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HIROTA METHOD AND SOLITON SOLUTIONS

Year 2022, , 157 - 172, 31.12.2022
https://doi.org/10.51477/mejs.1029348

Abstract

Solitons are an important class of solutions to nonlinear differential equations which appear in different areas of physics and applied mathematics. In this study we provide a general overview of the Hirota method which is one of the most powerful tool in finding the multi-soliton solutions of nonlinear wave and evaluation equations. Bright and dark soliton solutions of nonlinear Schrödinger equation are discussed in detail

References

  • [1] Russell, J.S., “Report on waves”, report of the 14th meeting of the British Association for the Advancement of science, John Murray, London, 311–390, 1845.
  • [2] Boussinesq, J., “Theorie de l’intumescence liquid appellee onde solitare ou de translation, se propageant dans un canal rectangulaire”, C. R. Acad. Sci., Paris, 1872.
  • [3] Korteweg, D.J., De Vries, G., “On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves”, Phil. Mag., 39(240), 422 -443, 1895.
  • [4] Fermi, A., Pasta, J., Ulam, S., “Studies of nonlinear problems”, I. Los Alamos Report LA- 1940, Los Alamos National Laboratory, May 1955.
  • [5] Zabusky, N.J., Kruskal, M.D., “Interaction of solitons in a collisionless plasma and the recurrence of initial states” Phys Rev. Lett., 15, 240-243, 1965.
  • [6] Miura, R.M., “Korteweg-de Vries equation and generalizations I. A remarkable explicit nonlinear transformation”, J. Math. Phys., 9, 1202-1204, 1968.
  • [7] Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura; R.M., “Method for solving the Korteweg-de Vries equation”, Phys. Rev. Lett., 19(19), 1095-1097, 1967.
  • [8] Gardner, C.S., “Korteweg-de Vries equation and generalizations IV. The Korteweg-de Vries equation as a Hamiltonian system”, J. Math. Phys., 12, 1548–1551, 1971. [9] Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura; R.M., “Korteweg-de Vries equation and generalizations VI. Methods for exact solution”, Comm. Pure Appl.Math., 27, 97-133, 1974.
  • [10] Zakharov, V.E., Faddeev, L.D., “Korteweg-de Vries equation: a completely integrable Hamiltonian system”, Funct. Anal. Appl., 5, 280-287, 1971.
  • [11] Hirota, R., “Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons”, Phys. Rev. Lett., 27 (18), 1192-1194, 1971.
  • [12] Hirota, R., The direct method in soliton theory, Cambridge University Press, 2004.
  • [13] Wadati, M., Konno, K., Ichikawa, Y. H., “ A generalization of inverse scattering method”, J. Phys. Soc. Japan, 46(6), 1965-1966, 1979.
  • [14] Matveev, V.B., Salle, M.A., Darboux transformation and soliton, Springer, Berlin, 1991.
