Abstract
In this work, we study a (3+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation for the nonlinear dispersive waves in an inhomogeneous medium. Bilinear form and N-soliton solutions are derived, where N is a positive integer. The higher-order breather and lump solutions are constructed based on the N-soliton solutions. Hybrid solutions comprising the solitons and breathers, breathers and lumps, as well as solitons and lumps are worked out. Amplitudes and velocities of the one solitons as well as periods of the first-order breathers are investigated. Amplitudes of the first-order lumps reach the maximum and minimum values at certain points given in the paper. Interactions between any two of those waves are discussed graphically.
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References
McAnally, M., Ma, W.X.: Explicit solutions and Darboux transformations of a generalized D-Kaup-Newell hierarchy. Nonlinear Dyn. 102, 2767–2782 (2020)
Kumar, S., Jiwari, R., Mittal, R.C., Awrejcewicz, J.: Dark and bright soliton solutions and computational modeling of nonlinear regularized long wave model. Nonlinear Dyn. (2021). https://doi.org/10.1007/s11071-021-06291-9
Sulaiman, T.A., Yusuf, A., Alquran, M.: Dynamics of optical solitons and nonautonomous complex wave solutions to the nonlinear Schrodinger equation with variable coefficients. Nonlinear Dyn. (2021). https://doi.org/10.1007/s11071-021-06284-8
Zhao, Z., He, L.: M-lump and hybrid solutions of a generalized (2+1)-dimensional Hirota-Satsuma-Ito equation. Appl. Math. Lett. 111, 106612 (2021)
dos Santos, M.C., Cardoso, W.B.: Influence of fourth-order dispersion on the Anderson localization. Nonlinear Dyn. 101, 611–618 (2020)
Kumar, S., Kumar, A., Wazwaz, A.M.: New exact solitary wave solutions of the strain wave equation in microstructured solids via the generalized exponential rational function method. Eur. Phys. J. Plus 135, 870 (2020)
Yang, K.L., Zhou, X.J., Wang, C.J., Fu, P., Xia, C.Y., Wang, L.: Traveling wave induced periodic synchronous patterns in coupled discontinuous systems and its potential application. Nonlinear Dyn. 102, 2783–2792 (2020)
Sergeev, K.S., Chetverikov, A.P., del Rio, E.: Dissipative discrete breathers in a chain of Rayleigh oscillators. Nonlinear Dyn. 102, 1813–1823 (2020)
Dasgupta, C., Maitra, S.: Envelope solitons and rogue waves in Jupiter’s magnetosphere. Phys. Plasmas 27, 102110 (2020)
Calzavara, M., Salasnich, L.: Dark solitons in the unitary bose gas. Symmetry 12(6), 957 (2020)
Li, P., Malomed, B.A., Mihalache, D.: Metastable soliton necklaces supported by fractional diffraction and competing nonlinearities. Opt. Exp. 28(23), 34472–34488 (2020)
Redor, I., Barthélemy, E., Mordant, N., Michallet, H.: Analysis of soliton gas with large-scale video-based wave measurements. Exp. Fluids 61, 216 (2020)
Wang, M., Tian, B., Sun, Y., Zhang, Z.: Lump, mixed lump-stripe and rogue wave-stripe solutions of a (3+1)-dimensional nonlinear wave equation for a liquid with gas bubbles. Comput. Math. Appl. 79(3), 576–587 (2020)
Xu, G.Q., Wazwaz, A.M.: Bidirectional solitons and interaction solutions for a new integrable fifth-order nonlinear equation with temporal and spatial dispersion. Nonlinear Dyn. 101, 581–595 (2020)
