Нелокальні перетворення з додатковими змінними. Примусові симетрії

В. А. Тичинін

doi:10.37863/umzh.v74i3.6995

В. А. Тичинін Приднiпров. держ. академiя будiвництва та архiтектури, Днiпро

DOI: https://doi.org/10.37863/umzh.v74i3.6995

Ключові слова: класичні симетрії Лі, нелокальні перетворення, нелокальні симетрії, додаткові змінні, принцип нелінійної суперпозиції, формули розмноження розв’язків, перетворення Беклунда

Анотація

УДК 517.9: 519.46

Запропоновано концепцiю нелокального перетворення з додатковими змiнними, яку розроблено i застосовано для пошуку додаткових симетрiй нелiнiйних диференцiальних рiвнянь в частинних похiдних. Розглянуто можливi схеми зв’язку диференцiальних рiвнянь за допомогою продовжених нелокальних перетворень цього типу, наведено кiлька прикладiв. Метод застосовано для побудови алгоритмiв i формул розмноження розв’язкiв з вiдомих, якi використовують додатковi симетрiї. Ці формули використано для знаходження точних розв'язків деяких нелінійних рівнянь.

Посилання

L. V. Ovsiannikov, Group analysis of differential equations, Acad. Press, New York (1982).

P. J. Olver, Applications of Lie groups to differential equations, Springer-Verlag, New York (1993), https://doi.org/10.1007/978-1-4612-4350-2

G. W. Bluman, S. Kumei, Symmetries and differential equations, Appl. Math. Sci., 81, New York, Springer-Verlag (1989), https://doi.org/10.1007/978-1-4757-4307-4

G. W. Bluman, J. J. Cole, The general similarity solution of the heat equation, J. Math. Mech., № 18, 1025 – 1042 (1968/69).

W. I. Fushchich, N. I. Serov, The conditional symmetry and reduction of the nonlinear heat equation (Russian), Docl. Acad. Nauk Ukrain. Ser. A, 1990, № 7, 24 – 27 (1990).

P. J. Olver, P. Rosenau, The construction of special solutions to partial differential equations, Phys. Lett. A, 114, № 3, 107 – 112 (1986), https://doi.org/10.1016/0375-9601(86)90534-7

W. I. Fushchich, A. G. Nikitin, Simmetriya uravnenij Maksvella, Nauk. dumka, Kiev (1983).

C. Rogers, W. F. Shadwick, Backlund transformations and their applications, Acad. Press, New York-London (1982).

G. W. Bluman, G. J. Reid, S. Kumei, New classes of symmetries for partial differential equations, J. Math. Phys., 29, № 4, 806 – 811 (1988), https://doi.org/10.1063/1.527974

I. M. Anderson, N. Kamran, P. J. Olver, Internal, external, and generalized symmetries, Adv. Math., 100, № 1, 53 – 100 (1993), https://doi.org/10.1006/aima.1993.1029

W. I. Fuschych, V. A. Tychynin, Preprint No 82.33, Akad. Nauk Ukr.SSR, Inst. Math., Kiev (1982).

V. A. Tychynin, Non-local symmetry and generating solutions for Harry – Dym-type equations, J. Phys. A: Math. Gen., 27, № 13, 4549 – 4556 (1994),http://stacks.iop.org/0305-4470/27/4549

V. A. Tychynin, O. V. Petrova, O. M. Tertyshnyk, Symmetries and generation of solutions for partial differential equations, SIGMA Symmetry Integrability Geom. Methods Appl., 3, Paper 019, 0702033, 14 p. (2007), https://doi.org/10.3842/SIGMA.2007.019

V. A. Tychynin, O. V. Petrova, Nonlocal symmetries and formulae for generation of solutions for a class of diffusion–convection equations, J. Math. Anal. Appl., 382, № 1, 20 – 33 (2011), https://doi.org/10.1016/j.jmaa.2011.04.022

E. G. Reyes, Nonlocal symmetries and the Kaup – Kupershmidt equation, J. Math. Phys., 46, № 7, 073507, 19 p. (2005), https://doi.org/10.1063/1.1939988

F. Galas, New nonlocal symmetries with pseudopotentials, J. Phys. A: Math. Gen., 25, № 15, L981–L986 (1992).

A. R. Forsyth, Theory of differential equations, Vol. 5, 6, Dover Publication, N.Y. (1959).

W. F. Ames, Nonlinear partial differential equations in engineering. Vol. 1, Acad. Press, New York (1965).

N. H. Ibragimov, R. L. Anderson, Lie – Backlund tangent transformations, J. Math. Anal. Appl., 59, № 1, 145 – 162 (1977), https://doi.org/10.1016/0022-247X(77)90098-1

H. D. Wahlquist, F. B. Estabrook, Backlund transformation for solution of the Korteweg – de Vries equation, Phys. Rev. Lett., 31, № 23, 1386 – 1389 (1973), https://doi.org/10.1103/PhysRevLett.31.1386

H. D. Wahlquist, F. B. Estabrook, Prolongation structures of nonlinear evolution equations, J. Math. Phys., 16, № 1, 1 – 7 (1975), https://doi.org/10.1063/1.522396

F. B. Estabrook, Moving frames and prolongation algebras, J. Math. Phys., 23, № 11, 2071 – 2076 (1982), https://doi.org/10.1063/1.525248

F. Pirani, D. Robinson, W. F. Shadwick, Jet bundle formulation of Backlund transformations to nonlinear evolution equations, D. Reidel Publ. Co, Dordrecht (1979).

