Interaction solutions to Hirota-Satsuma-Ito equation in (2+ 1)-dimensions

Abstract

Abundant exact interaction solutions, including lump-soliton, lumpkink, and lump-periodic solutions, are computed for the Hirota-Satsuma-Ito equation in (2+1)-dimensions, through conducting symbolic computations with Maple. The basic starting point is a Hirota bilinear form of the Hirota-Satsuma-Ito equation. A few three-dimensional plots and contour plots of three special presented solutions are made to shed light on the characteristic of interaction solutions.

Keywords

Symbolic computation / lump solution / interaction solution

References

[1] Ablowitz M J, Clarkson P A. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge: Cambridge Univ Press, 1991

[2] 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

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

[4] 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

[5] 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

[6] Drazin P G, Johnson R S. Solitons: An Introduction. Cambridge: Cambridge Univ Press, 1989

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

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

[9] Hietarinta J. Introduction to the Hirota bilinear method. In: Kosmann-Schwarzbach Y, Grammaticos B, Tamizhmani K M, eds. Integrability of Nonlinear Systems. Lecture Notes in Phys, Vol 495. Berlin: Springer, 1997, 95–103

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

[11] Ibragimov N H. A new conservation theorem. J Math Anal Appl, 2007, 333: 311–328

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

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

[14] 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

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

[16] 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

[17] Li X Y, Zhao Q L, Li Y X, Dong H H. Binary Bargmann symmetry constraint associated with 3 × 3 discrete matrix spectral problem. J Nonlinear Sci Appl, 2015, 8: 496–506

[18] 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

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

[20] 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

[21] 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

[22] Ma W X. Wronskian solutions to integrable equations. Discrete Contin Dyn Syst, 2009, Suppl: 506–515

[23] Ma W X. Conservation laws of discrete evolution equations by symmetries and adjoint symmetries. Symmetry, 2015, 7: 714–725

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

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

[26] Ma W X. Conservation laws by symmetries and adjoint symmetries. Discrete Contin Dyn Syst Ser S, 2018, 11: 707–721

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

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

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

[30] Ma W X. Lump and interaction solutions to linear (4+ 1)-dimensional PDEs. Acta Math Sci Ser B Engl Ed, 2019, 39: 498–508

[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(to appear)

[32] Ma W X, Fan E G. Linear superposition principle applying to Hirota bilinear equations. Comput Math Appl, 2011, 61: 950–959

[33] 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

[34] 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

[35] 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

[36] 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

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

[38] 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

[39] 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

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

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

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

[43] 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

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

[45] Ünsal Ö, Ma W X. Linear superposition principle of hyperbolic and trigonometric function solutions to generalized bilinear equations. Comput Math Appl, 2016, 71: 1242–1247

[46] 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

[47] Wazwaz A-M, El-Tantawy S A. New (3+ 1)-dimensional equations of Burgers type and Sharma-Tasso-Olver type: multiple-soliton solutions. Nonlinear Dynam, 2017, 87: 2457–2461

[48] Xu Z H, Chen H L, Dai Z D. Rogue wave for the (2+ 1)-dimensional Kadomtsev-Petviashvili equation. Appl Math Lett, 2014, 37: 34–38

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

[50] Yang J Y, Ma W X. Abundant interaction solutions of the KP equation. Nonlinear Dynam, 2017, 89: 1539–1544

[51] 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

[52] Yang J Y, Ma WX, Qin Z Y. Lump and lump-soliton solutions to the (2+1)-dimensional Ito equation. Anal Math Phys, 2018, 8: 427–436

[53] Yang J Y, Ma W X, Qin Z Y. Abundant mixed lump-soliton solutions to the BKP equation. East Asian J Appl Math, 2018, 8: 224–232

[54] 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

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

[56] Zhang Y, Dong H H, Zhang X E, Yang H W. Rational solutions and lump solutions to the generalized (3+ 1)-dimensional shallow water-like equation. Comput Math Appl, 2017, 73: 246–252

[57] Zhang Y, Sun S L, Dong H H. Hybrid solutions of (3+ 1)-dimensional Jimbo-Miwa equation. Math Probl Eng, 2017, 2017: 5453941

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

[59] Zhang J B, Ma W X. Mixed lump-kink solutions to the BKP equation. Comput Math Appl, 2017, 74: 591–596

[60] Zhao H Q, Ma W X. Mixed lump-kink solutions to the KP equation. Comput Math Appl, 2017, 74: 1399–1405

[61] Zhao Q L, Li X Y. A Bargmann system and the involutive solutions associated with a new 4-order lattice hierarchy. Anal Math Phys, 2016, 6: 237–254

[62] Zhou Y, Ma W X. Applications of linear superposition principle to resonant solitons and complexitons. Comput Math Appl, 2017, 73: 1697–1706

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature