| Citation: |
Bo Ren, Ji Lin, Zhi-Mei Lou. LUMPS AND THEIR INTERACTION SOLUTIONS OF A (2+1)-DIMENSIONAL GENERALIZED POTENTIAL KADOMTSEV-PETVIASHVILI EQUATION[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 935-945. doi: 10.11948/20190162
|
LUMPS AND THEIR INTERACTION SOLUTIONS OF A (2+1)-DIMENSIONAL GENERALIZED POTENTIAL KADOMTSEV-PETVIASHVILI EQUATION
-
Abstract
A (2+1)-dimensional generalized potential Kadomtsev-Petviashvili (gpKP) equation which possesses a Hirota bilinear form is constructed. The lump waves are derived by using a positive quadratic function solution. By combining an exponential function with a quadratic function, an interaction solution between a lump and a one-kink soliton is obtained. Furthermore, an interaction solution between a lump and a two-kink soliton is presented by mixing two exponential functions with a quadratic function. This type of lump wave just appears to a line $k_2x+k_3y+k_4t+k_5 \sim 0$. We call this kind of lump wave is a special rogue wave. Some visual figures are depicted to explain the propagation phenomena of these interaction solutions. -
-
References
[1] M. Chen, X. Li, Y. Wang and B. Li, A pair of resonance stripe solitons and lump solutions to a reduced (3+1)-dimensional nonlinear evolution equation, Commun. Theor. Phys., 2017, 67(6), 595-600. doi: 10.1088/0253-6102/67/6/595 [2] C. Dai, G. Zhou, R. Chen, X. Lai and J. Zheng, Vector multipole and vortex solitons in two-dimensional Kerr media, Nonlinear Dyn., 2017, 88(4), 2629-2635. [3] Z. Dai, J. Liu and Z. Liu, Exact periodic kink-wave and degenerative soliton solutions for potential Kadomtsev-Petviashvili equation, Commun. Nonlinear Sci. Numer. Simul., 2010, 15(9), 2331-2336. doi: 10.1016/j.cnsns.2009.09.037 [4] C. Dai, J. Liu, Y. Fan and D. Yu, Two-dimensional localized Peregrine solution and breather excited in a variable-coefficient nonlinear Schrodinger equation with partial nonlocality, Nonlinear Dyn., 2017, 88(2), 1373-1383. [5] Y. Dang, H. Li and J. Lin, Soliton solutions in nonlocal nonlinear coupler, Nonlinear Dyn. 2017, 88(1), 489-501. [6] P. G. Estévez, J. Prada and J. Villarroel, On an algorithmic construction of lump solutions in a 2+1 integrable equation, J. Phys. A Math. Theor., 2007, 40(26), 7213-7231. doi: 10.1088/1751-8113/40/26/008 [7] P. G. Estévez, E. Díaz, F. D. Adame, J. M. Cerveró and E. Diez, Lump solitons in a higher-order nonlinear equation in 2 + 1 dimensions, Phys. Rev. E, 2016. DOI: 10.1103/PhysRevE.93.062219. [8] É. Falcon, C. Laroche and S. Fauve, Observation of depression solitary surface waves on a thin fluid layer, Phys. Rev. Lett., 2002, 89, 204501. [9] X. Gao, S. Lou and X. Tang, Bosonization, singularity analysis, nonlocal symmetry reductions and exact solutions of supersymmetric KdV equation, J. High Ener. Phys., 2013, 5, 2013:29. [10] C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the korteweg-devries equation, Phys. Rev. Lett., 1967, 19(19), 1095-1097. doi: 10.1103/PhysRevLett.19.1095 [11] L. Huang and Y. Yue and Y. Chen, Localized waves and interaction solutions to a (3+1)-dimensional generalized KP equation, Comput. Math. Appl., 2018, 76(4), 831-844. [12] K. Imai, Dromion and lump solutions of the Ishimori-I equation, Prog. Theor. Phys., 1997, 98(5), 1013-1023. doi: 10.1143/PTP.98.1013 [13] C. Kharif, E. Pelinovsky and