A Two-Component Generalization of the Integrable rdDym Equation

We find a two-component generalization of the integrable case of rdDym equation. The reductions of this system include the general rdDym equation, the Boyer-Finley equation, and the deformed Boyer-Finley equation. Also we find a Bäcklund transformation between our generalization and Bodganov's...

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Datum:	2012
1. Verfasser:	Morozov, O.I.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2012
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/148387
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spelling	irk-123456789-1483872019-02-19T01:24:45Z A Two-Component Generalization of the Integrable rdDym Equation Morozov, O.I. We find a two-component generalization of the integrable case of rdDym equation. The reductions of this system include the general rdDym equation, the Boyer-Finley equation, and the deformed Boyer-Finley equation. Also we find a Bäcklund transformation between our generalization and Bodganov's two-component generalization of the universal hierarchy equation. 2012 Article A Two-Component Generalization of the Integrable rdDym Equation / O.I. Morozov // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 26 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35A30; 58H05; 58J70 DOI: http://dx.doi.org/10.3842/SIGMA.2012.051 http://dspace.nbuv.gov.ua/handle/123456789/148387 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We find a two-component generalization of the integrable case of rdDym equation. The reductions of this system include the general rdDym equation, the Boyer-Finley equation, and the deformed Boyer-Finley equation. Also we find a Bäcklund transformation between our generalization and Bodganov's two-component generalization of the universal hierarchy equation.
format	Article
author	Morozov, O.I.
spellingShingle	Morozov, O.I. A Two-Component Generalization of the Integrable rdDym Equation Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Morozov, O.I.
author_sort	Morozov, O.I.
title	A Two-Component Generalization of the Integrable rdDym Equation
title_short	A Two-Component Generalization of the Integrable rdDym Equation
title_full	A Two-Component Generalization of the Integrable rdDym Equation
title_fullStr	A Two-Component Generalization of the Integrable rdDym Equation
title_full_unstemmed	A Two-Component Generalization of the Integrable rdDym Equation
title_sort	two-component generalization of the integrable rddym equation
publisher	Інститут математики НАН України
publishDate	2012
url	http://dspace.nbuv.gov.ua/handle/123456789/148387
citation_txt	A Two-Component Generalization of the Integrable rdDym Equation / O.I. Morozov // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 26 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT morozovoi atwocomponentgeneralizationoftheintegrablerddymequation AT morozovoi twocomponentgeneralizationoftheintegrablerddymequation
first_indexed	2025-07-12T19:17:25Z
last_indexed	2025-07-12T19:17:25Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 051, 5 pages A Two-Component Generalization of the Integrable rdDym Equation? Oleg I. MOROZOV Institute of Mathematics and Statistics, University of Tromsø, Tromsø 90-37, Norway E-mail: Oleg.Morozov@uit.no Received May 26, 2012, in final form August 09, 2012; Published online August 11, 2012 http://dx.doi.org/10.3842/SIGMA.2012.051 Abstract. We find a two-component generalization of the integrable case of rdDym equa- tion. The reductions of this system include the general rdDym equation, the Boyer–Finley equation, and the deformed Boyer–Finley equation. Also we find a Bäcklund transformation between our generalization and Bodganov’s two-component generalization of the universal hierarchy equation. Key words: coverings of differential equations; Bäcklund transformations 2010 Mathematics Subject Classification: 35A30; 58H05; 58J70 1 Introduction Recent papers [3, 8, 16] provide two-component generalizations for the hyper-CR Einstein–Weil structure equation [6, 