Solitons and Hirota's bilinear method

Recent conference talk

Hirota's bilinear method and integrability (Bangalore, 18-29 February, 2008) (downloadable pdf-file)

Introduction to Hirota's bilinear method (in conference proceedings)

  • J.Hietarinta: Introduction to the bilinear method, in `` Integrability of Nonlinear Systems'', eds. Y. Kosman-Schwarzbach, B. Grammaticos and K.M. Tamizhmani, Lecture Notes in Physics Vol. 638 (Springer, 2004) pp. 95-103.
  • J.Hietarinta: Bäcklund transformations from the bilinear viewpoint, CRM Proceedings and Lecture Notes 29 245-251 (2001).
  • J.Hietarinta: Hirota's bilinear method and its generalization, Int. J. Mod. Phys. A 12, 43-51 (1997).
  • J.Hietarinta: Hirota's bilinear method and partial integrability, in ``Partially Integrable Equations in Physics'', eds. R. Conte and N. Boccara, (Kluwer, 1990), pp. 459-478.

    • Soliton solutions

  • C. Gilson, J. Hietarinta, J. Nimmo and Y. Ohta: Sasa-Satsuma higher-order nonlinear Schrödinger equation and its bilinearization and multisoliton solutions, Phys. Rev. E 68, 016614 (2003)
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  • J. Hietarinta: Comment on `Exact solutions of the two-dimensional Burgers equation', J. Phys. A: Math. Gen. 33 5157-5158 (2000).
  • R. Radhakrishnan, M. Lakshmanan and J. Hietarinta: Inelastic collision and switching of coupled bright solitons in optical fibers, Phys. Rev. E 56, 2213-2216 (1997).
  • J. Hietarinta, A. Ramani and B. Grammaticos: Continuous vacua in bilinear soliton equations, J. Phys. A: Math. Gen. 27, 3149-3158 (1994).

    • Dromions

  • J. Hietarinta: One-dromion solutions for generic classes of equations, Phys. Lett. A 149, 113-118 (1990).
  • J. Hietarinta and R. Hirota: Multidromion solutions to the Davey-Stewartson equation, Phys. Lett. A 145, 237-244 (1990).
  • J.Hietarinta: On the dromion masses, in ``Nonlinear Evolution Equations and Dynamical Systems, Proceedings NEEDS'91'', eds. M. Boiti, P. Martina and F. Pempinelli, (World Scientific, 1992), pp. 23-29.
  • J.Hietarinta: From an analytical formula to a movie by way of REDUCE and C, in ``Proceedings of the Workshop on Symbolic and Numeric Computation'', eds. H. Apiola, M. Laine and E. Valkeila, (Computing Centre, University of Helsinki, 1991) Research Reports N:o 17, pp. 117-126.
  • J.Hietarinta: 2+1 Dimensional Dromions and Hirota's Bilinear Method, in ``Solitons and Chaos'', eds. I. Antoniou and F.J. Lambert, (Springer, 1991), pp. 321-324.
  • J.Hietarinta: Dromion Solutions for Generic NLS- and KdV- Type Equations, in ``Nonlinear Evolution Equations and Dynamical Systems, NEEDS'90'', eds. V.G. Makhankov and O.K. Pashaev, (Springer, 1991), pp. 83-85.