Chen, S.S., Tian, B., Chai, J., Wu, X.Y., Du, Z.: Lax pair, binary Darboux transformations and dark-soliton interaction of a fifth-order defocusing nonlinear Schrödinger equation for the attosecond pulses in the optical fiber communication. Wave. Random Complex 30(3), 389–402 (2020)
Du, X.X., Tian, B., Yuan, Y.Q., Du, Z.: Symmetry reductions, group-invariant solutions, and conservation laws of a (2+1)-dimensional nonlinear Schrodinger equation in a Heisenberg ferromagnetic spin chain. Ann. Phys. (Berlin) 531(11), 1900198 (2019)
Chen, Y.Q., Tian, B., Qu, Q.X., Li, H., Zhao, X.H., Tian, H.Y., Wang, M.: Reduction and analytic solutions of a variable-coefficient Korteweg-de Vries equation in a fluid, crystal or plasma. Mod. Phys. Lett. B 34(26), 2050287 (2020)
Chen, Y.Q., Tian, B., Qu, Q.X., Li, H., Zhao, X.H., Tian, H.Y., Wang, M.: Ablowitz-Kaup-Newell-Segur system, conservation laws and Backlund transformation of a variable-coefficient Korteweg-de Vries equation in plasma physics, fluid dynamics or atmospheric science. Int. J. Mod. Phys. B 34(25), 2050226 (2020)
Gao, X.Y., Guo, Y.J., Shan, W.R.: Oceanic studies via a variable-coefficient nonlinear dispersive-wave system in the Solar System. Chaos Solitons Fract. 142, 110367 (2021)
Osborne, A.R., Resio, D.T., Costa, A., de León, S.P.: Highly nonlinear wind waves in Currituck Sound: dense breather turbulence in random ocean waves. Ocean Dyn. 69, 187–219 (2019)
Mukherjee, A.: Free surface lump wave dynamics of a saturated superfluid \(^4He\) film with nontrivial boundary condition at the substrate surface. Phys. Scr. 95, 095209 (2020)
Tian, H.Y., Tian, B., Yuan, Y.Q., Zhang, C.R.: Superregular solutions for a coupled nonlinear Schrödinger system in a two-mode nonlinear fiber. Phys. Scr. 96(4), 045213 (2021)
Adhikary, A., Hossain, M.B., Khan, T.Z., Soheli, S.J., Rahman, M.A.: Performance analysis of Q-factor on wavelengths and bit rates using optical solitons with dispersion management. J. Opt. 49, 533–542 (2020)
Bandelow, U., Amiranashvili, S., Pickartz, S.: Stabilization of optical pulse transmission by exploiting fiber nonlinearities. J. Lightwave Technol. 38(20), 5743–5747 (2020)
Zhang, C.R., Tian, B., Qu, Q.X., Liu, L., Tian, H.Y.: Vector bright solitons and their interactions of the couple Fokas-Lenells system in a birefringent optical fiber. Z. Angew. Math. Phys. 71, 18 (2020)
Wang, M., Tian, B., Hu, C.C., Liu, S.H.: Generalized Darboux transformation, solitonic interactions and bound states for a coupled fourth-order nonlinear Schrödinger system in a birefringent optical fiber. Appl. Math. Lett (2021). https://doi.org/10.1016/j.aml.2020.106936
Yang, D.Y., Tian, B., Qu, Q.X., Zhang, C.R., Chen, S.S., Wei, C.C.: Lax pair, conservation laws, Darboux transformation and localized waves of a variable-coefficient coupled Hirota system in an inhomogeneous optical fiber. Chaos Solitons Fract. (2021). https://doi.org/10.1016/j.chaos.2020.110487
Akhmediev, N., Soto-Crespo, J.M., Devine, N.: Breather turbulence versus soliton turbulence: Rogue waves, probability density functions, and spectral features. Phys. Rev. E 94, 022212 (2016)
Chen, S.S., Tian, B., Sun, Y., Zhang, C.R.: Generalized Darboux Transformations, Rogue Waves, and modulation instability for the coherently coupled nonlinear Schrödinger equations in nonlinear optics. Ann. Phys. (Berlin) 531(8), 1900011 (2019)
Wang, Y., Gao, B.: The dynamic behaviors between multi-soliton of the generalized (3+1)-dimensional variable coefficients Kadomtsev–Petviashvili equation. Nonlinear Dyn. 101, 2463–2470 (2020)
