### AIMS Mathematics

2021, Issue 5: 5370-5386. doi: 10.3934/math.2021316
Research article

# Variety interaction solutions comprising lump solitons for the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation

• Received: 05 January 2021 Accepted: 03 March 2021 Published: 12 March 2021
• MSC : 35Q51, 37K40

• This paper deals with localized waves in the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation in the incompressible fluid. Based on Hirota's bilinear method, N-soliton solutions related to CDGKS equation are constructed. Taking the special reduction, the exact expression of multiple localized wave solutions comprising lump soliton(s) are obtained by using the long wave limit method. A variety of interactions are illustrated analytically and graphically. The influence of parameters on propagation is analyzed and summarized. The results and phenomena obtained in this paper enrich the dynamic behavior of the evolution of nonlinear localized waves.

Citation: Jianhong Zhuang, Yaqing Liu, Ping Zhuang. Variety interaction solutions comprising lump solitons for the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation[J]. AIMS Mathematics, 2021, 6(5): 5370-5386. doi: 10.3934/math.2021316

### Related Papers:

• This paper deals with localized waves in the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation in the incompressible fluid. Based on Hirota's bilinear method, N-soliton solutions related to CDGKS equation are constructed. Taking the special reduction, the exact expression of multiple localized wave solutions comprising lump soliton(s) are obtained by using the long wave limit method. A variety of interactions are illustrated analytically and graphically. The influence of parameters on propagation is analyzed and summarized. The results and phenomena obtained in this paper enrich the dynamic behavior of the evolution of nonlinear localized waves.