    • The three-soliton condition as a search method

  • B. Grammaticos, A. Ramani and J. Hietarinta: A search for integrable bilinear equations: the Painlevé approach, J. Math. Phys. 31, 2572-2578 (1990).
  • J. Hietarinta: A search of bilinear equations passing Hirota's three-soliton condition: IV. Complex bilinear equations, J. Math. Phys. 29, 628-635 (1988).
  • J. Hietarinta: A search of bilinear equations passing Hirota's three-soliton condition: III. sine-Gordon-type bilinear equations, J. Math. Phys. 28, 2586-2592 (1987).
  • J. Hietarinta: A search of bilinear equations passing Hirota's three-soliton condition: II. mKdV-type bilinear equations, J. Math. Phys. 28, 2094-2101 (1987).
  • J. Hietarinta: A search of bilinear equations passing Hirota's three-soliton condition: I. KdV-type bilinear equations, J. Math. Phys. 28, 1732-1742 (1987).
  • J.Hietarinta: Searching for integrable PDE's by testing Hirota's three-soliton condition, in ``Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation, ISSAC'91'', ed. Stephen M. Watt, (Association for Computing Machinery, 1991), pp. 295-300.
  • J.Hietarinta: Equations That Pass Hirota's Three-Soliton Condition and Other Tests of Integrability, in ``Nonlinear Evolution Equations and Dynamical Systems'', eds. S. Carillo and O. Ragnisco, (Springer, 1990), pp. 46-50.
  • J.Hietarinta: Recent results from the search for bilinear equations having three-soliton solutions, in ``Nonlinear evolution equations: integrability and spectral methods'', eds. A. Degasperis, A.P. Fordy and M. Lakshmanan, (Manchester U.P., 1990) pp. 307-317.

    • Trilinear equations

  • J.Hietarinta: Gauge symmetry and the generalization of Hirota's bilinear method, J. Nonlin. Math. Phys. 3, 260-265 (1996).
  • B. Grammaticos, A. Ramani and J. Hietarinta: Multilinear operators: the natural extension of Hirota's bilinear formalism, Phys. Lett. A 190, 65-70 (1994).
  • J. Satsuma, K. Kajiwara, J. Matsukidaira and J. Hietarinta: Solution to the Broer-Kaup system through its trilinear form, J. Phys. Soc. Jpn. 61, 3096-3102 (1992).
  • J. Hietarinta and J. Satsuma: The trilinear equation as a 2+2 dimensional extension of the 1+1 dimensional relativistic Toda lattice, Phys. Lett. A 161, 267-273 (1991).
  • J.Hietarinta, B. Grammaticos and A. Ramani: Integrable Trilinear PDE's, in ``Nonlinear Evolution Equations & Dynamical Systems, NEEDS '94'', eds. V.G. Makhankov, A.R. Bishop and D.D. Holm, (World Scientific, 1995), pp. 54-63.
  • J.Hietarinta, K. Kajiwara, J. Matsukidaira and J. Satsuma: The Relativistic Toda Lattice and Its Trilinear Form, in ``Nonlinear Evolution Equations and Dynamical Systems, Proceedings NEEDS'91'', eds. M. Boiti, P. Martina and F. Pempinelli, (World Scientific, 1992), pp. 30-43.

    • Solitons in the dissipative Toda-lattice

  • J. Hietarinta, T. Kuusela and B. Malomed: Shock waves in the dissipative Toda lattice, J. Phys. A: Math. Gen. 28, 3015-3024 (1995).
  • T. Kuusela, J. Hietarinta and B.A. Malomed: Numerical study of solitons in the damped AC-driven Toda lattice, J. Phys. A: Math. Gen. 26, L21-L26 (1993).
  • T. Kuusela and J. Hietarinta: Numerical, experimental and analytical studies of the dissipative Toda lattice. II. Elastic scattering of solitary waves, Physica D 46, 1-9 (1990).
  • T. Kuusela and J. Hietarinta: Numerical, experimental and analytical studies of the dissipative Toda lattice. I. The behaviour of a single solitary wave, Physica D 41, 322-340 (1990).
  • T. Kuusela and J. Hietarinta: Elastic scattering of solitary waves in the strongly dissipative Toda lattice, Phys. Rev. Lett. 62, 700-703 (1989).
  • T. Kuusela and J. Hietarinta: Nonlinear electrical transmission line as a burst generator, Rev. Sci. Instrum. 62, 2266-2270 (1991).
  • T. Kuusela and J.Hietarinta: Quantitative study of the dissipative Toda lattice, preliminary results, RIMS Kokyuroku 650, 29-45 (1988).
  • T. Kuusela, J. Hietarinta, K. Kokko and R. Laiho: Soliton experiments in a nonlinear electrical transmission line, Eur. J. Phys. 8, 27-33 (1987).