# American Institute of Mathematical Sciences

October  2020, 13(10): 2941-2948. doi: 10.3934/dcdss.2020167

## A study on lump solutions to a (2+1)-dimensional completely generalized Hirota-Satsuma-Ito equation

 1 School of Mathematics, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China 2 Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China 3 Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia 4 Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA 5 School of Mathematics, South China University of Technology, Guangzhou 510640, China 6 College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China 7 International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa 8 School of Mathematics and Physical Science, Xuzhou Institute of Technology, Xuzhou 221008, Jiangsu, China

* Corresponding author: Wen-Xiu Ma

Received  November 2018 Revised  May 2019 Published  October 2020 Early access  December 2019

We aim to generalize the (2+1)-dimensional Hirota-Satsuma-Ito (HSI) equation, passing the three-soliton test, to a new one which still has diverse solution structures. We add all second-order derivative terms to the HSI equation but demand the existence of lump solutions. Such lump solutions are formulated in terms of the coefficients, except two, in the resulting generalized HSI equation. As an illustrative example, a special completely generalized HSI equation is given, together with a lump solution, and three 3d-plots and contour plots of the lump solution are made to elucidate the characteristics of the presented lump solutions.

Citation: Yufeng Zhang, Wen-Xiu Ma, Jin-Yun Yang. A study on lump solutions to a (2+1)-dimensional completely generalized Hirota-Satsuma-Ito equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2941-2948. doi: 10.3934/dcdss.2020167
##### References:
 [1] S.-T. Chen and W. X. Ma, Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation, Front. Math. China, 13 (2018), 525-534.  doi: 10.1007/s11464-018-0694-z. [2] H. H. Dong, Y. Zhang and X. E. Zhang, The new integrable symplectic map and the symmetry of integrable nonlinear lattice equation, Commun. Nonlinear Sci. Numer. Simulat., 36 (2016), 354-365.  doi: 10.1016/j.cnsns.2015.12.015. [3] B. Dorizzi, B. Grammaticos, A. Ramani and P. Winternitz, Are all the equations of the Kadomtsev-Petviashvili hierarchy integrable?, J. Math. Phys., 27 (1986), 2848-2852.  doi: 10.1063/1.527260. [4] L.-N. Gao, Y.-Y. Zi, Y.-H. Yin, W. X. Ma and X. Lü, Bäcklund transformation, multiple wave solutions and lump solutions to a (3 + 1)-dimensional nonlinear evolution equation, Nonlinear Dynam., 89 (2017), 2233-2240.  doi: 10.1007/s11071-017-3581-3. [5] C. R. Gilson and 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. [6] Harun-Or-Roshid and M. Z. Ali, Lump solutions to a Jimbo-Miwa like equation, (2016), arXiv: 1611.04478. [7] J. Hietarinta, Introduction to the Hirota bilinear method, Integrability of Nonlinear Systems, Lecture Notes in Phys., Springer, Berlin, 495 (1997), 95-103.  doi: 10.1007/BFb0113694. [8] J. Hietarinta, A search for bilinear equations passing Hirota's three-soliton condition Ⅰ: KdV-type bilinear equations, J. Math. Phys., 28 (1987), 1732-1742.  doi: 10.1063/1.527815. [9] R. Hirota, The Direct Method in Soliton Theory, Cambridge Tracts in Mathematics, 155. Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543043. [10] K. Imai, Dromion and lump solutions of the Ishimori equation, Prog. Theor. Phys., 98 (1997), 1013-1023. [11] D. J. Kaup, The lump solutions and the Bäcklund transformation for the three-dimensional three-wave resonant interaction, J. Math. Phys., 22 (1981), 1176-1181.  