22] syy = stx + sysxx − sxsxy, (1.1) Plebański’s second heavenly equation [25] sxz = sty + sxxsyy − s2xy (1.2) and the universal hierarchy equation [18, 19, 22] sxx = sxsty − stsxy. (1.3) Namely, equations (1.1)–(1.3) appear from systems syy = stx + (sy + r)sxx − sxsxy, ryy = rtx + (sy + r)rxx − sxrxy + r2x; (1.4) sxz = sty + sxxsyy − s2xy + r, rxz = rty + syyrxx + sxxryy − 2sxyrxy, (1.5) and sxx = er(sxsty − stsxy),( e−r ) xx = sxrty − strxy, (1.6) respectively, by substituting for r = 0. Other reductions for (1.4) are found in [7, 16]: when u = 0, system (1.4) gives the Khokhlov–Zabolotskaya (or dispersionless Kadomtsev–Petviashvili) equation vyy = vtx + vvxx + v2x, ?This paper is a contribution to the Special Issue “Geometrical Methods in Mathematical Physics”. The full collection is available at http://www.emis.de/journals/SIGMA/GMMP2012.html mailto:Oleg.Morozov@uit.no http://dx.doi.org/10.3842/SIGMA.2012.051 http://www.emis.de/journals/SIGMA/GMMP2012.html 2 O.I. Morozov while substituting for v = ux in (1.4) produces the normal form uyy = utx + (ux + uy)uxx − uxuxy, for the family of equations studied in [7]. Also, we note the reduction v = uy for system (1.4). This reduction yields equation uyy = utx − uxuxy studied in [9, 14, 17, 21]. As it was shown in [3], the reduction s = x for system (1.6) gives the Boyer–Finley equation rty = ( e−r ) xx . (1.7) The purpose of the present paper is to introduce the two-component generalization for equa- tion uty = uxuxy − uyuxx, (1.8) which is integrable in the following sense: it has the differential covering [2, 11, 12, 13] pt = (ux − λ)px, py = λ−1uypx (1.9) containing the non-removable parameter λ 6= 0 [20]. We show that reductions of the generaliza- tion include the general r-th dispersionless Dym equation [1] uty = uxuxy + κuyuxx, (1.10) the Boyer–Finley equation (1.7), and the deformed Boyer–Finley equation. Also we find a Bäck- lund transformation between our generalization and Bodganov’s two-component generaliza- tion (1.6) of the universal hierarchy equation (1.3). 2 The two-component generalization Along with the covering (1.9) equation (1.8) has the covering qt = (ux − q)qx, qy = uyq −1qx, (2.1) which can be obtained by the method of [20]. While the coverings (1.9) and (2.1) are not equivalent w.r.t. the pseudo-group of contact transformations, (2.1) can be derived from (1.9) by the following procedure, see, e.g., [24]. We consider the function p = p(t, x, y) from (1.9) to be defined implicitly by the equation q(t, x, y, p(t, x, y)) = λ with qp 6= 0. Then for (x1, x2, x3) = (t, x, y) we have qxi + qppxi = 0, so pxi = −qxi/qp. Substituting these into (1.9) yields (2.1). Our main observation in this paper is that the covering (2.1) allows the generalization qt = (ux − q + v)qx + vxq, qy = uyq −1qx + vy. (2.2) This system is compatible whenever the two-component system uty = (ux + v)uxy − uyuxx, (2.3) vty = (ux + v)vxy − uyvxx + vxvy (2.4) holds. In other words, (2.2) is a covering for system (2.3), (2.4). A Two-Component Generalization of the Integrable rdDym Equation 3 3 Reductions By the construction, we have the following reduction for system (2.2): Reduction A. Substituting for v = 0 in equations (2.3), (2.2) gives equations (1.8) and (2.1), while (2.4) becomes an identity. Also, we have three other reductions. Reduction B. If we put v = −(κ−1 + 1)ux, then (2.3) gets the form uty = −κ−1uxuxy − uyuxx, (3.1) while (2.4) is its differential consequence. The transformation u 7→ −κu maps (3.1) to (1.10). The corresponding reduction of (2.2) produces the covering of (1.10) studied in [20, 23]. Reduction C. Taking v = −ux in (2.3), (2.4), we obtain uty = −uyuxx and its differential consequence. Then we divide this equation by uy, differentiate w.r.t. y and put uy = −ew. This gives the Boyer–Finley equation [4] wty = (ew)xx (3.2) This equation is equation (1.7) in a different notation. Substituting for q = ep in the corre- sponding reduction of (2.2), we have the covering [10, 15, 26] for equation (3.2): pt = wt − eppx, py = ew−p(wx − px). Reduction D. Finally, when we put v = uy − ux into (2.3) and (2.4), we get the equation uty = uy (uxy − uxx) and its differential consequence. Then for uy = ew we have the deformed Boyer–Finley equa- tion [5] wty = (ew)xy − (ew)xx , (3.3) and the corresponding reduction of equations (2.2) with q = es gives the covering st = (es − ew)sx − wt, sy = ew(sx − wx + wy). for (3.3). This covering in other notations was found in [5, 20]. 