Horikis, T.P., Frantzeskakis, D.J.: Light meets water in nonlocal media: surface tension analogue in optics. Phys. Rev. Lett. 118, 243903 (2017)
Beji, S.: Kadomtsev-Petviashvili type equation for uneven water depths. Ocean Eng. 154, 226–233 (2018)
Yu, W.T., Zhang, H.X., Zhou, Q., Biswas, A., Alzahrani, A.K., Liu, W.J.: The mixed interaction of localized, breather, exploding and solitary wave for the (3+1)-dimensional Kadomtsev-Petviashvili equation in fluid dynamics. Nonlinear Dyn. 100, 1611–1619 (2020)
Zhao, X., Tian, B., Qu, Q.X., Yuan, Y.Q., Du, X.X., Chu, M.X.: Dark-dark solitons for the coupled spatially modulated Gross–Pitaevskii system in the Bose-Einstein condensation. Mod. Phys. Lett. B 34(26), 2050282 (2020)
Seadawy, A.R., El-Rashidy, K.: Dispersive solitary wave solutions of Kadomtsev–Petviashvili and modified Kadomtsev–Petviashvili dynamical equations in unmagnetized dust plasma. Results Phys. 8, 1216–1222 (2018)
Hunt, M.J., Vanden-Broeck, J.M., Papageorgiou, D.T.: Benjamin-Ono Kadomtsev–Petviashvili’s models in interfacial electro-hydrodynamics. Eur. J. Mech. B 65, 459–463 (2017)
Osman, M.S.: Nonlinear interaction of solitary waves described by multi-rational wave solutions of the (2+1)-dimensional Kadomtsev-Petviashvili equation with variable coefficients. Nonlinear Dyn. 87, 1209–1216 (2017)
Hu, C.C., Tian, B., Yin, H.M., Zhang, C.R., Zhang, Z.: Dark breather waves, dark lump waves and lump wave-soliton interactions for a (3+1)-dimensional generalized Kadomtsev-Petviashvili equation in a fluid. Comput. Math. Appl. 78(1), 166–177 (2019)
Gao, X.Y., Guo, W.J., Shan, W.R.: Magneto-optical/ferromagnetic-material computation: Backlund transformations, bilinear forms and N solitons for a generalized (3+1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili system. Appl. Math. Lett. 111, 106627 (2021)
Yin, H.M., Tian, B., Zhao, X.C., Zhang, C.R., Hu, C.C.: Breather-like solitons, rogue waves, quasi-periodic/chaotic states for the surface elevation of water waves. Nonlinear Dyn. 97, 21–31 (2019)
Kumar, M., Tiwari, A.K.: Some group-invariant solutions of potential Kadomtsev–Petviashvili equation by using Lie symmetry approach. Nonlinear Dyn. 92, 781–792 (2018)
Zhang, C.R., Tian, B., Sun, Y., Yin, H.M.: Binary Darboux transformation and vector-soliton-pair interactions with the negatively coherent coupling in a weakly birefringent fiber. EPL 127, 40003 (2019)
Du, X.X., Tian, B., Qu, Q.X., Yuan, Y.Q., Zhao, X.H.: Lie group analysis, solitons, self-adjointness and conservation laws of the modified Zakharov-Kuznetsov equation in an electron-positron-ion magnetoplasma. Chaos Solitons Fract. 134, 109709 (2020)
Liu, J.G., Eslami, M., Rezazadeh, H., Mirzazadeh, M.: Rational solutions and lump solutions to a non-isospectral and generalized variable-coefficient Kadomtsev–Petviashvili equation. Nonlinear Dyn. 95, 1027–1033 (2019)
Hanif Y., Saleem U.: Darboux transformation and multi-soliton solutions of the discrete sine-Gordon equation. Prog. Theor. Exp. Phys. 2020(6): 063A01 (2020)
Bertola, M., Yang, D.: The partition function of the extended r-reduced Kadomtsev–Petviashvili hierarchy. J. Phys. A 48, 195205 (2015)
Yuan, Y.Q., Tian, B., Qu, Q.X., Zhang, C.R., Du, X.X.: Lax pair, binary Darboux transformation and dark solitons for the three-component Gross-Pitaevskii system in the spinor Bose-Einstein condensate. Nonlinear Dyn. 99, 3001–3011 (2020)