doi: 10.1063/1.525042. [12] 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, 132 (2017), 465. doi: 10.1140/epjp/i2017-11747-6. [13] B. Konopelchenko and W. Strampp, The AKNS hierarchy as symmetry constraint of the KP hierarchy, Inverse Probl., 7 (1991), L17–L24. doi: 10.1088/0266-5611/7/2/002. [14] X. Lü, W. X. Ma, S.-T. Chen and C. M. Khalique, A note on rational solutions to a Hirota-Satsuma-like equation, Appl. Math. Lett., 58 (2016), 13-18.  doi: 10.1016/j.aml.2015.12.019. [15] X.-Y. Li, Q.-L. Zhao, Y.-X. Li and H.-H. Dong, Binary Bargmann symmetry constraint associated with 3$\times$3 discrete matrix spectral problem, J. Nonlinear Sci. Appl., 8 (2015), 496-506.  doi: 10.22436/jnsa.008.05.05. [16] X.-Y. Li and Q.-L. Zhao, A new integrable symplectic map by the binary nonlinearization to the super AKNS system, J. Geom. Phys., 121 (2017), 123-137.  doi: 10.1016/j.geomphys.2017.07.010. [17] J.-G. 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., 80 (2018), 71-78.  doi: 10.1016/j.aml.2018.01.010. [18] X. Lu, W. X. Ma, Y. Zhou and C. M. Khalique, Rational solutions to an extended Kadomtsev-Petviashvili like equation with symbolic computation, Comput. Math. Appl., 71 (2016), 1560-1567.  doi: 10.1016/j.camwa.2016.02.017. [19] W. X. Ma and Y. Zhou, Lump solutions to nonlinear partial differential equations via Hirota bilinear forms, J. Differential Equations, 264 (2018), 2633-2659.  doi: 10.1016/j.jde.2017.10.033. [20] W. X. Ma, Y. Zhou and R. Dougherty, Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations, Int. J. Mod. Phys. B, 30 (2016), 1640018, 16 pp. doi: 10.1142/S021797921640018X. [21] W. X. Ma, Lump solutions to the Kadomtsev-Petviashvili equation, Phys. Lett. A, 379 (2015), 1975-1978.  doi: 10.1016/j.physleta.2015.06.061. [22] W. X. Ma, Generalized bilinear differential equations, Stud. Nonlinear Sci., 2 (2011), 140-144. [23] W. X. Ma, Comment on the $3+1$ dimensional Kadomtsev-Petviashvili equations, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 2663-2666.  doi: 10.1016/j.cnsns.2010.10.003. [24] W. X. Ma, Riemann-Hilbert problems and $N$-soliton solutions for a coupled mKdV system, J. Geom. Phys., 132 (2018), 45-54.  doi: 10.1016/j.geomphys.2018.05.024. [25] W. X. Ma, Lump and interaction solutions of linear PDEs in $(3+1)$-dimensions, East Asian J. Appl. Math., 9 (2019), 185-194.  doi: 10.4208/eajam.100218.300318. [26] W. X. Ma, Lump-type solutions to the $(3+1)$-dimensional Jimbo-Miwa equation, Int. J. Nonlinear Sci. Numer. Simulat., 17 (2016), 355-359.  doi: 10.1515/ijnsns-2015-0050. [27] W. X. Ma, X. L. Yong and H.-Q. Zhang, 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. [28] W. X. Ma, Conservation laws by symmetries and adjoint symmetries, Discrete Contin. Dyn. Syst. Series-S, 11 (2018), 707-721.  doi: 10.3934/dcdss.2018044. [29] W. X. Ma, J. Li and C. M. Khalique, A study on lump solutions to a generalized Hirota-Satsuma-Ito equation in $(2+1)$-dimensions, Complexity, 2018 (2018), 9059858, 7 pp. doi: 10.1155/2018/9059858. [30] S. V. Manakov, V. E. Zakharov, L. A. Bordag and V. B. Matveev, Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction, Phys. Lett. A, 63 (1977), 205-206.  doi: 10.1016/0375-9601(77)90875-1. [31] S. Manukure, Y. Zhou and W. X. Ma, Lump solutions to a $(2+1)$-dimensional extended KP equation, Comput. Math. Appl., 75 (2018), 2414-2419.  doi: 10.1016/j.camwa.2017.12.030. [32] B. Ren, W. X. Ma and 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. [33] B. Ren, W. X. Ma and 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. [34] J. Satsuma and M. J. Ablowitz, Two-dimensional lumps in nonlinear dispersive systems, J. Math. Phys., 20 (1979), 1496-1503.  doi: 10.1063/1.524208. [35] Y. Sun, B. Tian, X.-Y. Xie, J. Chai and H. M. Yin, Rogue waves and lump solitons for a $(3+1)$-dimensional B-type Kadomtsev-Petviashvili equation in fluid dynamics, Wave Random Complex Media, 28 (2018), 544-552.  