4 Bäcklund transformations The substitution ux = −v + st sx , uy = −e −r sx , vx = rxst sx − rt, vy = −e −rrx sx (4.1) maps system (2.2) to system qt = ( st sx − q ) qx + ( strx sx − rt ) q, qy = −e −r qsx (qx + rxq) (4.2) found in [3]. This system is the two-component generalization of the covering qt = ( st sx − q ) qx, qy = − qx qsx . 4 O.I. Morozov of equation (1.3). The compatibility conditions for (4.2) coincide with (1.6). Solving (4.1) for st, sx, rt, rx yields st = −(ux + v) e−r uy , sx = −e −r uy , rt = vy uy , rx = (ux + v)vy uy − vx. (4.3) This system is compatible whenever equations (2.3), (2.4) are satisfied. Thus equations (4.1) define a Bäcklund transformation from (2.3), (2.4) to (1.6) with the inverse transformation (4.3). In particular, when v = 0 and r = 0, we have a Bäcklund transformation ux = st sx , uy = − 1 sx , between (1.1) and (1.3) with the inverse transformation st = −ux uy , sx = − 1 uy . Acknowledgments I am very grateful to M.V. Pavlov and A.G. Sergyeyev for the valuable discussions. Also I’d like to thank M. Marvan and A.G. Sergyeyev for the warm hospitality in Mathematical Insti- tute, Silezian University at Opava, Czech Republic, where this work was initiated and partially supported by the ESF project CZ.1.07/2.3.00/20.0002. References [1] B laszak M., Classical R-matrices on Poisson algebras and related dispersionless systems, Phys. Lett. A 297 (2002), 191–195. [2] Bocharov A.V., Chetverikov V.N., Duzhin S.V., Khor’kova N.G., Krasil’shchik I.S., Samokhin A.V., Torkhov Y.N., Verbovetsky A.M., Vinogradov A.M., Symmetries and conservation laws for differential equa- tions of mathematical physics, Translations of Mathematical Monographs, Vol. 182, American Mathematical Society, Providence, RI, 1999. [3] Bogdanov L.V., Non-Hamiltonian generalizations of the dispersionless 2DTL hierarchy, J. Phys. A: Math. Theor. 43 (2010), 434008, 8 pages, arXiv:1003.0287. [4] Boyer C.P., Finley III J.D., Killing vectors in self-dual, Euclidean Einstein spaces, J. Math. Phys. 23 (1982), 1126–1130. [5] Dryuma V.S., Non-linear multi-dimensional equations related to commuting vector fields, in Book of ab- stracts of International Conference “Differential Equations and Related Topics” dedicated to I.G. Petrovskii, XXII Joint Session of Moscow Mathematical Society and Petrovskii Seminar (Moscow, May 21–26, 2007), Moscow Lomonosov State University, Moscow, 2007, 78. [6] Dunajski M., A class of Einstein–Weyl spaces associated to an integrable system of hydrodynamic type, J. Geom. Phys. 51 (2004), 126–137, nlin.SI/0311024. [7] Dunajski M., An interpolating dispersionless integrable system, J. Phys. A: Math. Theor. 41 (2008), 315202, 9 pages, arXiv:0804.1234. [8] Dunajski M., Anti-self-dual four-manifolds with a parallel real spinor, Proc. R. Soc. Lond. Ser. A 458 (2002), 1205–1222, math.DG/0102225. [9] Ferapontov E.V., Moro A., Sokolov V.V., Hamiltonian systems of hydrodynamic type in 2 + 1 dimensions, Comm. Math. Phys. 285 (2009), 31–65, arXiv:0710.2012. [10] Kashaev R.M., Saveliev M.V., Savelieva S.A., Vershik A.M., On nonlinear equations associated with Lie algebras of diffeomorphism groups of two-dimensional manifolds, in Ideas and Methods in Mathematical Analysis, Stochastics, and Applications (Oslo, 1988), Cambridge University Press, Cambridge, 1992, 295– 307. http://dx.doi.org/10.1016/S0375-9601(02)00421-8 http://dx.doi.org/10.1088/1751-8113/43/43/434008 http://dx.doi.org/10.1088/1751-8113/43/43/434008 http://arxiv.org/abs/1003.0287 http://dx.doi.org/10.1063/1.525479 http://dx.doi.org/10.1016/j.geomphys.2004.01.004 http://arxiv.org/abs/nlin.SI/0311024 http://dx.doi.org/10.1088/1751-8113/41/31/315202 