Lump solutions to a generalized Hietarinta-type equation via symbolic computation

Abstract

Lump solutions are one of important solutions to partial differential equations, both linear and nonlinear. This paper aims to show that a Hietarinta-type fourth-order nonlinear term can create lump solutions with second-order linear dispersive terms. The key is a Hirota bilinear form. Lump solutions are constructed via symbolic computations with Maple, and specific reductions of the resulting lump solutions are made. Two illustrative examples of the generalized Hietarinta-type nonlinear equations and their lumps are presented, together with three-dimensional plots and density plots of the lump solutions.

Keywords

Soliton equation / lump solution / symbolic computation / Hirota derivative / dispersion relation

References

[1] Ablowitz M J, Segur H. Solitons and the Inverse Scattering Transform. Philadelphia: SIAM, 1981

[2] Batwa S, Ma W X. A study of lump-type and interaction solutions to a (3+ 1)-dimensional Jimbo-Miwa-like equation. Comput Math Appl, 2018, 76: 1576–1582

[3] Caudrey P J. Memories of Hirota’s method: application to the reduced Maxwell-Bloch system in the early 1970s. Philos Trans Roy Soc A, 2011, 369: 1215–1227

[4] Chen S J, Ma W X, Lü X. Bäcklund transformation, exact solutions and interaction behaviour of the (3+ 1)-dimensional Hirota-Satsuma-Ito-like equation. Commun Nonlinear Sci Numer Simul, 2020, 83: 105135

[5] Chen S J, Yin Y H, Ma W X, Lü X. Abundant exact solutions and interaction phenomena of the (2+ 1)-dimensional YTSF equation. Anal Math Phys, 2019, 9: 2329–2344

[6] Chen S T, Ma W X. Lumps solutions to a generalized Calogero-Bogoyavlenskii-Schiff equation. Comput Math Appl, 2018, 76: 1680–1685

[7] Chen S T, Ma W X. Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation. Front Math China, 2018, 13: 525–534

[8] Dong H H, Zhang Y, Zhang X E. The new integrable symplectic map and the symmetry of integrable nonlinear lattice equation. Commun Nonlinear Sci Numer Simul, 2016, 36: 354–365

[9] Dorizzi B, Grammaticos B, Ramani A, Winternitz P. Are all the equations of the Kadomtsev-Petviashvili hierarchy integrable? J Math Phys, 1986, 27: 2848–2852

[10] Gao L N, Zi Y Y, Yin Y H, Ma W X, Lü X. Bäcklund transformation, multiple wave solutions and lump solutions to a (3+ 1)-dimensional nonlinear evolution equation. Nonlinear Dynam, 2017, 89(3): 2233–2240

[11] Gilson C R, Nimmo J J C. Lump solutions of the BKP equation. Phys Lett A, 1990, 147: 472–476

[12] Harun-Or-Roshid, Ali M Z. Lump solutions to a Jimbo-Miwa like equation. arXiv: 1611.04478

[13] Hietarinta J. A search for bilinear equations passing Hirota’s three-soliton condition I-KdV-type bilinear equations. J Math Phys, 1987, 28: 1732–1742

[14] Hietarinta J. Introduction to the Hirota bilinear method. In: Kosmann-Schwarzbach Y, Grammaticos B, Tamizhmani K M, eds. Integrability of Nonlinear Systems. Berlin: Springer, 1997, 95–103

[15] Hirota R. The Direct Method in Soliton Theory. New York: Cambridge Univ Press, 2004

[16] Hua Y F, Guo B L, Ma W X, Lü X. Interaction behavior associated with a generalized (2+ 1)-dimensional Hirota bilinear equation for nonlinear waves. Appl Math Model, 2019, 74: 184–198

[17] Imai K. Dromion and lump solutions of the Ishimori-I equation. Prog Theor Phys, 1997, 98: 1013–1023

[18] Kaup D J. The lump solutions and the Bäcklund transformation for the three-dimensional three-wave resonant interaction. J Math Phys, 1981, 22: 1176–1181

[19] Kofane T C, Fokou M, Mohamadou A, Yomba E. Lump solutions and interaction phenomenon to the third-order nonlinear evolution equation. Eur Phys J Plus, 2017, 132: 465

[20] Konopelchenko B, Strampp W. The AKNS hierarchy as symmetry constraint of the KP hierarchy. Inverse Problems, 1991, 7: L17–L24