doi: 10.1080/17455030.2017.1367866. [36] W. Tan, H. P. Dai, Z. D. Dai and W. Y. Zhong, Emergence and space-time structure of lump solution to the $(2+1)$-dimensional generalized KP equation, Pramana-J. Phys., 89 (2017), 77. [37] Y. N. Tang, S. Q. Tao and Q. Guan, Lump solitons and the interaction phenomena of them for two classes of nonlinear evolution equations, Comput. Math. Appl., 72 (2016), 2334-2342.  doi: 10.1016/j.camwa.2016.08.027. [38] D.-S. Wang and Y. B. Yin, Symmetry analysis and reductions of the two-dimensional generalized Benney system via geometric approach, Comput. Math. Appl., 71 (2016), 748-757.  doi: 10.1016/j.camwa.2015.12.035. [39] H. Wang, Lump and interaction solutions to the $(2+1)$-dimensional Burgers equation, Appl. Math. Lett., 85 (2018), 27-34.  doi: 10.1016/j.aml.2018.05.010. [40] J.-P. Wu and X.-G. Geng, Novel Wronskian condition and new exact solutions to a $(3+1)$-dimensional generalized KP equation, Commun. Theoret. Phys., 60 (2013), 556-510.  doi: 10.1088/0253-6102/60/5/08. [41] P. X. Wu, Y. F. Zhang, I. Muhammad and Q. Q. Yin, Lump, periodic lump and interaction lump stripe solutions to the $(2+1)$-dimensional B-type Kadomtsev-Petviashvili equation, Modern Phys. Lett. B, 32 (2018), 1850106, 12 pp. doi: 10.1142/S0217984918501063. [42] J.-Y. Yang and W. X. Ma, Lump solutions of the BKP equation by symbolic computation, Int. J. Mod. Phys. B, 30 (2016), 1640028, 7 pp. doi: 10.1142/S0217979216400282. [43] J.-Y. Yang and W. X. Ma, Abundant interaction solutions of the KP equation, Nonlinear Dynam., 89 (2017), 1539-1544.  doi: 10.1007/s11071-017-3533-y. [44] J.-Y. Yang, W. X. Ma and Z. Y. Qin, Lump and lump-soliton solutions to the $(2+1)$-dimensional Ito equation, Anal. Math. Phys., 8 (2018), 427-436.  doi: 10.1007/s13324-017-0181-9. [45] J.Y. Yang, W. X. Ma and Z. Y. Qin, Abundant mixed lump-soliton solutions to the BKP equation, East Asian J. Appl. Math., 8 (2018), 224-232.  doi: 10.4208/eajam.210917.051217a. [46] Y.-H. Yin, W. X. Ma, J.-G. Liu and X. Lü, Diversity of exact solutions to a $(3+1)$-dimensional nonlinear evolution equation and its reduction, Comput. Math. Appl., 76 (2018), 1275-1283.  doi: 10.1016/j.camwa.2018.06.020. [47] X. L. Yong, W. X. Ma, Y. H. Huang and Y. Liu, Lump solutions to the Kadomtsev-Petviashvili I equation with a self-consistent source, Comput. Math. Appl., 75 (2018), 3414-3419.  doi: 10.1016/j.camwa.2018.02.007. [48] J.-P. Yu and Y.-L. Sun, Study of lump solutions to dimensionally reduced generalized KP equations, Nonlinear Dynam., 87 (2017), 2755-2763.  doi: 10.1007/s11071-016-3225-z. [49] X. E. Zhang, Y. Chen and Y. Zhang, Breather, lump and $X$ soliton solutions to nonlocal KP equation, Comput. Math. Appl., 74 (2017), 2341-2347.  doi: 10.1016/j.camwa.2017.07.004. [50] Y. Zhang, H. H. Dong, X. E. Zhang and H. W. Yang, Rational solutions and lump solutions to the generalized $(3+1)$-dimensional Shallow Water-like equation, Comput. Math. Appl., 73 (2017), 246-252.  doi: 10.1016/j.camwa.2016.11.009. [51] Y. Zhang, S. L. Sun and H. H. Dong, Hybrid solutions of $(3+1)$-dimensional Jimbo-Miwa equation, Math. Probl. Eng., 2017 (2017), 5453941, 15 pp. doi: 10.1155/2017/5453941. [52] Y. Zhang, Y. P. Liu and 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. [53] J.-B. Zhang and 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. [54] Q.-L. Zhao and X.-Y. Li, A Bargmann system and the involutive solutions associated with a new 4-order lattice hierarchy, Anal. Math. Phys., 6 (2016), 237-254.  doi: 10.1007/s13324-015-0116-2. [55] H.-Q. Zhao and W. X. Ma, Mixed lump-kink solutions to the KP equation, Comput. Math. Appl., 74 (2017), 1399-1405.  doi: 10.1016/j.camwa.2017.06.034.

