In oceanography, acoustics and hydrodynamics: an extended coupled \((2+1)\)-dimensional Burgers system. (English) Zbl 07834911
Summary: In oceanography, acoustics and hydrodynamics, people pay attention to the Burgers-type equations for different wave processes, one of which is an extended coupled \((2+1)\)-dimensional Burgers system hereby under investigation. Based on the scaling transformation, Bell polynomials, Hirota operators and symbolic computation, we structure out two hetero-Bäcklund transformations, each of which to a solvable linear partial differential equation, and construct two sets of the bilinear forms, with the relevant one- and two-soliton solutions. Results rely on the coefficients in the original system.
MSC:
| 35Qxx | Partial differential equations of mathematical physics and other areas of application |
| 37Kxx | Dynamical system aspects of infinite-dimensional Hamiltonian and Lagrangian systems |
| 37Jxx | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
Keywords:
oceanography; acoustics; hydrodynamics; Bell polynomials; hetero-Bäcklund transformations; bilinear forms; solitons; symbolic computationReferences:
| [1] | “Burgers’ equation”, https://encyclopedia.thefreedictionary.com/Burgers |
| [2] | Saad, K. M.; AL-Shareef, E. H.; Alomari, A. K.; Baleanu, D.; Gomez-Aguilar, J. F., On exact solutions for time-fractional Korteweg-de Vries and Korteweg-de Vries-Burger’s equations using homotopy analysis transform method, Chin. J. Phys., 63, 149, (2020) · Zbl 07829561 |
| [3] | Wazwaz, A. M., Multiple complex soliton solutions for the integrable KdV, fifth-order Lax, modified KdV, Burgers, and Sharma-Tasso-Olver equations, Chin. J. Phys., 59, 372, (2019) · Zbl 07823539 |
| [4] | Senol, M.; Tasbozan, O.; Kurt, A., Numerical solutions of fractional Burgers’ type equations with conformable derivative, Chin. J. Phys., 58, 75, (2019) · Zbl 07822234 |
| [5] | Osman, M. S.; Baleanu, D.; Adem, A. R.; Hosseini, K.; Mirzazadeh, M.; Eslami, M., Double-wave solutions and Lie symmetry analysis to the (2+1)-dimensional coupled Burgers equations, Chin. J. Phys., 63, 122, (2020) · Zbl 07829558 |
| [6] | Wang, G. W.; Liu, Y. X.; Han, S. X.; Wang, H.; Su, X., Generalized symmetries and mCK method analysis of the (2+1)-dimensional coupled Burgers equations, Symmetry (Basel), 11, 1473, (2019) |
| [7] | Wazwaz, A. M., Multiple-front solutions for the Burgers equation and the coupled Burgers equations, Appl. Math. Comput., 190, 1198, (2007) · Zbl 1123.65106 |
| [8] | Wang, G. W.; Xu, T. Z.; Biswas, A., Topological solitons and conservation laws of the coupled Burgers equations, Rom. Rep. Phys., 66, 274, (2014) |
| [9] | Srivastava, V. K.; Singh, S.; Awasthi, M. K., Numerical solutions of coupled Burgers’ equations by an implicit finite-difference scheme, AIP Adv., 3, 082131, (2013) |
| [10] | ul Islam, S.; Sarler, B.; Vertnik, R.; Kosec, G., Radial basis function collocation method for the numerical solution of the two-dimensional transient nonlinear coupled Burgers’ equations, Appl. Math. Model., 36, 1148, (2012) · Zbl 1243.76076 |
| [11] | Wang, G. W.; Fakhar, K.; Kara, A. H., Soliton solutions and group analysis of a new coupled (2+1)-dimensional Burgers equations, Acta Phys. Pol. B, 46, 923, (2015) · Zbl 1371.35261 |