[21] Li X Y, Zhao Q L. A new integrable symplectic map by the binary nonlinearization to the super AKNS system. J Geom Phys, 2017, 121: 123–137

[22] Liu J G, Zhou L, He Y. 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

[23] Liu M S, Li X Y, Zhao Q L. Exact solutions to Euler equation and Navier-Stokes equation. Z Angew Math Phys, 2019, 70: 43

[24] Lü X, Chen S T, Ma W X. Constructing lump solutions to a generalized Kadomtsev-Petviashvili-Boussinesq equation. Nonlinear Dynam, 2016, 86: 523–534

[25] Lü X, Ma W X, Chen S T, Khalique C M. A note on rational solutions to a Hirota-Satsuma-like equation. Appl Math Lett, 2016, 58: 13–18

[26] Lü X, Ma W X, Zhou Y, Khalique C M. Rational solutions to an extended Kadomtsev-Petviashvili like equation with symbolic computation. Comput Math Appl, 2016, 71: 1560–1567

[27] Ma W X. Lump solutions to the Kadomtsev-Petviashvili equation. Phys Lett A, 2015, 379: 1975–1978

[28] Ma W X. Lump-type solutions to the (3+ 1)-dimensional Jimbo-Miwa equation. Int J Nonlinear Sci Numer Simul, 2016, 17: 355–359

[29] Ma W X. Riemann-Hilbert problems and N-soliton solutions for a coupled mKdV system. J Geom Phys, 2018, 132: 45–54

[30] Ma W X. Abundant lumps and their interaction solutions of (3+ 1)-dimensional linear PDEs. J Geom Phys, 2018, 133: 10–16

[31] Ma W X. A search for lump solutions to a combined fourth-order nonlinear PDE in (2+ 1)-dimensions. J Appl Anal Comput, 2019, 9: 1319–1332

[32] Ma W X. Interaction solutions to Hirota-Satsuma-Ito equation in (2+ 1)-dimensions. Front Math China, 2019, 14: 619–629

[33] Ma W X. Lump and interaction solutions of linear PDEs in (3+ 1)-dimensions. East Asian J Appl Math, 2019, 9: 185–194

[34] Ma W X. Lump and interaction solutions to linear PDEs in (2+ 1)-dimensions via symbolic computation. Modern Phys Lett B, 2019, 33: 1950457

[35] Ma W X, Geng X G. Bäcklund transformations of soliton systems from symmetry constraints. CRM Proc Lecture Notes, 2011, 29: 313–323

[36] Ma W X, Li J, Khalique C M. A study on lump solutions to a generalized Hirota-Satsuma-Ito equation in (2+ 1)-dimensions. Complexity, 2018, 2018: 9059858

[37] Ma W X, Qin Z Y, Lü X. Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dynam, 2016, 84: 923–931

[38] Ma W X, Strampp W. An explicit symmetry constraint for the Lax pairs and the adjoint Lax pairs of AKNS systems. Phys Lett A, 1994, 185: 277–286

[39] Ma W X, Yong X L, Zhang H Q. Diversity of interaction solutions to the (2+ 1)-dimensional Ito equation. Comput Math Appl, 2018, 75: 289–295

[40] Ma W X, You Y. Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions. Trans Amer Math Soc, 2005, 357: 1753–1778

[41] Ma W X, Zhang L Q. Lump solutions with higher-order rational dispersion relations. Pramana-J Phys, 2020, 94: 43

[42] Ma W X, Zhou Y. Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J Differential Equations, 2018, 264: 2633–2659

[43] Ma W X, Zhou Y, Dougherty R. Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations. Internat J Modern Phys B, 2016, 30: 1640018

[44] Manakov S V, Zakharov V E, Bordag L A, Matveev V B. Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction. Phys Lett A, 1977, 63: 205–206

[45] Manukure S, Zhou Y, Ma W X. Lump solutions to a (2+ 1)-dimensional extended KP equation. Comput Math Appl, 2018, 75: 2414–2419

[46] Novikov S, Manakov S V, Pitaevskii L P, Zakharov V E. Theory of Solitons—The Inverse Scattering Method. New York: Consultants Bureau, 1984

