Abstract
The KdV hierarchy is a paradigmatic example of the rich mathematical structure underlying integrable systems and has far-reaching connections in several areas of theoretical physics. While the positive part of the KdV hierarchy is well known, in this paper we consider an affine Lie algebraic construction for its negative part. We show that the original Miura transformation can be extended to a gauge transformation that implies several new types of relations among the negative flows of the KdV and mKdV hierarchies. Contrary to the positive flows, such a “gauge-Miura” correspondence becomes degenerate whereby more than one negative mKdV model is mapped into a single negative KdV model. For instance, the sine-Gordon and another negative mKdV flow are mapped into a single negative KdV flow which inherits solutions of both former models. The gauge-Miura correspondence implies a rich degeneracy regarding solutions of these hierarchies. We obtain similar results for the generalized KdV and mKdV hierachies constructed with the affine Lie algebra \( \hat{s\ell}\left(r+1\right) \). In this case the first negative mKdV flow corresponds to an affine Toda field theory and the gauge-Miura correspondence yields its KdV counterpart. In particular, we show explicitly a KdV analog of the Tzitzéica-Bullough-Dodd model. In short, we uncover a rich mathematical structure for the negative flows of integrable hierarchies obtaining novel relations and integrable systems.
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References
R.M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Math. Phys. 9 (1968) 1202.
C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967) 1095 [INSPIRE].
C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Korteweg-de Vries equation and generalizations. VI. Methods for exact solution, Commun. Pure Appl. Math. 27 (1974) 97.
V.E. Zaharov and A.B. Sabat, Exact theory of two-dimensional selffocusing and one-dimensional selfmodulation of waves in nonlinear media, Zh. Eksp. Teor. Fiz. 61 (1971) 118 [INSPIRE].
E.K. Sklyanin and L.D. Faddeev, Quantum mechanical approach to completely integrable field theory models, Sov. Phys. Dokl. 23 (1978) 902 [INSPIRE].
E.K. Sklyanin, L.A. Takhtadzhyan and L.D. Faddeev, Quantum inverse problem method. I, Theor. Math. Phys. 40 (1979) 688.
E.K. Sklyanin, Quantum version of the method of inverse scattering problem, Zap. Nauchn. Semin. 95 (1980) 55 [INSPIRE].
V. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge University Press (1997).
P.D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math. 21 (1968) 467.
I.M. Gelfand and L.A. Dikii, Asymptotic behavior of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-de Vries equations, Russ. Math. Surveys 30 (1975) 77 [INSPIRE].
T. Miwa, M. Jimbo and E. Date, Solitons: differential equations, symmetries and infinite dimensional algebras, Cambridge University Press (2000).
V.G. Drinfeld and V.V. Sokolov, Lie algebras and equations of Korteweg-de Vries type, J. Sov. Math. 30 (1984) 1975 [INSPIRE].
A.N. Leznov and M.V. Savelev, Two-dimensional exactly and completely integrable dynamical systems (monopoles, instantons, dual models, relativistic strings, Lund Regge model, generalized Toda lattice, etc.), Commun. Math. Phys. 89 (1983) 59 [INSPIRE].
D.I. Olive and N. Turok, Local conserved densities and zero curvature conditions for Toda lattice field theories, Nucl. Phys. B 257 (1985) 277 [INSPIRE].
D.I. Olive, N. Turok and J.W.R. Underwood, Solitons and the energy momentum tensor for affine Toda theory, Nucl. Phys. B 401 (1993) 663 [INSPIRE].
D.I. Olive, N. Turok and J.W.R. Underwood, Affine Toda solitons and vertex operators, Nucl. Phys. B 409 (1993) 509 [hep-th/9305160] [INSPIRE].
O. Babelon and D. Bernard, Dressing symmetries, Commun. Math. Phys. 149 (1992) 279 [hep-th/9111036] [INSPIRE].
O. Babelon and D. Bernard, Affine solitons: a relation between tau functions, dressing and Bäcklund transformations, Int. J. Mod. Phys. A 8 (1993) 507 [hep-th/9206002] [INSPIRE].
O. Babelon, D. Bernard and M. Talon, Introduction to classical integrable systems, Cambridge University Press (2003).
H. Aratyn, L.A. Ferreira, J.F. Gomes and A.H. Zimerman, Kac-Moody construction of Toda type field theories, Phys. Lett. B 254 (1991) 372 [INSPIRE].