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##### References:
 [1] S.-T. Chen and W. X. Ma, Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation, Front. Math. China, 13 (2018), 525-534.  doi: 10.1007/s11464-018-0694-z. [2] H. H. Dong, Y. Zhang and X. E. Zhang, The new integrable symplectic map and the symmetry of integrable nonlinear lattice equation, Commun. Nonlinear Sci. Numer. Simulat., 36 (2016), 354-365.  doi: 10.1016/j.cnsns.2015.12.015. [3] B. Dorizzi, B. Grammaticos, A. Ramani and P. Winternitz, Are all the equations of the Kadomtsev-Petviashvili hierarchy integrable?, J. Math. Phys., 27 (1986), 2848-2852.  doi: 10.1063/1.527260. [4] L.-N. Gao, Y.-Y. Zi, Y.-H. Yin, W. X. Ma and X. Lü, Bäcklund transformation, multiple wave solutions and lump solutions to a (3 + 1)-dimensional nonlinear evolution equation, Nonlinear Dynam., 89 (2017), 2233-2240.  doi: 10.1007/s11071-017-3581-3. [5] C. R. Gilson and 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. [6] Harun-Or-Roshid and M. Z. Ali, Lump solutions to a Jimbo-Miwa like equation, (2016), arXiv: 1611.04478. [7] J. Hietarinta, Introduction to the Hirota bilinear method, Integrability of Nonlinear Systems, Lecture Notes in Phys., Springer, Berlin, 495 (1997), 95-103.  doi: 10.1007/BFb0113694. [8] J. Hietarinta, A search for bilinear equations passing Hirota's three-soliton condition Ⅰ: KdV-type bilinear equations, J. Math. Phys., 28 (1987), 1732-1742.  doi: 10.1063/1.527815. [9] R. Hirota, The Direct Method in Soliton Theory, Cambridge Tracts in Mathematics, 155. Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543043. [10] K. Imai, Dromion and lump solutions of the Ishimori equation, Prog. Theor. Phys., 98 (1997), 1013-1023. [11] D. J. Kaup, The lump solutions and the Bäcklund transformation for the three-dimensional three-wave resonant interaction, J. Math. Phys., 22 (1981), 1176-1181.  doi: 10.1063/1.525042. [12] 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, 132 (2017), 465. doi: 10.1140/epjp/i2017-11747-6. [13] B. Konopelchenko and W. Strampp, The AKNS hierarchy as symmetry constraint of the KP hierarchy, Inverse Probl., 7 (1991), L17–L24. doi: 10.1088/0266-5611/7/2/002. [14] X. Lü, W. X. Ma, S.-T. Chen and C. M. Khalique, A note on rational solutions to a Hirota-Satsuma-like equation, Appl. Math. Lett., 58 (2016), 13-18.  doi: 10.1016/j.aml.2015.12.019. [15] X.-Y. Li, Q.-L. Zhao, Y.-X. Li and H.-H. Dong, Binary Bargmann symmetry constraint associated with 3$\times$3 discrete matrix spectral problem, J. Nonlinear Sci. Appl., 8 (2015), 496-506.  doi: 10.22436/jnsa.008.05.05. [16] X.-Y. Li and Q.-L. Zhao, A new integrable symplectic map by the binary nonlinearization to the super AKNS system, J. Geom. Phys., 121 (2017), 123-137.  doi: 10.1016/j.geomphys.2017.07.010. [17] J.-G. 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., 80 (2018), 71-78.  doi: 10.1016/j.aml.2018.01.010. [18] X. Lu, W. X. Ma, Y. Zhou and C. M. Khalique, Rational solutions to an extended Kadomtsev-Petviashvili like equation with symbolic computation, Comput. Math. Appl., 71 (2016), 1560-1567.  