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## 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  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 & Continuous Dynamical Systems - S, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] Harun-Or-Roshid and M. Z. Ali, Lump solutions to a Jimbo-Miwa like equation, (2016), arXiv: 1611.04478. Google Scholar [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.  Google Scholar [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.  Google Scholar [9] R. Hirota, The Direct Method in Soliton Theory, Cambridge Tracts in Mathematics, 155. Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543043.  Google Scholar [10] K. Imai, Dromion and lump solutions of the Ishimori equation, Prog. Theor. Phys., 98 (1997), 1013-1023.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] W. X. Ma, Generalized bilinear differential equations, Stud. Nonlinear Sci., 2 (2011), 140-144.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [43] J.-Y. Yang and W. X. 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show all references

##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] Harun-Or-Roshid and M. Z. Ali, Lump solutions to a Jimbo-Miwa like equation, (2016), arXiv: 1611.04478. Google Scholar [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.  Google Scholar [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.  Google Scholar [9] R. Hirota, The Direct Method in Soliton Theory, Cambridge Tracts in Mathematics, 155. Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543043.  Google Scholar [10] K. Imai, Dromion and lump solutions of the Ishimori equation, Prog. Theor. Phys., 98 (1997), 1013-1023.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] W. X. Ma, Generalized bilinear differential equations, Stud. Nonlinear Sci., 2 (2011), 140-144.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [38] D.-S. Wang and Y. B. Yin, Symmetry analysis and reductions of the two-dimensional generalized Benney system via geometric approach, 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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