Abstract
It is shown that the relativistic invariance plays a key role in the study of integrable systems. Using the relativistically invariant sine-Gordon equation, the Tzitzeica equation, the Toda fields and the second heavenly equation as dual relations, some continuous and discrete integrable positive hierarchies such as the potential modified Korteweg-de Vries hierarchy, the potential Fordy-Gibbons hierarchies, the potential dispersionless Kadomtsev-Petviashvili-like (dKPL) hierarchy, the differential-difference dKPL hierarchy and the second heavenly hierarchies are converted to the integrable negative hierarchies including the sG hierarchy and the Tzitzeica hierarchy, the two-dimensional dispersionless Toda hierarchy, the two-dimensional Toda hierarchies and negative heavenly hierarchy. In (1+1)-dimensional cases the positive/negative hierarchy dualities are guaranteed by the dualities between the recursion operators and their inverses. In (2+1)-dimensional cases, the positive/negative hierarchy dualities are explicitly shown by using the formal series symmetry approach, the mastersymmetry method and the relativistic invariance of the duality relations. For the 4-dimensional heavenly system, the duality problem is studied firstly by formal series symmetry approach. Two elegant commuting recursion operators of the heavenly equation appear naturally from the formal series symmetry approach so that the duality problem can also be studied by means of the recursion operators.
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References
P.J. Coles, M. Berta, M. Tomamichel and S. Wehner, Entropic uncertainty relations and their applications, Rev. Mod. Phys. 89 (2017) 015002.
A. Slobozhanyuk, S.H. Mousavi, X. Ni, D. Smirnova, Y.S. Kivshar and A.B. Khanikaev, Three-dimensional all-dielectric photonic topological insulator, Nature Photon. 11 (2016) 130.
X. Dong, D. Harlow and A.C. Wall, Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality, Phys. Rev. Lett. 117 (2016) 021601 [arXiv:1601.05416] [INSPIRE].
N. Seiberg, T. Senthil, C. Wang and E. Witten, A Duality Web in 2+1 Dimensions and Condensed Matter Physics, Annals Phys. 374 (2016) 395 [arXiv:1606.01989] [INSPIRE].
A. Karch and D. Tong, Particle-Vortex Duality from 3d Bosonization, Phys. Rev. X 6 (2016) 031043 [arXiv:1606.01893] [INSPIRE].
X.N. Gao, S.Y. Lou and X.Y. Tang, Bosonization, singularity analysis, nonlocal symmetry reductions and exact solutions of supersymmetric KdV equation, JHEP 05 (2013) 029 [arXiv:1308.6695] [INSPIRE].
V.V. Bazhanov, S.L. Lukyanov and B.A. Runov, Bukhvostov-Lipatov model and quantum-classical duality, Nucl. Phys. B 927 (2018) 468 [arXiv:1711.09021] [INSPIRE].
E.S. Abers and B.W. Lee, Gauge Theories, Phys. Rept. 9 (1973) 1 [INSPIRE].
J.E. Kim, P. Langacker, M. Levine and H.H. Williams, A Theoretical and Experimental Review of the Weak Neutral Current: A Determination of Its Structure and Limits on Deviations from the Minimal SU(2) − L × U(1) Electroweak Theory, Rev. Mod. Phys. 53 (1981) 211 [INSPIRE].
S.Y. Lou and G.-J. Ni, Gaussian Effective Potential Method for SU(2) × U(1) Gauge Theory and Bounds on the Higgs Boson Mass, Phys. Rev. D 40 (1989) 3040 [INSPIRE].
ATLAS collaboration, Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1 [arXiv:1207.7214] [INSPIRE].
CMS collaboration, Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC, Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235] [INSPIRE].
P.A. Clarkson, Nonclassical symmetry reductions of the boussinesq equation, Chaos Solitons Fractals 5 (1995) 2261.
Y.-Q. Li, J.-C. Chen, Y. Chen and S.Y. Lou, Darboux transformations via lie point symmetries: KdV equation, Chin. Phys. Lett. 31 (2014) 010201.
S.-J. Liu, X.-Y. Tang and S.Y. Lou, Multiple Darboux-Bäcklund transformations via truncated Painlevé expansion and Lie point symmetry approach, Chin. Phys. B 27 (2018) 060201 [INSPIRE].
P.A. Clarkson and M.D. Kruskal, New similarity reductions of the boussinesq equation, J. Math. Phys. 30 (1989) 2201.
P.A. Clarkson and E.L. Mansfield, Algorithms for the nonclassical method of symmetry reductions, SIAM J. Appl. Math. 54 (1994) 1693.
