm, Appl. Math. Comput., 2012, 218(24), 11871-11879. [10] W. Ma, Riemann-Hilbert problems and N-soliton solutions for a coupled mKdV system, J. Geom. Phys., 2018, 132, 45-54. doi: 10.1016/j.geomphys.2018.05.024 [11] Y. Shi, Exact breather-type solutions and resonance-type solutions of the (2+1)-dimensional potential Burgers system, Romanian J. Phys., 2017, 62(5-6), Article no. 116, 1-16. [12] X. Tang and Z. Liang, Variable separation solutions for the (3+1)-dimensional Jimbo-Miwa equation, Phys. Lett. A, 2006, 351(6), 398-402. doi: 10.1016/j.physleta.2005.11.035 [13] Y. Tang, S. Tao, M. Zhou and Q. Guan, Interaction solutions between lump and other solitons of two classes of nonlinear evolution equations, Nonlinear Dynam., 2017, 89(2), 1-14. [14] A. M. Wazwaz and S. A. EI-Tantawy, Solving the (3+1)-dimensional KP-Boussinesq and BKP-Boussinesq equations by the simplified Hirota's method, Nonlinear Dynam., 2017, 88(4), 3017-3021. [15] A. M. Wazwaz, Multiple complex soliton solutions for integrable negative-order KdV and integrable negative-order modified KdV equations, Appl. Math. Lett., 2019, 88, 1-7. doi: 10.1016/j.aml.2018.08.004 [16] X. Wang, Y. Li and Y. Chen, Generalized Darboux transformation and localized waves in coupled Hirota equations, Wave Motion, 2014, 51(7), 1149-1160. doi: 10.1016/j.wavemoti.2014.07.001 [17] D. Wang, S. Yin, Y. Tian and Y. Liu, Integrability and bright soliton solutions to the coupled nonlinear Schrödinger equation with higher-order effects, Appl. Math. Comput., 2014, 229, 296-309. [18] A. M. Wazwaz and L. Kaur, Complex simplified Hirota's forms and Lie symmetry analysis for multiple real and complex soliton solutions of the modified KdV-Sine-Gordon equation, Nonlinear Dynam., 2019, 95(3), 2209-2215. doi: 10.1007/s11071-018-4686-z [19] C. Wang, Z. Dai and L. Liang, Exact three-wave solution for higher dimensional KdV-type equation, Appl. Math. Comput., 2010, 216(2), 501-505. [20] C. Wang and Z. Dai, Breather-type multi-solitary waves to the Kadomtsev-Petviashvili equation with positive dispersion, Appl. Math. Comput., 2014, 235, 332-337. [21] A. M. Wazwaz, Negative-order KdV equations in (3+1)-dimensions by using the KdV recursion operator, Waves Random Complex Media, 2017, 27(4), 768-778. doi: 10.1080/17455030.2017.1317115 [22] T. Xia, X. Chen and D. Chen, Darboux transformation and soliton-like solutions of nonlinear Schrödinger equations, Chaos Solitons Fractals, 2005, 26(3), 889-896. doi: 10.1016/j.chaos.2005.01.030 [23] M. Xu, S. Tian, J. Tu and T. Zhang, Lie symmetry analysis, conservation laws, solitary and periodic waves for a coupled Burgers equation, Superlattice Microst., 2017, 101, 415-428. doi: 10.1016/j.spmi.2016.11.050 [24] M. Xu, T. Xia and B. Hu, Riemann-Hilbert approach and N-soliton solutions for the Chen-Lee-Liu equation, Modern Phys. Lett. B, 2019, 33(2), 1950002. doi: 10.1142/S0217984919500027 [25] F. Yu, Localized analytical solutions and numerically stabilities of generalized Gross-Pitaevskii (GP(p, q)) equation with specific external potentials, Appl. Math. Lett., 2018, 85, 1-7. doi: 10.1016/j.aml.2018.05.003 [26] F. Yu