Gao, X.Y., Guo, Y.J., Shan, W.R.: Water-wave symbolic computation for the Earth, Enceladus and Titan: the higher-order Boussinesq-Burgers system, auto- and non-auto-Bäcklund transformations. Appl. Math. Lett. 104, 106170 (2020)
Gao, X.Y., Guo, Y.J., Shan, W.R.: Shallow water in an open sea or a wide channel: auto- and non-auto-Bäcklund transformations with solitons for a generalized (2+1)-dimensional dispersive long-wave system. Chaos Solitons Fract. 138, 109950 (2020)
Zhao, X., Tian, B., Tian, H.Y., Yang, D.Y.: Bilinear Bäcklund transformation, Lax pair and interactions of nonlinear waves for a generalized (2+1)-dimensional nonlinear wave equation in nonlinear optics/fluid mechanics/plasma physics. Nonlinear Dyn. 103, 1785–1794 (2021)
Yang, D.Y., Tian, B., Qu, Q.X., Li, H., Zhao, X.H., Chen, S.S., Wei, C.C.: Darboux-dressing transformation, semi-rational solutions, breathers and modulation instability for the cubic-quintic nonlinear Schrödiger system with variable coefficients in a non-Kerr medium, twin-core nonlinear optical fiber or waveguide. Phys. Scr. 96(4), 045210 (2021)
Wazwaz, A.M.: New (3+1)-dimensional Date-Jimbo-Kashiwara-Miwa equations with constant and time-dependent coefficients: Painlevé integrability. Phys. Lett. A 384(32), 126787 (2020)
Wazwaz, A.M.: A (2+1)-dimensional time-dependent Date-Jimbo-Kashiwara-Miwa equation: Painlevé integrability and multiple soliton solutions. Comput. Math. Appl. 79(4), 1145–1149 (2020)
Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations: IV. a new hierarchy of soliton equations of KP-type. Phys. D 4(3), 343–365 (1982)
Chauhan, A., Sharma, K., Arora, R.: Lie symmetry analysis, optimal system, and generalized group invariant solutions of the (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation. Math. Meth. Appl. Sci. 43(15), 8823–8840 (2020)
Kumar, S., Kumar, A.: Dynamical structures of solitons and some new types of exact solutions for the (2+1)-dimensional DJKM equation using Lie symmetry analysis. Mod. Phys. Lett. B 34, 2150015 (2020)
Ablowitz, M.J., Satsuma, J.: Solitons and rational solutions of nonlinear evolution equations. J. Math. Phys. 19, 2180 (1978)
Satsuma, J., Ablowitz, M.J.: Two-dimensional lumps in nonlinear dispersive systems. J. Math. Phys. 20, 1496 (1979)
Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)
Cao, Y.L., Malomed, B.A., He, J.S.: Two (2+1)-dimensional integrable nonlocal nonlinear Schrödinger equations: breather, rational and semi-rational solutions. Chaos Solitons Fract. 114, 99–107 (2018)
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We express our sincere thanks to the Editors, Referees and all the members of our discussion group for their valuable comments.
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This work has been supported by the National Natural Science Foundation of China under Grant No. 11272023, and by the Fundamental Research Funds for the Central Universities.
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Wang, D., Gao, YT., Yu, X. et al. Bilinear form, solitons, breathers, lumps and hybrid solutions for a (3+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation. Nonlinear Dyn 104, 1519–1531 (2021). https://doi.org/10.1007/s11071-021-06329-y
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DOI: https://doi.org/10.1007/s11071-021-06329-y