S.Y. Lou, Similarity solutions of the kadomtsev-petviashvili equation, J. Phys. A 23 (1990) L649.
S.Y. Lou, H.-Y. Yuan, D.-F. Chen and W.-Z. Chen, Similarity reductions of the KP equation by a direct method, J. Phys. A 24 (1991) 1455 [INSPIRE].
C.W. Cao and X.G. Geng, Neumann and Bargmann systems associated with the coupled KdV soliton hierarchy, J. Phys. A 23 (1991) 4117.
Y. Cheng and Y. shen Li, The constraint of the kadomtsev-petviashvili equation and its special solutions, Phys. Lett. A 157 (1991) 22.
B. Konopelchenko, J. Sidorenko and W. Strampp, (1+1)-dimensional integrable systems as symmetry constraints of (2+1)-dimensional systems, Phys. Lett. A 157 (1991) 17.
S.Y. Lou and X.-B. Hu, Infinitely many lax pairs and symmetry constraints of the KP equation, J. Math. Phys. 38 (1997) 6401.
S.Y. Lou and R.X. Yao, Primary branch solutions of first order autonomous scalar partial differential equations via lie symmetry approach, J. Nonlin. Math. Phys. 24 (2017) 379.
P.J. Olver, Evolution Equations Possessing Infinitely Many Symmetries, J. Math. Phys. 18 (1977) 1212 [INSPIRE].
S.Y. Lou, Integrable models constructed from the symmetries of the modified KdV equation, Phys. Lett. B 302 (1993) 261 [INSPIRE].
H. Aratyn, E. Nissimov and S. Pacheva, Method of squared eigenfunction potentials in integrable hierarchies of KP type, Commun. Math. Phys. 193 (1998) 493.
S.Y. Lou and X.-B. Hu, Non-local symmetries via darboux transformations, J. Phys. A 30 (1997) L95.
X.-P. Cheng, C.-L. Chen and S.Y. Lou, Interactions among different types of nonlinear waves described by the Kadomtsev-Petviashvili equation, Wave Motion 51 (2014) 1298.
X.-B. Hu, S.Y. Lou and X.-M. Qian, Nonlocal symmetries for bilinear equations and their applications, Stud. Appl. Math. 122 (2009) 305.
S.Y. Lou, X. Hu and Y. Chen, Nonlocal symmetries related to bäcklund transformation and their applications, J. Phys. A 45 (2012) 155209.
S.Y. Lou, Conformal invariance and integrable models, J. Phys. A 30 (1997) 4803.
B. Fuchssteiner, Mastersymmetries, higher order time-dependent symmetries and conserved densities of nonlinear evolution equations, Prog. Theor. Phys. 70 (1983) 1508.
S.Y. Lou, Generalized symmetries and W(infinity) algebras in three-dimensional Toda field theory, Phys. Rev. Lett. 71 (1993) 4099 [INSPIRE].
S.Y. Lou, Negative Kadomtsev-Petviashvili hierarchy, Phys. Scripta 57 (1998) 481.
D.J. Korteweg and G. de Vries, XLI. on the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. 39 (1895) 422.
D.G. Crighton, Applications of KdV, Acta Appl. Math. 39 (1995) 39.
C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura, Method for solving the Korteweg-deVries equation, Phys. Rev. Lett. 19 (1967) 1095 [INSPIRE].
P. Minnhagen, The two-dimensional Coulomb gas, vortex unbinding, and superfluid-superconducting films, Rev. Mod. Phys. 59 (1987) 1001 [INSPIRE].
K. Sawada and T. Kotera, A method for finding N-soliton solutions of the KdV equation and KdV-like equation, Prog. Theor. Phys. 51 (1974) 1355.
P.J. Caudrey, R.K. Dodd and J.D. Gibbon, A new hierarchy of Korteweg-de Vries equations, Proc. Roy. Soc. Lond. A 351 (1976) 407.
D.J. Kaup, On the inverse scattering problem for cubic eigenvalue problems of the class ψxxx + 6Qψx + 6Rψ = λψ, Stud. Appl. Math. 62 (1980) 189.
B.A. Kupershmidt, A Super Korteweg-De Vries Equation: An Integrable System, Phys. Lett. A 102 (1984) 213 [INSPIRE].
A.P. Fordy and J. Gibbons, Some remarkable nonlinear transformations, Phys. Lett. A 75 (1980) 325.
