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New integrable (3+1)-dimensional systems and contact geometry

Profile image of Artur SergyeyevArtur Sergyeyev

2018, Letters in Mathematical Physics

https://doi.org/10.1007/S11005-017-1013-4
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Abstract

We introduce a novel systematic construction for integrable (3+1)-dimensional dispersionless systems using nonisospectral Lax pairs that involve contact vector fields. In particular, we present new large classes of (3+1)-dimensional integrable dispersionless systems associated to the Lax pairs which are polynomial and rational in the spectral parameter.

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References (59)

  1. Ablowitz, M.J., Clarkson, P.A.: Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, Cambridge (1991)
  2. Adler, V.E., Shabat, A.B., Yamilov, R.I.: Symmetry approach to the integrability problem. Theor. Math. Phys. 125, no. 3, 1603-1661 (2000)
  3. Arsie, A., Lorenzoni, P.: Complex reflection groups, logarithmic connections and bi-flat F -manifolds. Lett. Math. Phys. 107, no. 10, 1919-1961 (2017) (arXiv:1604.04446)
  4. B laszak, M.: Classical R-matrices on Poisson algebras and related dispersionless systems. Phys. Lett. A 297, no. 3-4, 191-195 (2002)
  5. B laszak, M., Szablikowski, B.: Classical R-matrix theory of dispersionless systems. II. (2+1) dimension theory. J. Phys. A: Math. Gen. 35, no. 48, 10345-10364 (2002) (arXiv:nlin/0211018)
  6. Bobenko, A.I., Schief, W.K., Suris, Y.B.: On a discretization of confocal quadrics. I. An integrable systems approach. J. Int. Sys. 1, xyw005 (2015) (arXiv:1511.01777)
  7. Bogdanov, L.V., Konopelchenko, B.G.: Projective differential geometry of multidimensional dispersionless integrable hierarchies. J. Phys.: Conf. Ser. 482, 012005 (2014) (arXiv:1310.0203)
  8. Bruce, A.J., Grabowska, K., Grabowski, J.: Remarks on contact and Jacobi geometry. SIGMA 13, 059 (2017) (arXiv:1507.05405)
  9. Burtsev, S.P., Zakharov, V.E., Mikhailov, A.V.: Inverse scattering method with variable spectral parameter. Theor. Math. Phys. 70, no. 3, 227-240 (1987)
  10. Calderbank, D.M.J., Kruglikov, B.: Integrability via geometry: dispersionless differential equations in three and four dimensions. arXiv:1612.02753
  11. Calogero, F., Degasperis, A.: Spectral transform and solitons. Vol. I. Tools to solve and investigate nonlinear evolution equations. North-Holland Publishing Co., Amsterdam-New York (1982)
  12. Chvartatskyi, O., Dimakis, A., Müller-Hoissen, F.: Self-consistent sources for integrable equations via deformations of binary Darboux transformations. Lett. Math. Phys. 106, no. 8, 1139-1179 (2016) (arXiv:1510.05166)
  13. Dapić, N., Kunzinger, M., Pilipović, S.: Symmetry group analysis of weak solutions. Proc. London Math. Soc. 84, no. 3, 686-710 (2002) (arXiv:math/0104055)
  14. de León, M., Marrero J.C., Padrón E.: On the geometric prequantization of brackets. Rev. R. Acad. Cien. Serie A. Mat. 95, no. 1, 65-83 (2001)
  15. De Sole, A., Kac, V.G., Turhan, R.: A new approach to the Lenard-Magri scheme of integrability. Comm. Math. Phys. 330, no. 1, 107-122 (2014) (arXiv:1303.3438)
  16. Doktorov, E.V., Leble, S.B.: A dressing method in mathematical physics. Springer, Dordrecht (2007)
  17. Dubrovin, B.A., Novikov, S.P.: The Hamiltonian formalism of one-dimensional systems of hydrodynamic type and the Bogolyubov-Whitham averaging method. Sov. Math. Dokl. 270, no. 4, 665-669 (1983)