http://arxiv.org/abs/0804.1234 http://dx.doi.org/10.1098/rspa.2001.0918 http://arxiv.org/abs/math.DG/0102225 http://dx.doi.org/10.1007/s00220-008-0522-5 http://arxiv.org/abs/0710.2012 A Two-Component Generalization of the Integrable rdDym Equation 5 [11] Krasil’shchik I.S., Lychagin V.V., Vinogradov A.M., Geometry of jet spaces and nonlinear partial differential equations, Advanced Studies in Contemporary Mathematics, Vol. 1, Gordon and Breach Science Publishers, New York, 1986. [12] Krasil’shchik I.S., Vinogradov A.M., Nonlocal symmetries and the theory of coverings: an addendum to Vinogradov’s “Local symmetries and conservation laws”, Acta Appl. Math. 2 (1984), 79–96. [13] Krasil’shchik I.S., Vinogradov A.M., Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations, Acta Appl. Math. 15 (1989), 161–209. [14] Krichever I.M., The τ -function of the universal Whitham hierarchy, matrix models and topological field theories, Comm. Pure Appl. Math. 47 (1994), 437–475, hep-th/9205110. [15] Malykh A.A., Nutku Y., Sheftel M.B., Winternitz P., Invariant solutions of complex Monge–Ampère equation and gravitational instantons, Phys. Atomic Nuclei 61 (1989), 1986–1989. [16] Manakov S.V., Santini P.M., Cauchy problem on the plane for the dispersionless Kadomtsev–Petviashvili equation, JETP Lett. 83 (2006), 462–466, nlin.SI/0604023. [17] Mañas M., Medina E., Mart́ınez Alonso L., On the Whitham hierarchy: dressing scheme, string equations and additional symmetries, J. Phys. A: Math. Gen. 39 (2006), 2349–2381, nlin.SI/0509017. [18] Mart́ınez Alonso L., Shabat A.B., Energy-dependent potentials revisited: a universal hierarchy of hydrody- namic type, Phys. Lett. A 299 (2002), 359–365, nlin.SI/0202008. [19] Mart́ınez Alonso L., Shabat A.B., Hydrodynamic reductions and solutions of the universal hierarchy, Theoret. and Math. Phys. 140 (2004), 1073–1085, nlin.SI/0312043. [20] Morozov O.I., Contact integrable extensions of symmetry pseudo-groups and coverings of (2+1) dispersion- less integrable equations, J. Geom. Phys. 59 (2009), 1461–1475, arXiv:0809.1218. [21] Pavlov M.V., Classification of integrable Egorov hydrodynamic chains, Theoret. and Math. Phys. 138 (2004), 45–58, nlin.SI/0603055. [22] Pavlov M.V., Integrable hydrodynamic chains, J. Math. Phys. 44 (2003), 4134–4156, nlin.SI/0301010. [23] Pavlov M.V., The Kupershmidt hydrodynamic chains and lattices, Int. Math. Res. Not. (2006), Art. ID 46987, 43 pages, nlin.SI/0604049. [24] Pavlov M.V., Chang J.H., Chen Y.T., Integrability of the Manakov–Santini hierarchy, arXiv:0910.2400. [25] Plebański J.F., Some solutions of complex Einstein equations, J. Math. Phys. 16 (1975), 2395–2402. [26] Zakharov V.E., Integrable systems in multidimensional spaces, in Mathematical Problems in Theoretical Physics (Berlin, 1981), Lecture Notes in Phys., Vol. 153, Springer, Berlin, 1982, 190–216. http://dx.doi.org/10.1007/BF01405492 http://dx.doi.org/10.1007/BF00131935 http://dx.doi.org/10.1002/cpa.3160470403 http://arxiv.org/abs/hep-th/9205110 http://dx.doi.org/10.1134/S0021364006100080 http://arxiv.org/abs/nlin.SI/0604023 http://dx.doi.org/10.1088/0305-4470/39/10/008 http://arxiv.org/abs/nlin.SI/0509017 http://dx.doi.org/10.1016/S0375-9601(02)00662-X http://arxiv.org/abs/nlin.SI/0202008 http://dx.doi.org/10.1023/B:TAMP.0000036538.41884.57 http://dx.doi.org/10.1023/B:TAMP.0000036538.41884.57 http://arxiv.org/abs/nlin.SI/0312043 http://dx.doi.org/10.1016/j.geomphys.2009.07.009 http://arxiv.org/abs/0809.1218 http://dx.doi.org/10.1023/B:TAMP.0000010632.20218.62 http://arxiv.org/abs/nlin.SI/0603055 http://dx.doi.org/10.1063/1.1597946 http://arxiv.org/abs/nlin.SI/0301010 http://dx.doi.org/10.1155/IMRN/2006/46987 http://arxiv.org/abs/nlin.SI/0604049 http://arxiv.org/abs/0910.2400 http://dx.doi.org/10.1063/1.522505 http://dx.doi.org/10.1007/3-540-11192-1_38 1 Introduction 2 The two-component generalization 3 Reductions 4 Bäcklund transformations References

A Two-Component Generalization of the Integrable rdDym Equation

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