M.F. De Groot, T.J. Hollowood and J.L. Miramontes, Generalized Drinfeld-Sokolov hierarchies, Commun. Math. Phys. 145 (1992) 57 [INSPIRE].
T.J. Hollowood and J.L. Miramontes, Tau functions and generalized integrable hierarchies, Commun. Math. Phys. 157 (1993) 99 [hep-th/9208058] [INSPIRE].
J.L. Miramontes, Tau functions generating the conservation laws for generalized integrable hierarchies of KdV and affine Toda type, Nucl. Phys. B 547 (1999) 623 [hep-th/9809052] [INSPIRE].
L.A. Ferreira, J.L. Miramontes and J. Sanchez Guillen, Tau functions and dressing transformations for zero curvature affine integrable equations, J. Math. Phys. 38 (1997) 882 [hep-th/9606066] [INSPIRE].
A.B. Zamolodchikov, Infinite additional symmetries in two-dimensional conformal quantum field theory, Theor. Math. Phys. 65 (1985) 1205 [INSPIRE].
R. Sasaki and I. Yamanaka, Virasoro algebra, vertex operators, quantum sine-Gordon and solvable quantum field theories, Adv. Stud. Pure Math. 16 (1988) 271 [INSPIRE].
T. Eguchi and S.-K. Yang, Deformations of conformal field theories and soliton equations, Phys. Lett. B 224 (1989) 373 [INSPIRE].
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Commun. Math. Phys. 177 (1996) 381 [hep-th/9412229] [INSPIRE].
A.M. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981) 207 [INSPIRE].
M.R. Douglas, Strings in less than one-dimension and the generalized K−D−V hierarchies, Phys. Lett. B 238 (1990) 176 [INSPIRE].
D.J. Gross and A.A. Migdal, A nonperturbative treatment of two-dimensional quantum gravity, Nucl. Phys. B 340 (1990) 333 [INSPIRE].
T. Banks, M.R. Douglas, N. Seiberg and S.H. Shenker, Microscopic and macroscopic loops in nonperturbative two-dimensional gravity, Phys. Lett. B 238 (1990) 279 [INSPIRE].
E. Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys Diff. Geom. 1 (1991) 243 [INSPIRE].
M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Commun. Math. Phys. 147 (1992) 1 [INSPIRE].
R. Dijkgraaf, H.L. Verlinde and E.P. Verlinde, Loop equations and Virasoro constraints in nonperturbative 2D quantum gravity, Nucl. Phys. B 348 (1991) 435 [INSPIRE].
R. Dijkgraaf, Intersection theory, integrable hierarchies and topological field theory, in NATO ASI series, Springer, U.S.A. (1992), p. 95 [https://doi.org/10.1007/978-1-4615-3472-3_4].
C. Itzykson and J.B. Zuber, Combinatorics of the modular group. 2. The Kontsevich integrals, Int. J. Mod. Phys. A 7 (1992) 5661 [hep-th/9201001] [INSPIRE].
I. Krichever, O. Lipan, P. Wiegmann and A. Zabrodin, Quantum integrable systems and elliptic solutions of classical discrete nonlinear equations, Commun. Math. Phys. 188 (1997) 267 [hep-th/9604080] [INSPIRE].
V.V. Bazhanov and S.L. Lukyanov, Integrable structure of quantum field theory: classical flat connections versus quantum stationary states, JHEP 09 (2014) 147 [arXiv:1310.4390] [INSPIRE].
A. Alexandrov et al., Classical tau-function for quantum spin chains, JHEP 09 (2013) 064 [arXiv:1112.3310] [INSPIRE].
A. Maloney, G.S. Ng, S.F. Ross and I. Tsiares, Thermal correlation functions of KdV charges in 2D CFT, JHEP 02 (2019) 044 [arXiv:1810.11053] [INSPIRE].
A. Maloney, G.S. Ng, S.F. Ross and I. Tsiares, Generalized Gibbs ensemble and the statistics of KdV charges in 2D CFT, JHEP 03 (2019) 075 [arXiv:1810.11054] [INSPIRE].