and L. Li, Inverse scattering transformation and soliton stability for a nonlinear Gross-Pitaevskii equation with external potentials, Appl. Math. Lett., 2019, 91, 41-47. doi: 10.1016/j.aml.2018.11.026 [27] F. Yu, L. Feng and L. Li, Darboux transformations for super-Schrödinger equation, super-Dirac equation and their exact solutions, Nonlinear Dynam., 2017, 88(2), 1257-1271. doi: 10.1007/s11071-016-3308-x [28] X. Yong, G. Wang, W. Li, Y. Huang and J. Gao, On the Darboux transformation of a generalized inhomogeneous higher-order nonlinear Schrödinger equation, Nonlinear Dynam., 2017, 87(1), 75-82. doi: 10.1007/s11071-016-3026-4 [29] H. Zhang, J. Li, T. Xu, Y. Zhang, W. Hu and B. Tian, Optical soliton solutions for two coupled nonlinear Schrödinger systems via Darboux transformation, Phys. Scr., 2007, 76, 452-460. doi: 10.1088/0031-8949/76/5/009 [30] S. Zhang, S. Lou and C. Qu, New variable separation approach: application to nonlinear diffusion equations, J. Phys. A: Math. Gen., 2003, 36, 12223-12242. doi: 10.1088/0305-4470/36/49/006 [31] Z. Zhao, Z. Dai and C. Wang, Extend three-wave method for the (2+1)-dimensional Ito equation, Appl. Math. Comput., 2010, 217(5), 2295-2300. [32] Z. Zhao, Z. Dai and G. Mu, The breather-type and periodic-type solutions for the (2+1)-dimensional breaking soliton equation, Comput. Math. Appl., 2011, 61(8), 2048-2052. doi: 10.1016/j.camwa.2010.08.065 -
-
Figures(4)
Export File
Citation
Zhou-Zheng Kang, Tie-Cheng Xia. American Institute of Mathematical Sciences[J]. Journal of Applied Analysis & Computation, 2020, 10(2): 729-739. doi: 10.11948/20190128
Format
Content
DownLoad:
-
Figure 1. Plots of solution (2.6) with the parameters being fixed at
$ a_{{3}} = 2, a_{{4}} = 2, c_{{1}} = \frac {1}{2}, c_{{3}} = 1, c_{{4}} = 1 $ . (a) Perspective view of the wave with$ y = t = 0 $ ; (b) Density plot corresponding to Figure 1(a); (c) The wave along the$ x $ -axis with$ y = z = 0 $ at different times -
Figure 2. Plots of solution (2.7) with the parameters being fixed at
$ a_{{1}} = 1, a_{{2}} = 1, a_{{3}} = 1, a_{{4}} = 1, k_{{1}} = 6, k_{{4}} = 1, l_{{1}} = 1, l_{{4}} = 2, c_{{1}} = -3, c_{{4}} = 2 $ . (a) Perspective view of the wave with$ y = t = 0 $ ; (b) Density plot corresponding to Figure 2(a) -
Figure 3. Plots of solution (2.8) with the parameters being fixed at
$ a_{{1}} = \frac {4}{5}, a_{{3}} = \frac {1}{2}, a_{{4}} = \frac {1}{2}, k_{{4}} = -\frac{1}{3}, c_{{1}} = \frac {1}{3}, c_{{4}} = \frac{2}{3} $ . (a) Perspective view of the wave with$ y = t = 0 $ ; (b) Contour plot corresponding to Figure 3(a); (c) The wave along the$ x $ -axis with$ y = z = 0 $ at different times -
Figure 4. Plots of solution (2.10) with the parameters being fixed at
$ a_{{1}} = 1, a_{{2}} = 2, a_{{3}} = 1, c_{{1}} = -1, c_{{3}} = -2, c_{{4}} = -1, k_{{3}} = 1, k_{{4}} = -1, x = 0 $ . (a) Perspective view of the wave at$ t = -2 $ ; (b) Perspective view of the wave at$ t = 0 $ ; (c) Perspective view of the wave at$ t = 2 $ ; (d) Density plot corresponding to Figure 4(a); (e) Density plot corresponding to Figure 4(b); (f) Density plot corresponding to Figure 4(c).