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HIROTA METHOD AND SOLITON SOLUTIONS

Year 2022, Volume: 8 Issue: 2, 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, Volume: 8 Issue: 2, 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 Volume: 8 Issue: 2

Cite

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