| [12] | Ali, A.; ul Islam, S.; Haq, S., A computational meshfree technique for the numerical solution of the two-dimensional coupled Burgers’ equations, Int. J. Comput. Methods Eng. Sci. Mech., 10, 406, (2009) · Zbl 1423.35303 |
| [13] | Yan, Z. Y., A transformation with symbolic computation and abundant new soliton-like solutions for the (1+ 2)-dimensional generalized Burgers equation, J. Phys. A, 35, 9923, (2002) · Zbl 1040.35103 |
| [14] | Yan, Z. Y., Complex PT-symmetric nonlinear Schrodinger equation and Burgers equation, Phil. Trans. R. Soc. A, 371, 20120059, (2013) |
| [15] | Gao, X. Y.; Guo, Y. J.; Shan, W. R., Scaling and hetero-/auto-Backlund transformations with solitons of an extended coupled (2+1)-dimensional Burgers system for the wave processes in hydrodynamics and acoustics, Mod. Phys. Lett. B, 34, Article 2050389 pp., (2020) |
| [16] | Wang, J. Y.; Liang, Z. F.; Tang, X. Y., Infinitely many generalized symmetries and Painleve analysis of a (2+1)-dimensional Burgers system, Phys. Scr., 89, 025201, (2014) |
| [17] | Unsal, O.; Tascan, F., Soliton solutions, Backlund transformation and Lax pair for coupled Burgers system via Bell polynomials, Z. Naturforch. A, 70, 359, (2015) |
| [18] | Wazwaz, A. M., Multiple kink solutions for two coupled integrable (2+1)-dimensional systems, Appl. Math. Lett., 58, 1, (2016) · Zbl 1342.35314 |
| [19] | Motsepa, T.; Khalique, C. M., Conservation laws and solutions of a generalized coupled (2+1)-dimensional Burgers system, Comput. Math. Appl., 74, 1333, (2017) · Zbl 1394.35430 |
| [20] | Beg, O. A.; Beg, T. A.; Karim, I.; Khan, M. S.; Alam, M. M.; Ferdows, M.; Shamshuddin, M. D., Numerical study of magneto-convective heat and mass transfer from inclined surface with Soret diffusion and heat generation effects: A model for ocean magnetic energy generator fluid dynamics, Chin. J. Phys., 60, 167;, (2019), D. Wang, Y.T. Gao, C.C. Ding, C.Y. Zhang, Solitons and periodic waves for a generalized (3+1)-dimensional Kadomtsev-Petviashvili equation in fluid dynamics and plasma physics, Commun. Theor. Phys. 72 (2020) 115004; T.T. Jia, Y.T. Gao, G.F. Deng, L. Hu, Quintic time-dependent-coefficient derivative nonlinear Schrodinger equation in hydrodynamics or fiber optics: bilinear forms and dark/anti-dark/gray solitons, Nonlinear Dyn. 98 (2019) 269; J.J. Su, Y.T. Gao, C.C. Ding, Darboux transformations and rogue wave solutions of a generalized AB system for the geophysical flows, Appl. Math. Lett. 88 (2019) 201; Y. J. Feng, Y. T. Gao, L. Q. Li, T. T. Jia, Bilinear form, solitons, breathers and lumps of a (3+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation in ocean dynamics, fluid mechanics and plasma physics, Eur. Phys. J. Plus 135 (2020) 272; F. Y. Liu, Y. T. Gao, X. Yu, C. C. Ding, G. F. Deng, T. T. Jia, Painleve analysis, Lie group analysis and soliton-cnoidal, resonant, hyperbolic function and rational solutions for the modified Korteweg-de Vries-Calogero-Bogoyavlenskii-Schiff equation in fluid mechanics, Chaos Solitons Fract. 144 (2021) 110559 |