A. Slunyaev, Rogue waves in the ocean, Springer-Verlag, Berlin, 2009. [14] M. Kumar and A. K. Tiwari, Some group-invariant solutions of potential Kadomtsev-Petviashvili equation by using Lie symmetry approach, Nonlinear Dyn. 2018, 92(2), 781-792. [15] T. C. Kofane, M. Fokou, A. Mohamadou and E. Yomba, Lump solutions and interaction phenomenon to the third-order nonlinear evolution equation, Eur. Phys. J. Plus, 2017, 132(11), 465. doi: 10.1140/epjp/i2017-11747-6 [16] X. Li, Y. Wang, M. Chen and B. Li, Lump solutions and resonance stripe solitons to the (2+1)-dimensional Sawada-Kotera equation, Adv. Math. Phys., 2017, (2017) 1743789. [17] S. Lou and J. Lin, Rogue Waves in Nonintegrable KdV-Type Systems, Chin. Phys. Lett. 2018, 35, 13-16. [18] S. Manukure, Y. Zhou and W. Ma, Lump solutions to a (2+1)-dimensional extended KP equation, Comput. Math. Appl., 2018, 75(7), 2414-2419. doi: 10.1016/j.camwa.2017.12.030 [19] W. Ma, Lump solutions to the Kadomtsev-Petviashvili equation, Phys. Lett. A, 2015, 379(36), 1975-1978. doi: 10.1016/j.physleta.2015.06.061 [20] W. Ma, X. Yong and H. Zhang, Diversity of interaction solutions to the (2+1)-dimensional Ito equation, Comput. Math. Appl., 2018, 75(1), 289-295. [21] W. Ma, Z. Qin and X. Lü, Lump solutions to dimensionally reduced p-gKP and p-gBKP equations, Nonlinear Dyn. 2016, 84(2), 923-931. [22] W. Ma and Y. Zhou, Lump solutions to nonlinear partial differential equations via Hirota bilinear forms, J. Differ. Equations, 2018, 264(4), 2633-2659. doi: 10.1016/j.jde.2017.10.033 [23] W. Ma, Generalized bilinear differential equations, Stud. Nonlinear Sci., 2011, 2, 140-144. [24] D. E. Pelinovsky, Y. A. Stepanyants and Y.S. Kivshar, Self-focusing of plane dark solitons in nonlinear defocusing media, Phys. Rev. E, 1995, 51(5B), 5016-5026. [25] W. Peng, S. Tian and T. Zhang, Analysis on lump, lumpoff and rogue waves with predictability to the (2 +1)-dimensional B-type Kadomtsev-Petviashvili equation, Phys. Lett. A, 2018, 382(38), 2701-2708. doi: 10.1016/j.physleta.2018.08.002 [26] B. Ren, Interaction solutions for mKP equation with nonlocal symmetry reductions and CTE method, Phys. Scr., 2015, 90(6), 065206. doi: 10.1088/0031-8949/90/6/065206 [27] B. Ren, X. Cheng and J. Lin, The (2+1)-dimensional Konopelchenko-Dubrovsky equation: nonlocal symmetries and interaction solutions, Nonlinear Dyn., 2016, 86(3), 1855-1862. [28] B. Ren, Symmetry reduction related with nonlocal symmetry for Gardner equation, Commun. Nonli. Sci. Numer. Simulat., 2017, 42, 456-463. doi: 10.1016/j.cnsns.2016.06.017 [29] B. Ren, Dynamics behavior of lumps and interaction solutions of a (3+1)-dimensional partial differential equation, Complexity, 2019, 2019, 9512531. [30] B. Ren, W. Ma and J. Yu, Rational solutions and their interaction solutions of the (2+1)-dimensional modified dispersive water wave equation, Comput. Math. Appl., 2019, 77(8), 2086-2095. doi: 10.1016/j.camwa.2018.12.010 [31] B. Ren, W. Ma and J. Yu, Characteristics and interactions of solitary and lump waves of a (2+1)-dimensional coupled nonlinear partial differential equation, Nonlinear Dyn., 2019, 96(1), 717-727. [32] B. Ren, J. Yu and X. Liu, Nonlocal symmetries and interaction solutions for potential Kadomtsev-Petviashvili equation, Commun. Theor. Phys., 2016, 65(3), 341-346. doi: 10.1088/0253-6102/65/3/341 [33] L. Stenflo and M. Marklund, Rogue waves in the atmosphere, J. Plasma Phys., 2010, 76(3-4), 293-295. doi: 10.1017/S0022377809990481 [34] J. Satsuma and M. J. Ablowitz, Two-dimensional lumps in nonlinear dispersive systems, J. Math. Phys., 1979, 20(7), 1496-1503. doi: 10.1063/1.524208 [35] X. Tang, S. Lou and Y. Zhang, Localized exicitations in (2+1)-dimensional systems, Phys. Rev. E, 2002, 66, 046601. doi: 10.1103/PhysRevE.66.046601 [36] X. Tang, S. Liu, Z. Liang and J. Wang, A general nonlocal variable coefficient KdV equation with shifted parity and delayed time reversal, Nonlinear Dyn., 2018, 94(1), 693-702. [37] D. Wang, Y. Shi, W. Feng and L. Wen, Dynamical and energetic instabilities of F=2 spinor Bose-Einstein condensates in an optical lattice, Physica D 2017, 351-352, 30-41. doi: 10.1016/j.physd.2017.04.002 [38] D. Wang, B. Guo and X. Wang, Long-time asymptotics of the focusing Kundu-Eckhaus equation with nonzero boundary conditions, J. Differ. Equations, 2019, 266(9), 5209-5253. doi: 10.1016/j.jde.2018.10.053 [39] J. Wang, H. An and B. Li, Non-traveling lump solutions and mixed lump-kink solutions to (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation, Mod. Phys. Lett. B, 2019, 33(22), 1950262. doi: 10.1142/S0217984919502622 [40] J. Weiss, M. Tabor and G. Carnevale, The Painlevé property for partial differential equations, J. Math. Phys., 1983, 24, 522. doi: 10.1063/1.525721 [41] J. Yu and Y. Sun, Lump solutions to dimensionally reduced Kadomtsev-Petviashvili-like equations, Nonlinear Dyn., 2017, 87(2), 1405-1412. [42] J. Yu and Y. Sun, A direct Backlund transformation for a (3+1)-dimensional Kadomtsev-Petviashvili-Boussinesq-like equation, Nonlinear Dyn., 2017, 90(4), 2263-2268. doi: 10.1007/s11071-017-3799-0 [43] J. Yang, W. Ma and Z. Qin, Lump and lump-soliton solutions to the (2+1)-dimensional Ito equation, Anal. Math. Phys., 2018, 8, 427-436. doi: 10.1007/s13324-017-0181-9 [44] H. Zhang and W. Ma, Lump solutions to the (2+1)-dimensional Sawada-Kotera equation, Nonlinear Dyn., 2017, 87(4), 2305-2310. [45] X. Zhang and Y. Chen, Rogue wave and a pair of resonance stripe solitons to a reduced (3+1)-dimensional Jimbo-Miwa equation, Commun. Nonlinear Sci. Numer. Simul., 2017, 52, 24-31. doi: 10.1016/j.cnsns.2017.03.021 -
-
Figures(3)
Export File
Citation
Bo Ren, Ji Lin, Zhi-Mei Lou. LUMPS AND THEIR INTERACTION SOLUTIONS OF A (2+1)-DIMENSIONAL GENERALIZED POTENTIAL KADOMTSEV-PETVIASHVILI EQUATION[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 935-945. doi: 10.11948/20190162
Format
Content
DownLoad:
-
Figure 1. Profile of the solution (2.11). (a) 3-dimensional plot with the time
$ t = 0 $ , (b) the corresponding density plot, (c) the contour plot about the moving path described by the straight line (2.14), i.e.,$ y = \frac{2}{7} x + \frac{5}{14} $ . -
Figure 2. Profile of the solution (3.4). (a) 3-dimensional plot with the time
$ t = 0 $ , (b) the corresponding density plot, (c) the contour plot about the moving path described by the straight line (2.14), i.e.,$ y = x + \frac{2}{7} $ . -
Figure 3. Profile of the solution (3.9). (a) 3-dimensional plot with the time
$ t = 0 $ , (b) the corresponding density plot, (c) the contour plot about the moving path described by the straight line (2.14), i.e.,$ y = -\frac{6}{13}x - \frac{291}{169} - \frac{25\sqrt{3}}{169} $ .