  • [15] Ito, M., “An extension of nonlinear evolution equatios of the KdV (mKdV) type to higher orders”, J. Phys. Soc. Japan, 49(2), 771–778, 1980.
  • [16] Freeman, N.C., Nimmo, J.J.C., “Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: The wronskian technique”, Phys.Lett. A., 95(1), 1–3, 1983; Freeman, N.C., Nimmo, J.J.C., “A method of obtaining the n-soliton solutions of the Boussinesq equation in terms of a wronskian”, Phys.Lett. A., 95(1), 4–6, 1983.
  • [17] Nimmo, J.J.C., “A bilinear Backlund transformation for the nonlinear Schrodinger equation”, Phys.Lett. A., 99(6-7), 279–280, 1983
  • [18] Nimmo, J.J.C., Yilmaz, H., “Binary Darboux transformation for the Sasa-Satsuma equation”, J. Phys. A., 48(42), 2015. [19] Hietarinta, J., “A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV type bilinear equations”, J. Math. Phys., 28, 1732, 2094, 1987; Hietarinta, J., “Hirota’s bilinear method and its generalization”, Inter. J. Mod. Phys. A, 12(1), 43-51, 1997.
  • [20] Hietarinta, J., “Introduction to the bilinear method”in: Kosman-Schwarzbach, Y., Tamizhmani, K.M., Grammaticos, B. (eds), Integrability of nonlinear systems. Lecture notes in physics, 638, 95-105, Springer, Berlin, 2004.
  • [21] Taniuti, T., Yajima, N., “Perturbation method for a nonlinear wave modulation I. ”, J.Math. Phys., 10, 1369-1372, 1969.
  • [22] Debnath, L., Nonlinear partial differential equations for scientists and engineers, Birkhauser, 2012.
  • [23] Zakharov, V.E., Shabat, A.B., “Exact theory of two-dimensional self-focusing and one -dimensional self-modulation of waves in nonlinear media”, Soviet Physics JETP, 34, 62 -69, 1972
  • [24] Hirota, R., “Exact envelope-soliton solutions of a nonlinear wave equation”, J. Math. Phys., 14, 805-809, 1973.
  • [25] Hasegawa, A., Tappert, F., “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion”, Appl. Phys. Lett., 23, 171-172, 1973.
  • [26] Ablowitz, M.J., Nonlinear dispersive waves. Asymptotic analysis and solitons, Cambridge University Press, 2011.
  • [27] Weiner, A.M. et al., “Experimental observation of the fundamental dark soliton in optical fibers”, Phys. Rev. Lett., 61, 2445, 1988.
  • [28] Tlidi, M., Gelens, L., “High-order dispersion stabilizes dark dissipative solitons in all-fiber cavities”, Opt. Lett., 35, 306-308, 2010.
  • [29] Carr, L.D., Brand, J., Burger, S., Sanpera, A., “Dark-soliton creation in Bose-Einstein condensates”, Phys. Rev. A, 63, 051601(R), 2001.
  • [30] Shukla, P.K., Eliasson, B., “Formation and dynamics of dark-solitons and vortices in quantum electron plasmas”, Phys. Rev.Lett., 96, 245001, 2006.
  • [31] Heidemann, R. et al., “Dissipative dark soliton in a complex plasma”, Phys. Rev.Lett., 102, 135002, 2009.
Year 2022, , 157 - 172, 31.12.2022
https://doi.org/10.51477/mejs.1029348