| [21] | Melikhov, I. F.; Popov, I. Y., Model of cell membrane in ultrasonic field, Chin. J. Phys., 65, 334;, (2020), S.H. Liu, B. Tian, Q.X. Qu, H.Li, X.H. Zhao, X.X. Du, S.S. Chen, Breather, lump, shock and travelling-wave solutions for a (3+1)-dimensional generalized Kadomtsev-Petviashvili equation in fluid mechanics and plasma physics, Int. J. Comput. Math. https://doi.org/10.1080/00207160.2020.1805107, in press (2021); D. Wang, Y.T. Gao, J.J. Su, C.C. Ding, Bilinear forms and soliton solutions for a (2+1)-dimensional variable-coefficient nonlinear Schrodinger equation in an optical fiber, Mod. Phys. Lett. B 34 (2020) 2050336; C.C. Hu, B. Tian, H.M. Yin, C.R. Zhang, Z. Zhang, 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 (2019) 166; Y.Q. Chen, B. Tian, Q.X. Qu, H. Li, X.H. Zhao, H.Y. Tian, M. Wang, 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 (2020) 2050226 |
| [22] | Abdelsalam, S. I.; Mekheimer, K. S.; Zaher, A. Z., Alterations in blood stream by electroosmotic forces of hybrid nanofluid through diseased artery: Aneurysmal/stenosed segment, Chin. J. Phys., 67, 314;, (2020), M. Wang, B. Tian, Y. Sun, H.M. Yin, Z. Zhang, Mixed lump-stripe, bright rogue wave-stripe, dark rogue wave-stripe and dark rogue wave solutions of a generalized Kadomtsev-Petviashvili equation in fluid mechanics, Chin. J. Phys. 60 (2019) 440; X. Zhao, B. Tian, H.Y. Tian, D.Y. Yang, Bilinear Backlund 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. in press (2021), https://doi.org/10.1007/s11071-020-06154-9; C.C. Ding, Y.T. Gao, G.F. Deng, Breather and hybrid solutions for a generalized (3+1)-dimensional B-type Kadomtsev-Petviashvili equation for the water waves, Nonlinear Dyn. 97 (2019) 2023; G.F. Deng, Y.T. Gao, C.C. Ding, J.J. Su, Solitons and breather waves for the generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt system in fluid mechanics, ocean dynamics and plasma physics, Chaos Solitons Fract. 140 (2020) 110085; Y.J. Feng, Y.T. Gao, T.T. Jia, L.Q. Li, Soliton interactions of a variable-coefficient three-component AB system for the geophysical flows, Mod. Phys. Lett. B 33 (2019) 1950354 |
| [23] | Nganbe II, M. M.; Nyobe, E. N.; Hona, J.; Pemha, E., Heat transfer and circular flow around a hydrodynamic turning point through a porous annular tube, Chin. J. Phys., 61, 316;, (2019), M. Wang, B. Tian, Y. Sun, Z. Zhang, 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 (2020) 576; M. Wang, B. Tian, Q.X. Qu, X.X. Du, C.R. Zhang, Z. Zhang, Lump, lumpoff and rogue waves for a (2+1)-dimensional reduced Yu-Toda-Sasa-Fukuyama equation in a lattice or liquid, Eur. Phys. J. Plus 134 (2019) 578; Y.Q. Chen, B. Tian, Q.X. Qu, H. Li, X.H. Zhao, H.Y. Tian, M. Wang, Reduction and analytic solutions of a variable-coefficient Korteweg-de Vries equation in a fluid, crystal or plasma, Mod. Phys. Lett. B 34 (2020) 2050287. · Zbl 1443.76233 |
| [24] | Tahir, M.; Awan, A. U.; Osman, M. S.; Baleanu, D.; Alqurashi, M. M., Abundant periodic wave solutions for fifth-order Sawada-Kotera equations, Results Phys., 17, 103105, (2020) |
| [25] | Osman, M. S., One-soliton shaping and inelastic collision between double solitons in the fifth-order variable-coefficient Sawada-Kotera equation, Nonlinear Dyn., 96, 1491, (2019) · Zbl 1437.35603 |
| [26] | Liu, J. G.; Zhu, W. H.; Osman, M. S.; Ma, W. X., An explicit plethora of different classes of interactive lump solutions for an extension form of 3D-Jimbo-Miwa model, Eur. Phys. J. Plus, 135, 412, (2020) |