  18. Dunajski, M., Grant, J.D.E., Strachan, I.A.B.: Multidimensional integrable systems and deformations of Lie algebra homomorphisms. J. Math. Phys. 48, no. 9, 093502 (2007) (arXiv:nlin/0702040)
  19. Dunajski, M.: Solitons, Instantons and Twistors. Oxford Univ. Press, Oxford (2010).
  20. Dunajski, M., Ferapontov, E.V., Kruglikov, B.: On the Einstein-Weyl and conformal self-duality equations. J. Math. Phys. 56, 083501 (2015) (arXiv:1406.0018)
  21. Ferapontov, E.V., Khusnutdinova, K.R.: On integrability of (2+1)-dimensional quasilinear systems. Comm. Math. Phys. 248, 187-206 (2004) (arXiv:nlin/0305044)
  22. Ferapontov, E.V., Khusnutdinova, K.R., Klein, C.: On linear degeneracy of integrable quasilinear systems in higher dimensions. Lett. Math. Phys. 96, no. 1-3, 5-35 (2011) (arXiv:0909.5685)
  23. Ferapontov, E.V., Lorenzoni, P., Savoldi, A.: Hamiltonian operators of Dubrovin-Novikov type in 2D. Lett. Math. Phys. 105, no. 3, 341-377 (2015) (arXiv:1312.0475)
  24. Ferapontov, E.V., Moro, A., Novikov, V.S.: Integrable equations in 2+1 dimensions: deformations of dispersionless limits. J. Phys. A: Math. Theor. 42, no. 34, 345205 (2009) (arXiv:0903.3586)
  25. Fokas, A.S.: Symmetries and integrability. Stud. Appl. Math. 77, no. 3, 253-299 (1987)
  26. Fuchssteiner, B.: Mastersymmetries, higher order time-dependent symmetries and conserved densities of nonlinear evolution equations. Progr. Theoret. Phys. 70, no. 6, 1508-1522 (1983)
  27. Grundland, A.M., Sheftel, M.B., Winternitz, P.: Invariant solutions of hydrodynamic-type equations. J. Phys. A: Math. Gen. 33, 8193-8215 (2000)
  28. Guay-Paquet, M., Harnad, J.: 2D Toda τ -functions as combinatorial generating functions. Lett. Math. Phys. 105, no. 6, 827-852 (2015) (arXiv:1405.6303)
  29. Hentosh, O.E., Prykarpatsky, Y.A., Blackmore, D., Prykarpatski, A.K.: Lie-algebraic structure of Lax-Sato integrable heavenly equations and the Lagrange-d'Alembert principle. J. Geom. Phys. 120, 208-227 (2017)
  30. Khesin, B., Tabachnikov, S.: Contact complete integrability. Regul. Chaotic Dyn. 15, no. 4-5, 504-520 (2010) (arXiv:0910.0375)
  31. Kodama, Y., Gibbons, J.: A method for solving the dispersionless KP hierarchy and its exact solutions. II. Phys. Lett. A 135, no. 3, 167-170 (1989)
  32. Konopelchenko, B.G.: Introduction to multidimensional integrable equations. The inverse spectral transform in 2 + 1 dimensions. Plenum Press, New York (1992)
  33. Konopelchenko, B.G., Martínez Alonso, L.: Dispersionless scalar integrable hierarchies, Whitham hierarchy, and the quasiclassical ∂-dressing method. J. Math. Phys. 43, no. 7, 3807-3823 (2002) (arXiv:nlin/0105071)
  34. Krasil'shchik, J., Verbovetsky, A.: Geometry of jet spaces and integrable systems. J. Geom. Phys. 61, no. 9, 1633-1674 (2011) (arXiv:1002.0077)
  35. Krichever, I.M.: The dispersionless Lax equations and topological minimal models. Comm. Math. Phys. 143, 415-429 (1992)
  36. Kruglikov, B., Morozov, O.: Integrable dispersionless PDEs in 4D, their symmetry pseudogroups and deformations. Lett. Math. Phys. 105, no. 12, 1703-1723 (2015) (arXiv:1410.7104)
  37. Kushner, A., Lychagin, V., Rubtsov, V.: Contact geometry and non-linear differential equations. Cambridge Univ. Press, Cambridge (2007)
  38. Lin, C.C., Reissner, E., Tsien, H.S.: On two-dimensional non-steady motion of a slender body in a compressible fluid. J. Math. and Phys. 27, 220-231 (1948)