doi: 10.1016/j.camwa.2016.02.017. [19] W. X. Ma and Y. Zhou, Lump solutions to nonlinear partial differential equations via Hirota bilinear forms, J. Differential Equations, 264 (2018), 2633-2659.  doi: 10.1016/j.jde.2017.10.033. [20] W. X. Ma, Y. Zhou and R. Dougherty, Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations, Int. J. Mod. Phys. B, 30 (2016), 1640018, 16 pp. doi: 10.1142/S021797921640018X. [21] W. X. Ma, Lump solutions to the Kadomtsev-Petviashvili equation, Phys. Lett. A, 379 (2015), 1975-1978.  doi: 10.1016/j.physleta.2015.06.061. [22] W. X. Ma, Generalized bilinear differential equations, Stud. Nonlinear Sci., 2 (2011), 140-144. [23] W. X. Ma, Comment on the $3+1$ dimensional Kadomtsev-Petviashvili equations, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 2663-2666.  doi: 10.1016/j.cnsns.2010.10.003. [24] W. X. Ma, Riemann-Hilbert problems and $N$-soliton solutions for a coupled mKdV system, J. Geom. Phys., 132 (2018), 45-54.  doi: 10.1016/j.geomphys.2018.05.024. [25] W. X. Ma, Lump and interaction solutions of linear PDEs in $(3+1)$-dimensions, East Asian J. Appl. Math., 9 (2019), 185-194.  doi: 10.4208/eajam.100218.300318. [26] W. X. Ma, Lump-type solutions to the $(3+1)$-dimensional Jimbo-Miwa equation, Int. J. Nonlinear Sci. Numer. Simulat., 17 (2016), 355-359.  doi: 10.1515/ijnsns-2015-0050. [27] W. X. Ma, X. L. Yong and H.-Q. Zhang, 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. [28] W. X. Ma, Conservation laws by symmetries and adjoint symmetries, Discrete Contin. Dyn. Syst. Series-S, 11 (2018), 707-721.  doi: 10.3934/dcdss.2018044. [29] W. X. Ma, J. Li and C. M. Khalique, A study on lump solutions to a generalized Hirota-Satsuma-Ito equation in $(2+1)$-dimensions, Complexity, 2018 (2018), 9059858, 7 pp. doi: 10.1155/2018/9059858. [30] S. V. Manakov, V. E. Zakharov, L. A. Bordag and V. B. Matveev, Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction, Phys. Lett. A, 63 (1977), 205-206.  doi: 10.1016/0375-9601(77)90875-1. [31] S. Manukure, Y. Zhou and W. X. Ma, Lump solutions to a $(2+1)$-dimensional extended KP equation, Comput. Math. Appl., 75 (2018), 2414-2419.  doi: 10.1016/j.camwa.2017.12.030. [32] B. Ren, W. X. Ma and 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. [33] B. Ren, W. X. Ma and 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. [34] J. Satsuma and M. J. Ablowitz, Two-dimensional lumps in nonlinear dispersive systems, J. Math. Phys., 20 (1979), 1496-1503.  doi: 10.1063/1.524208. [35] Y. Sun, B. Tian, X.-Y. Xie, J. Chai and H. M. Yin, Rogue waves and lump solitons for a $(3+1)$-dimensional B-type Kadomtsev-Petviashvili equation in fluid dynamics, Wave Random Complex Media, 28 (2018), 544-552.  doi: 10.1080/17455030.2017.1367866. [36] W. Tan, H. P. Dai, Z. D. Dai and W. Y. Zhong, Emergence and space-time structure of lump solution to the $(2+1)$-dimensional generalized KP equation, Pramana-J. Phys., 89 (2017), 77. [37] Y. N. Tang, S. Q. Tao and Q. Guan, Lump solitons and the interaction phenomena of them for two classes of nonlinear evolution equations, Comput. Math. Appl., 72 (2016), 2334-2342.  