G. Tzitzéica, Sur une nouvelle classes de surfaces, C.R. Acad. Sci. Paris 144 (1907) 1257.
G. Tzitzéica, Sur une nouvelle classes de surfaces, C.R. Acad. Sci. Paris 150 (1910) 955, 1227.
R.K. Dodd and R.K. Bullough, Polynomial Conserved Densities for the sine-Gordon Equations, Proc. Roy. Soc. Lond. A 352 (1977) 481 [INSPIRE].
A.V. Mikhailov, lntegrability ofa two-dimensional generalization of the Toda chain, JETP Lett. 30 (1979) 414.
V.E. Zakharov and A.B. Shabat, A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem, Funct. Anal. Appl. 8 (1974) 226.
M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Method For Solving The Sine-Gordon Equation, Phys. Rev. Lett. 30 (1973) 1262 [INSPIRE].
M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Nonlinear-Evolution Equations of Physical Significance, Phys. Rev. Lett. 31 (1973) 125 [INSPIRE].
A.M. Polyakov, Quark Confinement and Topology of Gauge Groups, Nucl. Phys. B 120 (1977) 429 [INSPIRE].
R. Daviet and N. Dupuis, Nonperturbative functional renormalization-group approach to the sine-Gordon model and the Lukyanov-Zamolodchikov conjecture, Phys. Rev. Lett. 122 (2019) 155301 [arXiv:1812.01908] [INSPIRE].
S. Samuel, The Grand Partition Function in Field Theory with Applications to sine-Gordon, Phys. Rev. D 18 (1978) 1916 [INSPIRE].
F. Buijnsters, A. Fasolino and M. Katsnelson, Motion of domain walls and the dynamics of kinks in the magnetic peierls potential, Phys. Rev. Lett. 113 (2014) 217202.
P. Minnhagen, A. Rosengren and G. Grinstein, Screening properties of a classical two-dimensional coulomb gas from the sine-gordon equation, Phys. Rev. B 18 (1978) 1356.
A.A. Boris, A. Rydh, T. Golod, H. Motzkau, A.M. Klushin and V.M. Krasnov, Evidence for nonlocal electrodynamics in planar josephson junctions, Phys. Rev. Lett. 111 (2013) 117002.
S.R. Coleman, The Quantum sine-Gordon Equation as the Massive Thirring Model, Phys. Rev. D 11 (1975) 2088 [INSPIRE].
S. Mandelstam, Soliton Operators for the Quantized sine-Gordon Equation, Phys. Rev. D 11 (1975) 3026 [INSPIRE].
P. Minnhagen, New renormalization equations for the kosterlitz-thouless transition, Phys. Rev. B 32 (1985) 3088.
A. Luther, Eigenvalue spectrum of interacting massive fermions in one-dimension, Phys. Rev. B 14 (1976) 2153 [INSPIRE].
R. Sasaki and R.K. Bullough, Geometric Theory of Local and Nonlocal Conservation Laws for the sine-Gordon Equation, Proc. Roy. Soc. Lond. A 376 (1981) 401 [INSPIRE].
S.Y. Lou, Abundant symmetries for the (1+1)-dimensional classical Liouville field theory, J. Math. Phys. 35 (1994) 2336 [INSPIRE].
S.Y. Lou, Soliton molecules and asymmetric solitons in three fifth order systems via velocity resonance, J. Phys. Comm. 4 (2020) 041002.
S.Y. Lou, Symmetries of the kadomtsev-petviashvili equation, J. Phys. A 26 (1993) 4387.
S.Y. Lou, Symmetry algebras of the potential Nizhnik-Novikov-Veselov model, J. Math. Phys. 35 (1994) 1755.
S.Y. Lou, Symmetries and algebras of the integrable dispersive long wave equations in (2+1)-dimensional spaces, J. Phys. A 27 (1994) 3235.
S.Y. Lou and X.-M. Qian, Generalized symmetries and algebras of the two-dimensional differential-difference toda equation, J. Phys. A 27 (1994) L641.
V.E. Zakharov, A.V. Odesskii, M. Cisternino and M. Onorato, Five-wave classical scattering matrix and integrable equations, Theor. Math. Phys. 180 (2014) 759.
J.D. Finley, III and J.F. Plebanski, The Classification of All H Spaces Admitting a Killing Vector, J. Math. Phys. 20 (1979) 1938 [INSPIRE].
M. Mineev-WEinstein, P.B. Wiegmann and A. Zabrodin, Integrable structure of interface dynamics, Phys. Rev. Lett. 84 (2000) 5106 [nlin/0001007] [INSPIRE].