A. Dymarsky and K. Pavlenko, Generalized Gibbs ensemble of 2d CFTs at large central charge in the thermodynamic limit, JHEP 01 (2019) 098 [arXiv:1810.11025] [INSPIRE].
A. Dymarsky and K. Pavlenko, Exact generalized partition function of 2D CFTs at large central charge, JHEP 05 (2019) 077 [arXiv:1812.05108] [INSPIRE].
A. Dymarsky and K. Pavlenko, Generalized eigenstate thermalization hypothesis in 2D conformal field theories, Phys. Rev. Lett. 123 (2019) 111602 [arXiv:1903.03559] [INSPIRE].
A. Dymarsky, A. Kakkar, K. Pavlenko and S. Sugishita, Spectrum of quantum KdV hierarchy in the semiclassical limit, JHEP 09 (2022) 169 [arXiv:2208.01062] [INSPIRE].
A. Pérez, D. Tempo and R. Troncoso, Boundary conditions for general relativity on AdS3 and the KdV hierarchy, JHEP 06 (2016) 103 [arXiv:1605.04490] [INSPIRE].
C. Erices, M. Riquelme and P. Rodríguez, BTZ black hole with Korteweg-de Vries-type boundary conditions: thermodynamics revisited, Phys. Rev. D 100 (2019) 126026 [arXiv:1907.13026] [INSPIRE].
A. Dymarsky and S. Sugishita, KdV-charged black holes, JHEP 05 (2020) 041 [arXiv:2002.08368] [INSPIRE].
D. Grumiller and W. Merbis, Near horizon dynamics of three dimensional black holes, SciPost Phys. 8 (2020) 010 [arXiv:1906.10694] [INSPIRE].
M. Cárdenas, F. Correa, K. Lara and M. Pino, Integrable systems and spacetime dynamics, Phys. Rev. Lett. 127 (2021) 161601 [arXiv:2104.09676] [INSPIRE].
M. Lenzi and C.F. Sopuerta, Darboux covariance: a hidden symmetry of perturbed Schwarzschild black holes, Phys. Rev. D 104 (2021) 124068 [arXiv:2109.00503] [INSPIRE].
M. Lenzi and C.F. Sopuerta, Black hole greybody factors from Korteweg-de Vries integrals: theory, Phys. Rev. D 107 (2023) 044010 [arXiv:2212.03732] [INSPIRE].
J.F. Gomes, A.L. Retore and A.H. Zimerman, Miura and generalized Bäcklund transformation for KdV hierarchy, J. Phys. A 49 (2016) 504003 [arXiv:1610.02303] [INSPIRE].
J.M.C. Ferreira, J.F. Gomes, G.V. Lobo and A.H. Zimmermann, Gauge Miura and Bäcklund transformations for generalized An-KdV hierarchies, J. Phys. A 54 (2021) 435201 [arXiv:2106.00741] [INSPIRE].
J.F. Gomes, G. Starvaggi Franca, G.R. de Melo and A.H. Zimerman, Negative even grade mKdV hierarchy and its soliton solutions, J. Phys. A 42 (2009) 445204 [arXiv:0906.5579] [INSPIRE].
K. Sawada and T. Kotera, A method for finding N-soliton solutions of the K.d.V. equation and K.d.V.-like equation, Prog. Theor. Phys. 51 (1974) 1355.
P.J. Caudrey, R.K. Dodd and J.D. Gibons, A new hierarchy of Korteweg-de Vries equations, Proc. Roy. Soc. Lond. A 351 (1976) 407.
A.P. Fordy and J. Gibbons, Factorization of operators. 1. Miura transformations, J. Math. Phys. 21 (1980) 2508 [INSPIRE].
J.M. Verosky, Negative powers of Olver recursion operators, J. Math. Phys. 32 (1991) 1733.
B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Physica D 95 (1996) 229.
A.N.W. Hone, The associated Camassa-Holm equation and the KdV equation, J. Phys. A 32 (1999) L307.
Z. Qiao and E. Fan, Negative-order Korteweg-de Vries equations, Phys. Rev. E 86 (2012) 016601.
R. Camassa and D.D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993) 1661 [patt-sol/9305002] [INSPIRE].
A. Degasperis, D.D. Holm and A.N.W. Hone, A new integrable equation with peakon solutions, Theor. Math. Phys. 133 (2002) 1463.