| [27] | Osman, M. S.; Inc, M.; Liu, J. G.; Hosseini, K.; Yusuf, A., Different wave structures and stability analysis for the generalized (2+1)-dimensional Camassa-Holm-Kadomtsev-Petviashvili equation, Phys. Scr., 95, 035229, (2020) |
| [28] | Osman, M. S.; Wazwaz, A. M., A general bilinear form to generate different wave structures of solitons for a (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation, Math. Methods Appl. Sci., 42, 6277, (2019) · Zbl 1434.35144 |
| [29] | Javid, A.; Raza, N.; Osman, M. S., Multi-solitons of thermophoretic motion equation depicting the wrinkle propagation in substrate-supported graphene sheets, Commun. Theor. Phys., 71, 362, (2019) |
| [30] | Pandir, Y.; Ekin, A., Dynamics of combined soliton solutions of unstable nonlinear Schrodinger equation with new version of the trial equation method, Chin. J. Phys., 67, 534;, (2020), S.S. Chen, B. Tian, J. Chai, X.Y. Wu, Z. Du, Lax pair, binary Darboux transformations and dark-soliton interaction of a fifth-order defocusing nonlinear Schrodinger equation for the attosecond pulses in the optical fiber communication, Wave. Random Complex 30 (2020) 389; Y.Q. Yuan, B. Tian, Q.X. Qu, C.R. Zhang, X.X. Du, Lax pair, binary Darboux transformation and dark solitons for the three-component Gross-Pitaevskii system in the spinor Bose-Einstein condensate, Nonlinear Dyn. 99 (2020) 3001; X. Zhao, B. Tian, Q.X. Qu, Y.Q. Yuan, X.X. Du, M.X. Chu, Dark-dark solitons for the coupled spatially modulated Gross-Pitaevskii system in the Bose-Einstein condensation, Mod. Phys. Lett. B 34 (2020) 2050282 · Zbl 1498.78032 |
| [31] | Yildirim, Y.; Biswas, A.; Ekici, M.; Gonzalez-Gaxiola, O.; Khan, S.; Triki, H.; Moraru, L.; Alzahrani, A. K.; Belic, M. R., Optical solitons with Kudryashov’s model by a range of integration norms, Chin. J. Phys., 66, 660;, (2020), X.X. Du, B. Tian, Q.X. Qu, Y.Q. Yuan, X.H. Zhao, 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 (2020) 109709; Z. Du, B. Tian, H.P. Chai, X.H. Zhao, Dark-bright semi-rational solitons and breathers for a higher-order coupled nonlinear Schrodinger system in an optical fiber, Appl. Math. Lett. 102 (2020) 106110 · Zbl 1483.35177 |
| [32] | Kudryashov, N. A., Periodic and solitary waves in optical fiber Bragg gratings with dispersive reflectivity, Chin. J. Phys., 66, 401;, (2020), S.H. Liu, B. Tian, Q.X. Qu, C.R. Zhang, C.C. Hu, M. Wang, Lump, mixed lump-stripe, mixed rogue wave-stripe and breather wave solutions for a (3+1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation, Mod. Phys. Lett. B 34 (2020) 2050243; L.Q. Li, Y.T. Gao, L. Hu, T.T. Jia, C.C. Ding, Y.J. Feng, Bilinear form, soliton, breather, lump and hybrid solutions for a (2+1)-dimensional Sawada-Kotera equation, Nonlinear Dyn. 100 (2020) 2729; C.R. Zhang, B. Tian, Q.X. Qu, L. Liu, H.Y. Tian, Vector bright solitons and their interactions of the couple Fokas-Lenells system in a birefringent optical fiber, Z. Angew. Math. Phys. 71 (2020) 18; J.J. Su, Y.T. Gao, G.F. Deng, T.T. Jia, Solitary waves, breathers, and rogue waves modulated by long waves for a model of a baroclinic shear flow, Phys. Rev. E 100 (2019) 042210; G.F. Deng, Y.T. Gao, J.J. Su, C.C. Ding, T.T. Jia, Solitons and periodic waves for the (2+1)-dimensional generalized Caudrey-Dodd-Gibbon-Kotera-Sawada equation in fluid mechanics, Nonlinear Dyn. 99 (2020) 1039 |