Q.H. Park, Extended conformal symmetries in real heavens, Phys. Lett. B 236 (1990) 423.
M. Mañas and L.M. Aloson, A hodograph transformation which applies to the Boyer-Finley equation, Phys. Lett. A 320 (2004) 383.
C.P. Boyer and J.D. Finley, III, Killing Vectors in Selfdual, Euclidean Einstein Spaces, J. Math. Phys. 23 (1982) 1126 [INSPIRE].
R.S. Ward, Einstein-Weyl spaces and SU(infinity) Toda fields, Class. Quant. Grav. 7 (1990) L95 [INSPIRE].
S.V. Manakov and P.M. Santini, The dispersionless 2d toda equation: dressing, cauchy problem, longtime behaviour, implicit solutions and wave breaking, J. Phys. A 42 (2009) 095203.
G. Darboux, Lecons Sur la théorie générale des Surfaces. II, Gauthier-Villars, Paris, France (1888).
D. Levi and P. Winternitz, Symmetries and conditional symmetries of differential difference equations, J. Math. Phys. 34 (1993) 3713 [INSPIRE].
C. Cao, X. Geng and Y. Wu, From the special 2+1 toda lattice to the kadomtsev-petviashvili equation, J. Phys. A 32 (1999) 8059.
H.-W. Tam, X.-B. Hu and X.-M. Qian, Remarks on several 2+1 dimensional lattices, J. Math. Phys. 43 (2002) 1008.
J.F. Plebanski, Some solutions of complex Einstein equations, J. Math. Phys. 16 (1975) 2395 [INSPIRE].
S. Manakov and P. Santini, Inverse scattering problem for vector fields and the cauchy problem for the heavenly equation, Phys. Lett. A 359 (2006) 613.
S.V. Manakov and P.M. Santini, On the solutions of the second heavenly and pavlov equations, J. Phys. A 42 (2009) 404013.
B.G. Konopelchenko, W.K. Schief and A. Szereszewski, Self-dual Einstein spaces and the general heavenly equation. Eigenfunctions as coordinates, Class. Quant. Grav. 38 (2021) 045007 [arXiv:2008.07261] [INSPIRE].
M. Dunajski and L.J. Mason, HyperKähler hierarchies and their twistor theory, Commun. Math. Phys. 213 (2000) 641 [math/0001008] [INSPIRE].
M. Dunajski and L.J. Mason, Twistor theory of hyperKähler metrics with hidden symmetries, J. Math. Phys. 44 (2003) 3430 [math/0301171] [INSPIRE].
F. Neyzi, Y. Nutku and M.B. Sheftel, Multi-Hamiltonian structure of Plebanski’s second heavenly equation, J. Phys. A 38 (2005) 8473 [nlin/0505030] [INSPIRE].
A.P. Fordy and J. Gibbons, Integrable Nonlinear Klein-Gordon Equations And Toda Lattices, Commun. Math. Phys. 77 (1980) 21 [INSPIRE].
K. Pohlmeyer, Integrable Hamiltonian Systems and Interactions Through Quadratic Constraints, Commun. Math. Phys. 46 (1976) 207 [INSPIRE].
F. Lund and T. Regge, Unified approach to strings and vortices with soliton solutions, Phys. Rev. D 14 (1976) 1524.
B.S. Getmanov, New Lorentz-invariant system with exact multisoliton solutions, JETP Lett. 25 (1977) 119.
V.E. Zakharov and A.V. Mikhailov, Relativistically Invariant Two-Dimensional Models in Field Theory Integrable by the Inverse Problem Technique (in Russian), Sov. Phys. JETP 47 (1978) 1017 [INSPIRE].
A.V. Mikhailov, Integrability of the two-dimensional Thirring model, JETP Lett. 23 (1976) 320.
R.S. Ward, Ansatze for Selfdual Yang-Mills Fields, Commun. Math. Phys. 80 (1981) 563 [INSPIRE].
B. Doubrov, E.V. Ferapontov, B. Kruglikov and V.S. Novikov, On a class of integrable systems of Monge-Ampère type, J. Math. Phys. 58 (2017) 063508 [arXiv:1701.02270] [INSPIRE].
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Lou, S.Y., Hu, X.B. & Liu, Q.P. Duality of positive and negative integrable hierarchies via relativistically invariant fields. J. High Energ. Phys. 2021, 58 (2021). https://doi.org/10.1007/JHEP07(2021)058
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DOI: https://doi.org/10.1007/JHEP07(2021)058