Abstract
By using the recently discovered new invariant properties of the ansatz of R. Hirota's method, we prove that the classes of linear fractional solutions to some nonlinear equations are closed. This allows us to construct new solutions for a chosen class of dissipative equations. This algorithm is similar to the method of “dressing” the solutions of integrable equations. The equations thus obtained imply a “compatibility ” condition and are known as a nonlinear Lax pair with variable coefficients. So we propose a method for constructing such pairs. To construct solutions of a more complicated form, we propose to use the “property of zero denominators and factorized brackets,” which has been discovered experimentally. The expressions thus constructed are said to be “quasi-invariant.” They allow us to find true relations between the functions contained in the ansatz, to correct the ansatz, and to construct a solution. We present some examples of new solutions constructed following this approach. Such solutions can be used for majorizing in comparison theorems and for modeling phase processes and process in neurocomputers. A program for computing solutions by methods of computer algebra is written. These techniques supplement the classical methods for constructing solutions by using their group properties.
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REFERENCES
K. A. Volosov, “Invariant properties of the ansatz of Hirota's method for quasilinear parabolic equations,” in: International conference “Differential Equations and Related Topics.” XX Joint Session of Petrovskii Seminar and Moscow Mathematical Society, Moscow, 2001, p. 433.
K. A. Volosov, “Tools for mathematical modeling,” in: The Third International Conference, St.-Petersburg, 2001.
V. G. Danilov, V. P. Maslov, and K. A. Volosov, Mathematical Modelling of Heat and Mass Transfer Processes, Kluwer, Dordrecht-Boston-London, 1995.
V. G. Danilov and P. Yu. Subochev, “Kink solutions in the KPP-Fisher equation,” Mat. Zametki [Math. Notes], 50 (1991), no. 3, 152–154.
R. Hirota, “Exact solution of the Korteweg-de Vries equations for the multiple collisions of solitons,” J. Phys. Soc. Japan., 33 (1972), 1459.
R. Bullough and Ph. Caudry (editors), Solitons, Springer-Verlag, Heidelberg, 1980.
S. P. Novikov (editor), Waves in Active and Nonlinear Media in Applications to Electronics [Russian translation of collected papers], Mir, Moscow, 1977.
J. Goldstone and R. Jaskiw, “Quantization of nonlinear waves,” Phys. Rev. D, 11 (1975), 1486.
R. F. Dashen, B. Hasslacher, and A. Neveu, “Nonperturbative methods and extended-hadron models in field theory, I-III,” Phys. Rev. D, 10 (1974), 4114–4129, 4130–4137, 4138–4142.
M. J. Ablowitz and A. Zeppetella, “Explicit solutions of Fisher's equations for a special wave speed,” Bull. Math. Biol., 41 (1979), 835–840.
V. G. Danilov, G. A. Omel′yanov, and E. V. Radkevich, “Justification of the asymptotic solution for the phase field system and the modified Stefan problem,” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 186 (1996), no. 12, 63–80.
M. J. Ablowitz and H. Segur, Solution for Inverse Scattering Transform, SIAM, Philadelphia, 1981.
V. P. Maslov, V. G. Danilov, and K. A. Volosov, Mathematical Modelling of Heat and Mass Transfer Processes (Evolution of Dissipative Structures), With Appendix written by N. A. Kolobov, [in Russian], Nauka, Moscow, 1987.
K. A. Volosov, V. G. Danilov, and A. M. Loginov, “Exact self-similar and two-phase solutions of systems of semilinear parabolic equations,” Teoret. Mat. Fiz. [Theoret. and Math. Phys.], 101 (1994), no. 2, 189–199.
R. A. Fisher, “The wave of advance of advantageous genes,” Annals of Eugenics, 7 (1937), 355–369.
V. G. Danilov, G. A. Omel′yanov, and E. V. Radkevich, “Asymptotic solution of the phase field system and the modified Stefan problem,” Differentsial′nye Uravneniya, 31 (1995), no. 3, 483–491
V. G. Danilov, G. A. Omel′yanov, and E. V. Radkevich, “Initial data's regularization of the modified Stefan problem,” Mat. Zametki [Math. Notes], 57 (1995), no. 5, 793–795.
V. P. Maslov and G. A. Omel′yanov, “Asymptotic soliton-like solution of equations with small dispersion,” Uspekhi Mat. Nauk [Russian Math. Surveys], 36 (1981), no. 3, 63–126.
V. P. Maslov and G. A. Omel′yanov, “Hugoniot-type conditions for infinitely narrow solutions of the equation for simple waves,” Sibirsk. Mat. Zh. [Siberian Math. J.], 24 (1983), no. 5, 172–182.
K. A. Volosov, V. G. Danilov, N. A. Kolobov, and V. P. Maslov, “Localized solitory waves,” Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.], 287 (1986), no. 6, 535–538.
V. F. Zaitsev and A. D. Polyanin, Reference Book. Ordinary Differential Equations [in Russian], Fizmatlit, Moscow, 2001.
N. V. Belotelov and A. I. Lobanov, “Population models of nonlinear diffusion,” Matematicheskoe Modelirovanie, 9 (1997), no. 12, 43–56.
A. I. Lobanov and T. K. Starozhilova, “Qualitative study of the initial stage of formation of not-inequilibrium structures in a 'reaction-diffusion' type model” Matematicheskoe Modelirovanie, 9 (1997), no. 12, 23–26.
E. V. Mel′nikova, “Nonlinear dynamics of epidemic spreading,” Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinamika, 6 (1998), no. 2, 110–116.
A. I. Kozhonov, “Boundary-value problem for a class of parabolic equations arising in the description of desalination process,” Proc. Siberian Division of Academy of Sciences of the USSR. Institute for Hydrodynamics [in Russian], (1978), no. 36, 38–46.
K. A. Volosov, Invariant Properties of the Ansatz of R. Hirota' Method. New Informational Technologies [in Russian], Materials of the Fourth Seminar, MGIEM (technical university), Moscow, 2001.
V. V. Pukhnachev, “Equivalence transformations and hidden symmetry of evolution equations,” Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.], 294 (1987), no. 3, 535–538.
V. A. Galaktionov and S. A. Posashkov, “Exact solutions and invariant spaces for nonlinear equations of gradient diffusion,” Zh. Vychisl. Mat. i Mat. Fiz. [Comput. Math. and Math. Phys.], 34 (1994), 374–383.
B. H. Gilding and R. Kersner, “The characterization of reaction-convection-diffusion processes by travelling waves,” J. Differential Equations, 124 (1996), no. 1, 27–79.
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Volosov, K.A. A Property of the Ansatz of Hirota's Method for Quasilinear Parabolic Equations. Mathematical Notes 71, 339–354 (2002). https://doi.org/10.1023/A:1014846823941
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DOI: https://doi.org/10.1023/A:1014846823941