  39. Ma, W.-X., Bullough, R.K., Caudrey, P.J., Fushchych, W.I.: Time-dependent symmetries of variable-coefficient evolution equations and graded Lie algebras. J. Phys. A: Math. Gen. 30, no. 14, 5141-5149 (1997)
  40. Manakov, S.V., Santini, P.M.: Integrable dispersionless PDEs arising as commutation condition of pairs of vector fields. J. Phys.: Conf. Ser. 482, 012029 (2014) (arXiv:1312.2740)
  41. Manakov, S.V., Santini, P.M.: Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking. J. Phys. A: Math. Theor. 44, no. 34, 345203 (2011) (arXiv:1011.2619)
  42. Mañas, M., Martínez Alonso, L.: A hodograph transformation applicable to a large class of PDEs. Theor. Math. Phys. 137, 1544-1549 (2003)
  43. Matveev, V.B., Salle, M.A.: Darboux transformations and solitons. Springer, Berlin (1991)
  44. Matveev, V.B.: Darboux transformations, covariance theorems and integrable systems. In: L.D. Faddeev's Seminar on Mathematical Physics, pp. 179-209. Amer. Math. Soc., Providence, RI (2000)
  45. Mikhailov, A.V., Yamilov, R.I.: Towards classification of (2+1)-dimensional integrable equations. Integrability conditions. I. J. Phys. A: Math. Gen. 31, no. 31, 6707-6715 (1998)
  46. Mokhov, O.I.: Classification of nonsingular multidimensional Dubrovin-Novikov brackets. Funct. Anal. Appl. 42, no. 1, 33-44 (2008) (arXiv:math/0611785)
  47. Odesskii, A.V., Sokolov, V.V.: Integrable pseudopotentials related to generalized hypergeometric functions. Sel. Math. (N.S.) 16, 145-172 (2010) (arXiv:0803.0086)
  48. Olver, P.J.: Applications of Lie groups to differential equations. Springer, N.Y. (1993)
  49. Olver, P.J., Sanders, J.A., Wang, J.P.: Ghost symmetries. J. Nonlin. Math. Phys. 9, suppl. 1, 164-172 (2002)
  50. Previato, E.: Sigma function and dispersionless hierarchies. In: XXIX Workshop on Geometric Methods in Physics, AIP Conf. Proc., vol. 1307, pp. 140-156. Amer. Inst. Phys., Melville, New York (2010)
  51. Rosenhaus, V.: Boundary conditions and conserved densities for potential Zabolotskaya-Khokhlov equation. J. Nonlinear Math. Phys. 13, no. 2, 255-270 (2006)
  52. Sergyeyev, A.: A new class of (3+1)-dimensional integrable systems related to contact geometry. arXiv:1401.2122v3
  53. Takasaki, K., Takebe, T.: Integrable hierarchies and dispersionless limit, Rev. Math. Phys. 7, 743-808 (1995) (arXiv:hep-th/9405096)
  54. Takasaki, K.: Differential Fay identities and auxiliary linear problem of integrable hierarchies. In: Exploring new structures and natural constructions in mathematical physics, pp. 387-441. Math. Soc. Jap., Tokyo (2011) (arXiv:0710.5356)
  55. Wolf, T.: A comparison of four approaches to the calculation of conservation laws. European J. Appl. Math. 13, no. 2, 129-152 (2002) (arXiv:cs/0301027v2)
  56. Zabolotskaya, E.A., Khokhlov, R.V.: Quasi-plane waves in the nonlinear acoustics of confined beams. Sov. Phys. Acoust. 15, 35-40 (1969)
  57. Zakharov, V.E., Shabat, A.B.: Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II. Funct. Anal. Appl. 13, 166-174 (1979)
  58. Zakharov, V.E.: Multidimensional integrable systems. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), pp. 1225-1243. PWN, Warsaw (1984)
  59. Zakharov, V.E.: Dispersionless limit of integrable systems in (2+1) dimensions. In: Singular Limits of Dispersive Waves, pp. 165-174. Plenum Press, N.Y. (1994)

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