| [33] | Tsiganov, A. V., Backlund transformations for the jacobi system on an ellipsoid, Theor. Math. Phys., 192, 1350;, (2017), A.V. Tsiganov, Simultaneous separation for the Neumann and Chaplygin systems, Regul. Chaotic. Dyn. 20 (2015) 74; X.Y. Gao, Y.J. Guo, W.R. Shan, Shallow water in an open sea or a wide channel: Auto- and non-auto-Backlund transformations with solitons for a generalized (2+1)-dimensional dispersive long-wave system, Chaos Solitons Fract. 138 (2020) 109950; X.Y. Gao, Y.J. Guo, W.R. Shan, Hetero-Backlund transformation and similarity reduction of an extended (2+1)-dimensional coupled Burgers system in fluid mechanics, Phys. Lett. A 384 (2020) 126788 · Zbl 1386.37057 |
| [34] | Gao, X. Y.; Guo, Y. J.; Shan, W. R., Viewing the Solar System via a variable-coefficient nonlinear dispersive-wave system, Acta Mech., 231, 4415;, (2020), X.Y. Gao, Y.J. Guo, W.R. Shan, Oceanic studies via a variable-coefficient nonlinear dispersive-wave system in the Solar System, Chaos Solitons Fract. 141 (2020) 110367; X. Y. Gao, Y. J. Guo, W. R. Shan, Water-wave symbolic computation for the Earth, Enceladus and Titan: the higher-order Boussinesq-Burgers system, auto- and non-auto-Backlund transformations, Appl. Math. Lett. 104 (2020) 106170 · Zbl 1451.86003 |
| [35] | Kaplan, M., Two different systematic techniques to find analytical solutions of the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation, Chin. J. Phys., 56, 2523;, (2018), Y. Shen, B. Tian, S.H. Liu, D.Y. Yang, Bilinear Backlund transformation, soliton and breather solutions for a (3+1)-dimensional generalized Kadomtsev-Petviashvili equation in fluid dynamics and plasma physics, Phys. Scr. https://doi.org/10.1088/1402-4896/abdf0d, in press (2021); Y.Q. Yuan, B. Tian, Q.X. Qu, X.H. Zhao, X.X. Du, Periodic-wave and semirational solutions for the (2+1)-dimensional Davey-Stewartson equations on the surface water waves of finite depth, Z. Angew. Math. Phys. 71 (2020) 46; M. Wang, B. Tian, C. C. Hu, S. H. Liu, Generalized Darboux transformation, solitonic interactions and bound states for a coupled fourth-order nonlinear Schrodinger system in a birefringent optical fiber, Appl. Math. Lett. doi: 10.1016/j.aml.2020.106936, in press (2021); L. Hu, Y.T. Gao, S.L. Jia, J.J. Su, G.F. Deng, Solitons for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation for an irrotational incompressible fluid via the Pfaffian technique, Mod. Phys. Lett. B 33 (2019) 1950376; H.M. Yin, B. Tian, X.H. Zhao, Chaotic breathers and breather fission/fusion for a vector nonlinear Schrodinger equation in a birefringent optical fiber or wavelength division multiplexed system, Appl. Math. Comput. 368 (2020) 124768; S.S. Chen, B. Tian, Y. Sun, C.R. Zhang, Generalized Darboux transformations, rogue waves, and modulation instability for the coherently coupled nonlinear Schrodinger equations in nonlinear optics, Ann. Phys. (Berlin) 531 (2020) 1900011; H. Y. Tian, B. Tian, Y. Q. Yuan, C. R. Zhang, Superregular solutions for a coupled nonlinear Schrodinger system in a two-mode nonlinear fiber, Phys. Scr. doi: 10.1088/1402-4896/abd793, in press (2021) |