doi: 10.1016/j.camwa.2016.08.027. [38] D.-S. Wang and Y. B. Yin, Symmetry analysis and reductions of the two-dimensional generalized Benney system via geometric approach, Comput. Math. Appl., 71 (2016), 748-757.  doi: 10.1016/j.camwa.2015.12.035. [39] H. Wang, Lump and interaction solutions to the $(2+1)$-dimensional Burgers equation, Appl. Math. Lett., 85 (2018), 27-34.  doi: 10.1016/j.aml.2018.05.010. [40] J.-P. Wu and X.-G. Geng, Novel Wronskian condition and new exact solutions to a $(3+1)$-dimensional generalized KP equation, Commun. Theoret. Phys., 60 (2013), 556-510.  doi: 10.1088/0253-6102/60/5/08. [41] P. X. Wu, Y. F. Zhang, I. Muhammad and Q. Q. Yin, Lump, periodic lump and interaction lump stripe solutions to the $(2+1)$-dimensional B-type Kadomtsev-Petviashvili equation, Modern Phys. Lett. B, 32 (2018), 1850106, 12 pp. doi: 10.1142/S0217984918501063. [42] J.-Y. Yang and W. X. Ma, Lump solutions of the BKP equation by symbolic computation, Int. J. Mod. Phys. B, 30 (2016), 1640028, 7 pp. doi: 10.1142/S0217979216400282. [43] J.-Y. Yang and W. X. Ma, Abundant interaction solutions of the KP equation, Nonlinear Dynam., 89 (2017), 1539-1544.  doi: 10.1007/s11071-017-3533-y. [44] J.-Y. Yang, W. X. Ma and Z. Y. Qin, Lump and lump-soliton solutions to the $(2+1)$-dimensional Ito equation, Anal. Math. Phys., 8 (2018), 427-436.  doi: 10.1007/s13324-017-0181-9. [45] J.Y. Yang, W. X. Ma and Z. Y. Qin, Abundant mixed lump-soliton solutions to the BKP equation, East Asian J. Appl. Math., 8 (2018), 224-232.  doi: 10.4208/eajam.210917.051217a. [46] Y.-H. Yin, W. X. Ma, J.-G. Liu and X. Lü, Diversity of exact solutions to a $(3+1)$-dimensional nonlinear evolution equation and its reduction, Comput. Math. Appl., 76 (2018), 1275-1283.  doi: 10.1016/j.camwa.2018.06.020. [47] X. L. Yong, W. X. Ma, Y. H. Huang and Y. Liu, Lump solutions to the Kadomtsev-Petviashvili I equation with a self-consistent source, Comput. Math. Appl., 75 (2018), 3414-3419.  doi: 10.1016/j.camwa.2018.02.007. [48] J.-P. Yu and Y.-L. Sun, Study of lump solutions to dimensionally reduced generalized KP equations, Nonlinear Dynam., 87 (2017), 2755-2763.  doi: 10.1007/s11071-016-3225-z. [49] X. E. Zhang, Y. Chen and Y. Zhang, Breather, lump and $X$ soliton solutions to nonlocal KP equation, Comput. Math. Appl., 74 (2017), 2341-2347.  doi: 10.1016/j.camwa.2017.07.004. [50] Y. Zhang, H. H. Dong, X. E. Zhang and H. W. Yang, Rational solutions and lump solutions to the generalized $(3+1)$-dimensional Shallow Water-like equation, Comput. Math. Appl., 73 (2017), 246-252.  doi: 10.1016/j.camwa.2016.11.009. [51] Y. Zhang, S. L. Sun and H. H. Dong, Hybrid solutions of $(3+1)$-dimensional Jimbo-Miwa equation, Math. Probl. Eng., 2017 (2017), 5453941, 15 pp. doi: 10.1155/2017/5453941. [52] Y. Zhang, Y. P. Liu and X. Y. Tang, $M$-lump solutions to a $(3+1)$-dimensional nonlinear evolution equation, Comput. Math. 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Profiles of $u$ when $x = 0, 25, 50$: 3d plots (top) and contour plots (bottom)
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