S. Burger et al., Dark solitons in Bose-Einstein condensates, Phys. Rev. Lett. 83 (1999) 5198 [cond-mat/9910487] [INSPIRE].
A.M. Weiner et al., Experimental observation of the fundamental dark soliton in optical fibers, Phys. Rev. Lett. 61 (1988) 2445.
C. Becker et al., Oscillations and interactions of dark and dark-bright solitons in Bose-Einstein condensates, Nature Phys. 4 (2008) 496.
D. Delande and K. Sacha, Many-body matter-wave dark soliton, Phys. Rev. Lett. 112 (2014) 040402.
B. Basnet et al., Soliton walls paired by polar surface interactions in a ferroelectric nematic liquid crystal, Nature Commun. 13 (2022) 3932.
J. Kopyciński, Maciej Łebek, W. Górecki and K. Pawłowski, Ultrawide dark solitons and droplet-soliton coexistence in a dipolar Bose gas with strong contact interactions, Phys. Rev. Lett. 130 (2023) 043401.
H. Aratyn, J.F. Gomes and A.H. Zimerman, Integrable hierarchy for multidimensional Toda equations and topological anti-topological fusion, J. Geom. Phys. 46 (2003) 21 [Erratum ibid. 46 (2003) 201] [hep-th/0107056] [INSPIRE].
H. Aratyn, L.A. Ferreira, J.F. Gomes and A.H. Zimerman, A new deformation of W -infinity and applications to the two loop WZNW and conformal affine Toda models, Phys. Lett. B 293 (1992) 67 [hep-th/9201024] [INSPIRE].
L.A. Ferreira, J.F. Gomes, A.H. Zimerman and A. Schwimmer, Comments on two loop Kac-Moody algebras, Phys. Lett. B 274 (1992) 65 [hep-th/9110032] [INSPIRE].
A. Fring, G. Mussardo and P. Simonetti, Form-factors of the elementary field in the Bullough-Dodd model, Phys. Lett. B 307 (1993) 83 [hep-th/9303108] [INSPIRE].
P. Dorey, S. Faldella, S. Negro and R. Tateo, The Bethe ansatz and the Tzitzeica-Bullough-Dodd equation, Phil. Trans. Roy. Soc. Lond. A 371 (2013) 20120052 [arXiv:1209.5517] [INSPIRE].
H.W. Braden, E. Corrigan, P.E. Dorey and R. Sasaki, Affine Toda field theory and exact S matrices, Nucl. Phys. B 338 (1990) 689 [INSPIRE].
A.V. Mikhailov, M.A. Olshanetsky and A.M. Perelomov, Two-dimensional generalized Toda lattice, Commun. Math. Phys. 79 (1981) 473 [INSPIRE].
I.M. Krichever and S.P. Novikov, Holomorphic bundles over algebraic curves and non-linear equations, Russ. Math. Surv. 35 (1980) 53.
B.A. Dubrovin, Theta functions and non-linear equations, Russ. Math. Surv. 36 (1981) 11.
E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Transformation groups for soliton equations. V. Quasiperiodic solutions of the orthogonal KP equation, Publ. Res. Inst. Math. Sci. 18 (1982) 1111.
D.I. Olive, Kac-Moody algebras: an introduction for physicists, in the proceedings of the Proc. Winter school “geometry and physics”. Circolo matematico di palermo, Palermo, Italy (1985), p. 177.
J.F. Cornwell, Group theory in physics, volume 3, Academic Press (1989).
Acknowledgments
We are indebted to the referee from JHEP for posing the important question about commuting flows of integrable hierarchies, besides careful analysis of the paper. JFG and AHZ thank CNPq and FAPESP for support. YFA thanks FAPESP for financial support under grant #2021/00623-4 and #2022/13584-0. GVL is supported by CAPES. GF thanks UC Berkeley, where this work was partially completed, and in particular MI Jordan for support. This research was financed in part by CAPES (finance Code 001).
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Adans, Y.F., França, G., Gomes, J.F. et al. Negative flows of generalized KdV and mKdV hierarchies and their gauge-Miura transformations. J. High Energ. Phys. 2023, 160 (2023). https://doi.org/10.1007/JHEP08(2023)160
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DOI: https://doi.org/10.1007/JHEP08(2023)160