Abstract

References

  • [1] Russell, J.S., “Report on waves”, report of the 14th meeting of the British Association for the Advancement of science, John Murray, London, 311–390, 1845.
  • [2] Boussinesq, J., “Theorie de l’intumescence liquid appellee onde solitare ou de translation, se propageant dans un canal rectangulaire”, C. R. Acad. Sci., Paris, 1872.
  • [3] Korteweg, D.J., De Vries, G., “On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves”, Phil. Mag., 39(240), 422 -443, 1895.
  • [4] Fermi, A., Pasta, J., Ulam, S., “Studies of nonlinear problems”, I. Los Alamos Report LA- 1940, Los Alamos National Laboratory, May 1955.
  • [5] Zabusky, N.J., Kruskal, M.D., “Interaction of solitons in a collisionless plasma and the recurrence of initial states” Phys Rev. Lett., 15, 240-243, 1965.
  • [6] Miura, R.M., “Korteweg-de Vries equation and generalizations I. A remarkable explicit nonlinear transformation”, J. Math. Phys., 9, 1202-1204, 1968.
  • [7] Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura; R.M., “Method for solving the Korteweg-de Vries equation”, Phys. Rev. Lett., 19(19), 1095-1097, 1967.
  • [8] Gardner, C.S., “Korteweg-de Vries equation and generalizations IV. The Korteweg-de Vries equation as a Hamiltonian system”, J. Math. Phys., 12, 1548–1551, 1971. [9] Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura; R.M., “Korteweg-de Vries equation and generalizations VI. Methods for exact solution”, Comm. Pure Appl.Math., 27, 97-133, 1974.
  • [10] Zakharov, V.E., Faddeev, L.D., “Korteweg-de Vries equation: a completely integrable Hamiltonian system”, Funct. Anal. Appl., 5, 280-287, 1971.
  • [11] Hirota, R., “Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons”, Phys. Rev. Lett., 27 (18), 1192-1194, 1971.
  • [12] Hirota, R., The direct method in soliton theory, Cambridge University Press, 2004.
  • [13] Wadati, M., Konno, K., Ichikawa, Y. H., “ A generalization of inverse scattering method”, J. Phys. Soc. Japan, 46(6), 1965-1966, 1979.
  • [14] Matveev, V.B., Salle, M.A., Darboux transformation and soliton, Springer, Berlin, 1991.
  • [15] Ito, M., “An extension of nonlinear evolution equatios of the KdV (mKdV) type to higher orders”, J. Phys. Soc. Japan, 49(2), 771–778, 1980.
  • [16] Freeman, N.C., Nimmo, J.J.C., “Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: The wronskian technique”, Phys.Lett. A., 95(1), 1–3, 1983; Freeman, N.C., Nimmo, J.J.C., “A method of obtaining the n-soliton solutions of the Boussinesq equation in terms of a wronskian”, Phys.Lett. A., 95(1), 4–6, 1983.
  • [17] Nimmo, J.J.C., “A bilinear Backlund transformation for the nonlinear Schrodinger equation”, Phys.Lett. A., 99(6-7), 279–280, 1983
  • [18] Nimmo, J.J.C., Yilmaz, H., “Binary Darboux transformation for the Sasa-Satsuma equation”, J. Phys. A., 48(42), 2015. [19] Hietarinta, J., “A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV type bilinear equations”, J. Math. Phys., 28, 1732, 2094, 1987; Hietarinta, J., “Hirota’s bilinear method and its generalization”, Inter. J. Mod. Phys. A, 12(1), 43-51, 1997.
  • [20] Hietarinta, J., “Introduction to the bilinear method”in: Kosman-Schwarzbach, Y., Tamizhmani, K.M., Grammaticos, B. (eds), Integrability of nonlinear systems. Lecture notes in physics, 638, 95-105, Springer, Berlin, 2004.
  • [21] Taniuti, T., Yajima, N., “Perturbation method for a nonlinear wave modulation I. ”, J.Math. Phys., 10, 1369-1372, 1969.
  • [22] Debnath, L., Nonlinear partial differential equations for scientists and engineers, Birkhauser, 2012.
  • [23] Zakharov, V.E., Shabat, A.B., “Exact theory of two-dimensional self-focusing and one -dimensional self-modulation of waves in nonlinear media”, Soviet Physics JETP, 34, 62 -69, 1972
  • [24] Hirota, R., “Exact envelope-soliton solutions of a nonlinear wave equation”, J. Math. Phys., 14, 805-809, 1973.
  • [25] Hasegawa, A., Tappert, F., “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion”, Appl. Phys. Lett., 23, 171-172, 1973.
  • [26] Ablowitz, M.J., Nonlinear dispersive waves. Asymptotic analysis and solitons, Cambridge University Press, 2011.
  • [27] Weiner, A.M. et al., “Experimental observation of the fundamental dark soliton in optical fibers”, Phys. Rev. Lett., 61, 2445, 1988.
  • [28] Tlidi, M., Gelens, L., “High-order dispersion stabilizes dark dissipative solitons in all-fiber cavities”, Opt. Lett., 35, 306-308, 2010.
  • [29] Carr, L.D., Brand, J., Burger, S., Sanpera, A., “Dark-soliton creation in Bose-Einstein condensates”, Phys. Rev. A, 63, 051601(R), 2001.
  • [30] Shukla, P.K., Eliasson, B., “Formation and dynamics of dark-solitons and vortices in quantum electron plasmas”, Phys. Rev.Lett., 96, 245001, 2006.
  • [31] Heidemann, R. et al., “Dissipative dark soliton in a complex plasma”, Phys. Rev.Lett., 102, 135002, 2009.
There are 29 citations in total.

Details

Primary Language English
Subjects Mathematical Physics
Journal Section Review
Authors

Barış Yapışkan 0000-0003-2783-9394

Publication Date December 31, 2022
Submission Date November 30, 2021
Acceptance Date August 25, 2022
Published in Issue Year 2022

Cite

IEEE B. Yapışkan, “HIROTA METHOD AND SOLITON SOLUTIONS”, MEJS, vol. 8, no. 2, pp. 157–172, 2022, doi: 10.51477/mejs.1029348.

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