 [1] C. S. Gardner, The Korteweg-de Vries equation and generalizations. VI. Method for exact solution, Comm. Pure Appl. Math., 27 (1974), 97-133. doi: 10.1002/cpa.3160270108 [2] C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura, Method for solving the KdV equation, Phys. Rev. Lett., 19 (1967), 1095-1097. doi: 10.1103/PhysRevLett.19.1095 [3] D. S. Wang, B. L. Guo, X. L. Wang, Long-time asymptotics of the focusing Kundu-Eckhaus equation with nonzero boundary conditions, J. Differ. Equ., 266 (2019), 5209-5253. doi: 10.1016/j.jde.2018.10.053 [4] C. Rogers, W. Shadwick, B$\ddot{a}$cklund transformations and applications, New York: Academic Press, 1982. [5] D. S. Wang, J. Liu, Integrability aspects of some two-component KdV systems, Appl. Math. Lett., 79 (2018), 211-219. doi: 10.1016/j.aml.2017.12.018 [6] H. C. Hu, Q. P. Liu, New Darboux transformation for Hirota-Sastuma coupled KdV system, Chaos Soliton Fract., 17 (2003), 921-928. doi: 10.1016/S0960-0779(02)00309-0 [7] L. Xu, D. S. Wang, X. Y. Wen, Y. L. Jiang, Exotic localised vector waves in a two-component nonlinear wave system, J. Nonlinear Sci., 30 (2020), 537-564. doi: 10.1007/s00332-019-09581-0 [8] D. S. Wang, Q. Li, X. Y. Wen, L. Liu, Matrix spectral problems and integrability aspects of the Blaszak-Marciniak lattice equations, Reports Math. Phys., 86 (2020), 325-353. doi: 10.1016/S0034-4877(20)30087-2 [9] R. Hirota, J. Satsuma, A simple structure of superpositon formula of Backlund transformation, J. Phys. Soc. Japan, 45 (1978), 1741-1750. doi: 10.1143/JPSJ.45.1741 [10] R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., 27 (1971), 1192-1194. doi: 10.1103/PhysRevLett.27.1192 [11] N. C. Freeman, J. J. C. Nimmo, Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petmiashvili equations: the Wronskian technique, Phys. Lett. A, 95 (1983), 1-3. doi: 10.1016/0375-9601(83)90764-8 [12] J. C. Chen, Y. Chen, B. F. Feng, K. I. Maruno, Breather to the Yajima-Oikawa equation, arXiv: 1712.00945. [13] M. J. Ablowitz, A. Ramini, H. Segur, A connection between nonlinear evolution equations and ordinary differential equations of P-type. II, J. Math. Phys., 21 (1980), 1006-1015. doi: 10.1063/1.524548 [14] J. Weiss, B$\ddot{a}$cklund transformation and the Painlev$\acute{\mathrm{e}}$ property, J. Math. Phys., 27 (1986), 1296-1305. [15] R. Dickson, F. Gesztesy, K. Unterkofler, Algebra-geometric solutions of the Boussinesq hierarchy, Rev. Math. Phys., 11 (1999), 823-879. doi: 10.1142/S0129055X9900026X [16] C. Q. Dai, Y. Y. Wang, Coupled spatial periodic waves and solitons in the photovoltaic photorefractive crystals, Nonlinear Dyn., 102 (2020), 1733-1741. doi: 10.1007/s11071-020-05985-w [17] C. Q. Dai, Y. Y. Wang, J. F. Zhang, Managements of scalar and vector rogue waves in a partially nonlocal nonlinear medium with linear and harmonic potentials, Nonlinear Dyn., 102 (2020), 379-391. doi: 10.1007/s11071-020-05949-0 [18] W. X. Ma, X. L. Yong, H. Q. Zhang, Diversity of interaction solutions to the (2+1)-dimensional Ito equation, Comput. Math. Appl., 75 (2018), 289-295. doi: 10.1016/j.camwa.2017.09.013 [19] B. Ren, W. X. Ma, J. Yu, Characteristics and interactions of solitary and lump waves of a (2+1)-dimensional coupled nonlinear partial differential equation, Nonlinear Dyn., 96 (2019), 717-727. doi: 10.1007/s11071-019-04816-x [20] B. Ren, W. X. Ma, J. Yu, Rational solutions and their interaction solutions of the (2+1)-dimensional modified dispersive water wave equation, Comput. Math. Appl., 77 (2019), 2086-2095. doi: 10.1016/j.camwa.2018.12.010 [21] M. S. Osman, A. M. Wazwaz, A general bilinear form to generate different wave structures of solitons for a (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation, Math. Method Appl. Sci., 42 (2019), 6277-6283. doi: 10.1002/mma.5721 [22] A. M. Wazwaz, S. A. El-Tantawy, Solving the (3+1)-dimensional KP-Boussinesq and BKP-Boussinesq equations by the simplified Hirota's method, Nonlinear Dyn., 88 (2017), 3017-3021. doi: 10.1007/s11071-017-3429-x [23] Y. Liu, B. Li, A. M. Wazwaz, Novel high-order breathers and rogue waves in the Boussinesq equation via determinants. Math. Meth. Appl. Sci., 43 (2020), 3701-3715. [24] J. G. Liu, W. H. Zhu, L. Zhou, Breather wave solutions for the Kadomtsev-Petviashvili equationwith variable coefficients in a fluid based on the variable-coefficient three-wave approach, Math. Meth. Appl. Sci., 43 (2020), 458-465. doi: 10.1002/mma.5899 [25] J. Rao, D. Mihalache, Y. Cheng, J. He, Lump-soliton solutions to the Fokas system, Phys. Lett. A, 383 (2019), 1138-1142. doi: 10.1016/j.physleta.2018.12.045 [26] Y. Q. Liu, X. Y. Wen, D. S. Wang, Novel interaction phenomena of localized waves in the generalized (3+1)-dimensional KP equation, Comput. Math. Appl., 78 (2019), 1-19. doi: 10.1016/j.camwa.2019.03.005 [27] Y. Q. Liu, X. Y. Wen, D. S. Wang, The N-soliton solution and localized wave interaction solutions of the (2+1)-dimensional generalized Hirota-Satsuma-Ito equation, Comput. Math. Appl., 77 (2019), 947-966. doi: 10.1016/j.camwa.2018.10.035 [28] J. J. Wu, Y. Q. Liu, L. H. Piao, J. H. Zhuang, D. S. Wang, Nonlinear localized waves resonance and interaction solutions of the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation, Nonlinear Dyn., 100 (2020), 1527-1541. doi: 10.1007/s11071-020-05573-y [29] Y. Zhou, M. Solomon, W. X. Ma, Lump and lump-soliton solutions to the Hirota-Satsuma-Ito equation, Commun. Nonlinear Sci. Numer. Simulat., 68 (2019), 56-62. doi: 10.1016/j.cnsns.2018.07.038 [30] S. T. Chen, W. X. Ma, Lump solutions of a generalized Calogero-Bogoyavlenskii-Schiff equation, Comput. Math. Appl., 76 (2018), 1680-1685. doi: 10.1016/j.camwa.2018.07.019 [31] J. B. Zhang, W. X. Ma, Mixed lump-kink solutions to the BKP equation, Comput. Math. Appl., 74 (2017), 591-596. doi: 10.1016/j.camwa.2017.05.010 [32] B. Konopelchenko, V. Dubrovsky, Some new integrable nonlinear evolution equations in 2+1 dimensions, Phys. Lett. A, 102 (1984), 15-17. doi: 10.1016/0375-9601(84)90442-0 [33] K. Sawada, T. Kotera, A Method for finding N-Soliton solutions of the K.d.V. equation and K.d.V.-Like equation, Prog. Theor. Phys., 51 (1974), 1355-1367. doi: 10.1143/PTP.51.1355 [34] P. J. Caudrey, R. K. Dodd, J. D. Gibbon, A new hierarchy of Korteweg-de Vries equations, P. Roy. Soc. A-Math. Phy., 351 (1976), 407-422. [35] R. K. Dodd, J. D. Gibbon, The prolongation structure of a higher order Korteweg-de Vries equation, P. Roy. Soc. A-Math. Phy., 358 (1978), 287-296. [36] S. Y. Lou, Abundant symmetries for the 1+1 dimensional classical Liouville field theory, J. Math. Phys., 35 (1994), 2336-2348. doi: 10.1063/1.530556 [37] E. Date, M. Jimbo, M. Kashiwara, T. Miwa, KP hierarchies of orthogonal and symplectic type -transformation groups for soliton equations VI, J. Phys. Soc. Jpn., 50 (1981), 3813-3818. doi: 10.1143/JPSJ.50.3813 [38] M. Jimbo, T. Miwa, Solitons and infinite dimensional Lie algebras, Publ. Res. I. Math. Sci., 19 (1983), 943-1001. doi: 10.2977/prims/1195182017 [39] X. B. Hu, S. H. Li, The partition function of the Bures ensemble as the $\tau$-function of BKP and DKP hierarchies: continuous and discrete, J. Phys. A-Math. Theor., 50 (2017), 285201. doi: 10.1088/1751-8121/aa7395 [40] X. G. Geng, Darboux transformation of two-dimensional Sawada-Kotera equation, Appl. Math. J. Chinese Univ., 4 (1989), 494-497. (in Chinese) [41] C. W. Cao, Y. T. Wu, X. G. Geng, On quasi-periodic solutions of the 2+1 dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation, Phys. Lett. A, 256 (1999), 59-65. doi: 10.1016/S0375-9601(99)00201-7 [42] L. Wang, D. Q. Xian, Homoclinic breather-wave solutions, periodic-wave solutions and kink solitary-wave solutions for CDGKS equation, Chin. J. Quantum Elect., 29 (2012), 417-420. (in Chinese) [43] X. R. Kang, D. Q. Xian, Z. D. Dai, Non-traveling wave solutions for the (2+1)-D Caudrey-Dodd-Gibbon-Kotera-Sawada equation, Int. J. Numer. Method. H, 25 (2015), 617-628. doi: 10.1108/HFF-03-2013-0086 [44] X. G. Geng, G. L. He, L. H. Wu, Riemann theta function solutions of the Caudrey-Dodd-Gibbon-Sawada-Kotera hierarchy, J. Geom. Phys., 140 (2019), 85-103. doi: 10.1016/j.geomphys.2019.01.005 [45] C. R. Gilson, J. J. C. Nimmo, Lump solutions of the BKP equation, Phys. Lett. A, 147 (1990), 472-476. doi: 10.1016/0375-9601(90)90609-R [46] J. Manafian, M. Lakestani, N-lump and interaction solutions of localized waves to the (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation, J. Gemo. Phys., 150 (2020), 103598. doi: 10.1016/j.geomphys.2020.103598 [47] X. P. Cheng, Y. Q. Yang, B. Ren, J. Y. Wang, Interaction behavior between solitons and (2+1)-dimensional CDGKS waves, Wave Motion, 86 (2019), 150-161. doi: 10.1016/j.wavemoti.2018.08.008 [48] W. Q. Peng, S. F. Tian, L. Zou, T. T. Zhang, Characteristics of the solitary waves and lump waves with interaction phenomena in a (2+1)-dimensional generalized Caudrey-Dodd-Gibbon-Kotera-Sawada equation, Nonlinear Dyn., 93 (2018), 1841-1851. doi: 10.1007/s11071-018-4292-0 [49] X. H. Meng, The periodic solitary wave solutions for the (2+1)-dimensional fifth-order KdV equation, J. Appl. Math. Phys., 2 (2014), 639-643. doi: 10.4236/jamp.2014.27070 [50] Z. H. Yang, A series of exact solutions of (2+1)-dimensional CDGKS equation, Commun. Theor. Phys., 46 (2006), 807-811. doi: 10.1088/0253-6102/46/5/008 [51] J. H. Zhuang, Y. Q. Liu, X. Chen, J. J. Wu, X. Y. Wen, Diverse solitons and interaction solutions for the (2+1)-dimensional CDGKS equation, Mod. Phys. Lett. B, 33 (2019), 1950174. [52] L. Liu, X. Y. Wen, D. S. Wang, A new lattice hierarchy: Hamiltonian structures, symplectic map and N-fold Darboux transformation, Appl. Math. Model., 67 (2019), 201-218. doi: 10.1016/j.apm.2018.10.030 [53] J. Satsuma, M. J. Ablowitz, Two-dimensional lumps in nonlinear dispersive systems, J. Math. Phys., 20 (1979), 1496-1503. doi: 10.1063/1.524208 [54] Y. Zhang, Y. P. Liu, X. Y. Tang, M-lump solutions to a (3+1)-dimensional nonlinear evolution equation, Comput. Math. Appl., 76 (2018), 592-601. doi: 10.1016/j.camwa.2018.04.039
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