[47] Ren B, Ma W X, Yu J. Characteristics and interactions of solitary and lump waves of a (2+ 1)-dimensional coupled nonlinear partial differential equation. Nonlinear Dynam, 2019, 96: 717–727

[48] Ren Y W, Tao M S, Dong H H, Yang H W. Analytical research of (3+ 1)-dimensional Rossby waves with dissipation effect in cylindrical coordinate based on Lie symmetry approach. Adv Difference Equ, 2019, 2019: 13

[49] Satsuma J, Ablowitz M J. Two-dimensional lumps in nonlinear dispersive systems. J Math Phys, 1979, 20: 1496–1503

[50] Sun Y, Tian B, Xie X Y, Chai J, Yin H M. Rogue waves and lump solitons for a (3+ 1)-dimensional B-type Kadomtsev-Petviashvili equation in fluid dynamics. Waves Random Complex Media, 2018, 28: 544–552

[51] Tan W, Dai H P, Dai Z D, Zhong W Y. Emergence and space-time structure of lump solution to the (2+ 1)-dimensional generalized KP equation. Pramana-J Phys, 2017, 89: 77

[52] Tang Y N, Tao S Q, Guan Q. Lump solitons and the interaction phenomena of them for two classes of nonlinear evolution equations. Comput Math Appl, 2016, 72: 2334–2342

[53] Tang Y N, Tao S Q, Zhou M L, Guan Q. Interaction solutions between lump and other solitons of two classes of nonlinear evolution equations. Nonlinear Dynam, 2017, 89: 429–442

[54] Wang D S, Yin Y B. Symmetry analysis and reductions of the two-dimensional generalized Benney system via geometric approach. Comput Math Appl, 2016, 71: 748–757

[55] Wang H. Lump and interaction solutions to the (2+ 1)-dimensional Burgers equation. Appl Math Lett, 2018, 85: 27–34

[56] Wu J P, Geng X G. Novel Wronskian condition and new exact solutions to a (3+ 1)-dimensional generalized KP equation. Commun Theor Phys (Beijing), 2013, 60: 556–560

[57] Xu H N, Ruan W R, Zhang Y, Lü X. Multi-exponential wave solutions to two extended Jimbo-Miwa equations and the resonance behavior. Appl Math Lett, 2020, 99: 105976

[58] Xu X X. A deformed reduced semi-discrete Kaup-Newell equation, the related integrable family and Darboux transformation. Appl Math Comput, 2015, 251: 275–283

[59] Yang J Y, Ma W X. Lump solutions of the BKP equation by symbolic computation. Internat J Modern Phys B, 2016, 30: 1640028

[60] Yang J Y, Ma W X. Abundant lump-type solutions of the Jimbo-Miwa equation in (3+ 1)-dimensions. Comput Math Appl, 2017, 73: 220–225

[61] Yang Q Q, Zhao Q L, Li X Y. Explicit solutions and conservation laws for a new integrable lattice hierarchy. Complexity, 2019, 2019: 5984356

[62] Yin Y H, Ma W X, Liu J G, Lü X. Diversity of exact solutions to a (3+ 1)-dimensional nonlinear evolution equation and its reduction. Comput Math Appl, 2018, 76: 1225–1283

[63] Yong X L, Ma W X, Huang Y H, Liu Y. Lump solutions to the Kadomtsev-Petviashvili I equation with a self-consistent source. Comput Math Appl, 2018, 75: 3414–3419

[64] Yu J P, Sun Y L. Study of lump solutions to dimensionally reduced generalized KP equations. Nonlinear Dynam, 2017, 87: 2755–2763

[65] Zhang H Q, Ma W X. Lump solutions to the (2+ 1)-dimensional Sawada-Kotera equation. Nonlinear Dynam, 2017, 87: 2305–2310

[66] Zhang Y, Liu Y P, Tang X Y. M-lump solutions to a (3+ 1)-dimensional nonlinear evolution equation. Comput Math Appl, 2018, 76: 592–601

[67] Zhao Z L, He L C, Gao Y B. Rogue wave and multiple lump solutions of the (2+ 1)-dimensional Benjamin-Ono equation in fluid mechanics. Complexity, 2019, 2019: 8249635

[68] Zhong Y D, Zhao Q L, Li X Y. Explicit solutions to a coupled integrable lattice equation. Appl Math Lett, 2019, 98: 359–364

RIGHTS & PERMISSIONS

Higher Education Press