| [36] | Kumar, D.; Seadawy, A. R.; Joardar, A. K., Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology, Chin. J. Phys., 56, 75;, (2018), Z. Du, B. Tian, Q.X. Qu, X.Y. Wu, X.H. Zhao, Vector rational and semi-rational rogue waves for the coupled cubic-quintic nonlinear Schrodinger system in a non-Kerr medium, Appl. Numer. Math. 153 (2020) 179; X.X. Du, B. Tian, Y.Q. Yuan, Z. Du, 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 (2019) 1900198; C.R. Zhang, B. Tian, Y. Sun, H.M. Yin, Binary Darboux transformation and vector-soliton-pair interactions with the negatively coherent coupling in a weakly birefringent fiber, EPL 127 (2019) 40003; T.T. Jia, Y.T. Gao, X. Yu, L.Q. Li, Lax pairs, Darboux transformation, bilinear forms and solitonic interactions for a combined Calogero-Bogoyavlenskii-Schiff-type equation. Appl. Math. Lett. 114 (2021) 106702. |
| [37] | Ray, S. S., New double periodic exact solutions of the coupled Schrodinger-Boussinesq equations describing physical processes in laser and plasma physics, Chin. J. Phys., 55, 2039;, (2017), Y. Shen, B. Tian, C.R. Zhang, H.Y. Tian, S.H. Liu, Breather-wave, periodic-wave and travelling-wave solutions for a (2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli equation for an incompressible fluid, Mod. Phys. Lett. B No. MPLB-D-20-00563R1, in press (2021); D.Y. Yang, B. Tian, Q.X. Qu, C.R. Zhang, S.S. Chen, C.C. Wei, Lax pair, conservation laws, Darboux transformation and localized waves of a variable-coefficient coupled Hirota system in an inhomogeneous optical fiber, Chaos Solitons Fract. in press (2021) https://doi.org/10.1016/j. chaos.2020.110487; D.Y. Yang, B. Tian, Q.X. Qu, H. Li, X.H. Zhao, S.S. Chen, C.C. Wei, Darboux-dressing transformation, semi-rational solutions, breathers and modulation instability for the cubic-quintic nonlinear Schrodiger system with variable coefficients in a non-Kerr medium, twin-core nonlinear optical fiber or waveguide, Phys. Scr., 96 (2021) 045210; C.C. Ding, Y.T. Gao, G.F. Deng, D. Wang, Lax pair, conservation laws, Darboux transformation, breathers and rogue waves for the coupled nonautonomous nonlinear Schrodinger system in an inhomogeneous plasma, Chaos Solitons Fract. 133 (2020) 109580; H.M. Yin, B. Tian, X.C. Zhao, Magnetic breathers and chaotic wave fields for a higher-order (2+1)-dimensional nonlinear Schrodinger-type equation in a Heisenberg ferromagnetic spin chain, J. Magn. Magn. Mater. 495 (2020) 165871 |
| [38] | Bell, E. T., Exponential polynomials, Ann. Math., 35, 258, (1934) · Zbl 0009.21202 |
| [39] | Lambert, F.; Loris, I.; Springael, J.; Willox, R., On a direct bilinearization method: Kaup’s higher-order water wave equation as a modified nonlocal Boussinesq equation, J. Phys. A, 27, 5325, (1994) · Zbl 0845.35088 |
| [40] | Wang, Y. F.; Tian, B.; Jiang, Y., Soliton fusion and fission in a generalized variable-coefficient fifth-order Korteweg-de Vries equation in fluids, Appl. Math. Comput., 292, 448, (2017) · Zbl 1410.76058 |
| [41] | Matveev, V. B.; Salle, M. A., Darboux transformations and solitons, (1991), Springer: Springer Berlin · Zbl 0744.35045 |
| [42] | Wadati, M., Wave propagation in nonlinear lattice I, J. Phys. Soc. Jpn., 38, 673, (1975) · Zbl 1334.82022 |
| [43] | Caruello, F.; Tabor, M., Painleve expansions for nonintegrable evolution equations, Phys. D, 39, 77, (1989) · Zbl 0687.35093 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