R. Hermann, The pseudopotentials of Estabrook and Wahlquist, the geometry of solutions and the theory of connections, Phys. Rev. Lett., № 36, 835 – 836 (1976), https://doi.org/10.1103/PhysRevLett.36.835

M. J. Ablowitz, H. Segur, Solitons and the inverse scattering transform, SIAM, Philadelphia (1981).

I. Sh. Akhatov, R. K. Gazizov, N. H. Ibragimov, Nonlocal symmetries, A heuristic approach (Russian), Translated in J. Soviet Math., 55, № 1, 3 – 83 (1991); Itogi Nauki i Tekhniki. Current Problems in Mathematics. Newest Results (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, № 34, 3 – 84 (1989).

J. R. King, Some non-local transformations between nonlinear diffusion equations, J. Phys. A: Math. Gen., № 23, 5441 – 5464 (1990).

N. Euler, Multipotentialisations and iterating-solution formulae: the Krichever – Novikov equation, J. Nonlinear Math. Phys., 16, suppl. 1, 93 – 106 (2009), https://doi.org/10.1142/S1402925109000340

G. W. Bluman, P. Doran-Wu, The use of factors to discover potential systems or linearizations, Acta Appl. Math., № 41, 21 – 43 (1995), https://doi.org/10.1007/BF00996104

G. W. Bluman, Nonlocal extensions of similarity methods, J. Nonlinear Math. Phys., 15, Suppl. 1, 1 – 24 (2008), https://doi.org/10.2991/jnmp.2008.15.s1.1

G. W. Bluman, A. F. Cheviakov, Nonlocally related systems, linearization and nonlocal symmetries for the nonlinear wave equation, J. Math. Anal. Appl., № 333, 93 – 111 (2007), https://doi.org/10.1016/j.jmaa.2006.10.091

G. W. Bluman, A. F. Cheviakov, N. M. Ivanova, Framework for nonlocally related partial differential equation systems and nonlocal symmetries: Extension, simplification, and examples, J. Math. Phys., № 47, 113505, 1 – 23 (2006), https://doi.org/10.1063/1.2349488

R. O. Popovych, N. M. Ivanova, Hierarchy of conservation laws of diffusion-convection equations, J. Math. Phys., № 46, 043502, 1 – 22 (2005); https://doi.org/10.1063/1.1865813.

I. S. Krasil’shchik, A. M. Vinogradov, Nonlocal trends in the geometry of differential equations: symmetries, conservation laws and Backlund transformations, Acta Appl. Math., 15, no. 1-2, 161 – 209 (1989), https://doi.org/10.1007/BF00131935

N. M. Ivanova, R. O. Popovych, C. Sophocleous, O. O. Vaneeva, Conservation laws and hierarchies of potential symmetries for certain diffusion equations, Physica A, 388, no. 4, 343 – 356 (2008), https://doi.org/10.1016/j.physa.2008.10.018

A. Clarkson, A. S. Fokas, M. J. Ablowitz, Hodograph transformations of linearizable partial differential equations, SIAM J. Appl. Math., 49, no. 4, 1188 – 1209 (1989), https://doi.org/10.1137/0149071

W. I. Fuschych, V. A. Tychynin, Exact solutions and superposition principle for nonlinear wave equation, Docl. Acad. Nauk Ukr. Ser. A, № 5, 32 – 36 (1990).

V. A. Tychynin, New nonlocal symmetries of diffusion-convection equations and their connection with generalized hodograph transformation, Symmetry, 7, № 4, 1751 – 1767 (2015); https://doi.org/10.3390/sym7041751.

W. Rzeszut, V. Vladimirov, O. M. Tertyshnyk, V. A. Tychynin, Linearizability and nonlocal superposition for

nonlinear transport equation with memory, Rep. Math. Phys., 72, № 2, 235 – 252 (2013), https://doi.org/10.1016/S0034-4877(14)60016-1

V. A. Tychynin, On construction of new exact solutions for nonlinear equations via known particular solutions (Russian), Symmetry and solutions of equations of mathematical physics (Russian), vi, Akad. Nauk Ukrain. SSR, Inst. Math., Kiev, 86 – 89 (1989).

V. A. Tychynin, Non-local symmetries and solutions for some classes of nonlinear equations of mathematical physics, Thesis. IM NASU/V. A. Tychynin (1994).

V. A. Tychynin, Adjoint solutions and superposition principle for linearizable Krichever – Novikov equation, Symmetry and Integrability of Equations of Mathematical Physics, Collection of Works of Institute of Mathematics, Kyiv, vol. 16, № 1, 181 – 192 (2019).

A. Jeffrey, Applied partial differential equations. An introduction, Acad. Press, New York (2003).

G. W. Bluman, P. Doran, Wu the use of factors to discover potential systems or linearisations, Acta Appl. Math., 41, no. 1-3, 21 – 43 (1995).

Tichinin V. A., O. N. Tertyshnik, Nelokal'noe razmnozhenie reshenij odnogo nelinejnogo telegrafnogo uravneniya, Zbirnik prac' Drugogo Vseukraїns'kogo naukovogo seminaru „Ukrains'ka shkola grupovogo analizu diferencial'nih rivnyan': zdobutki i perspektivi”,19 – 20.10, 129 – 140 (2012).

V. I. Fuschych, V. A. Tychinin, N. I. Serov, Formula razmnozheniya reshenij uravnenij Kortevega – de Friza, Ukr. mat. zhurn., 44, № 5, 716 – 719 (1992).