Ricerche di Matematica

, Volume 68, Issue 2, pp 503–512 | Cite as

The Gross–Pitaevskii equation: Bäcklund transformations and admitted solutions

  • Sandra CarilloEmail author
  • Federico Zullo


Bäcklund transformations are applied to study the Gross–Pitaevskii equation. Supported by previous results, a class of Bäcklund transformations admitted by this equation are constructed. Schwarzian derivative as well as its invariance properties turn out to represent a key tool in the present investigation. Examples and explicit solutions of the Gross–Pitaevskii equation are obtained.


Nonlinear ordinary differential equations Gross–Pitaevskii equation Bäcklund transformations Schwarzian derivative 

Mathematics Subject Classification

35A24 49K15 37K35 35Q55 



The financial support of GNFM-INdAM, INFN, Sapienza Università di Roma and Unversità di Brescia are gratefully acknowledged.


  1. 1.
    Borovkova, O., Lobanov, V.E., Malomed, B.A.: Solitons supported by singular spatial modulation of the Kerr nonlinearity. Phys. Rev. A 85, 023845 (2012). CrossRefGoogle Scholar
  2. 2.
    Carillo, S.: A novel Bäcklund invariance of a nonlinear differential equation. J. Math. Anal. Appl. 252(2), 828–839 (2000). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Carillo, S., Lo Schiavo, M., Schiebold, C.: Bäcklund transformations and non Abelian nonlinear evolution equations: a novel Bäcklund chart. Symmetry Integrability Geom. Methods Appl. (SIGMA) 12, 087 (2016). (17 pages)CrossRefzbMATHGoogle Scholar
  4. 4.
    Carillo, S., Lo Schiavo, M., Porten, E., Schiebold, C.: A novel noncommutative KdV-type equation, its recursion operator, and solitons. J. Math. Phys. 59(3), 3053–3060 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Carillo, S., Zullo, F.: Ermakov-Pinney and Emden-Fowler equations: new solutions from novel Bäcklund transformations. Theor. Math. Phys. 196(3), 1268–1281 (2018)CrossRefGoogle Scholar
  6. 6.
    Ermakov, V.: Second order differential equations. Conditions of complete integrability, Universita Izvestia Kiev Series III 9, (1880), 1–25. English translation: A.O. Harin, Redactor: P.G.L. Leach, Applied Analysis and Discrete Mathematics, 2, 123–145 (2008)Google Scholar
  7. 7.
    Fuchssteiner, B., Carillo, S.: The action-angle transformation for soliton equations. Physica A 166, 651–676 (1990). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gbutzmann, S., Ritschel, U.: Analytic solution of Emden-Fowler equation and critical adsorption in spherical geometry. Z. Phys. B 96, 391–393 (1995). CrossRefGoogle Scholar
  9. 9.
    Goenner, H., Havas, P.: Exact solutions of the generalized Lane–Emden equation. J. Math. Phys. 41, 7029 (2000). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gu, C., Hu, H., Zhou, Z.: Darboux Transformations in Integrable Systems. Theory and Their Applications to Geometry. Mathematical Physics Studies, vol. 26. Springer, Dordrecht (2005)zbMATHGoogle Scholar
  11. 11.
    Güngör, F., Torres, P.J.: Lie point symmetry analysis of a second order differential equation with singularity. J. Math. Anal. Appl. 451(2), 976–989 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hawkins, R.L., Lidsey, J.E.: Ermakov-Pinney equation in scalar field cosmologies. Phys. Rev. D 66, 023523 (2002). MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hertl, E.: Spherically symmetric nonstatic perfect fluid solutions with shear. Gen. Relativ. Gravit. 28(8), 919–934 (1996). MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kivshar, Y.S., Agrawal, G.P.: Optical Solitons. From Fibers to Photonic Crystals. Academic Press, San Diego (2003)Google Scholar
  15. 15.
    Major, F., Gheorghe, V.N., Werth, G.: Charged Particle Traps—Physics and Techniques of Charged Particle Field Confinement. Springer, Berlin (2005)Google Scholar
  16. 16.
    Moser, J., Veselov, A.P.: Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun. Math. Phys. 139, 217–243 (1991). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Nicolin, A.I.: Resonant wave formation in Bose–Einstein condensates. Phys. Rev. E 84, 056202 (2011). CrossRefGoogle Scholar
  18. 18.
    Ragnisco, O., Zullo, F.: Bäcklund transformations for the trigonometric Gaudin magnet. Symmetry Integrability Geom. Methods Appl. (SIGMA) 6, 012 (2010). CrossRefzbMATHGoogle Scholar
  19. 19.
    Ragnisco, O., Zullo, F.: Bäcklund transformations as exact integrable time discretizations for the trigonometric Gaudin model. J. Phys. A 43, 434029 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ragnisco, O., Zullo, F.: Bäcklund transformation for the Kirchhoff top. Symmetry Integrability Geom. Methods Appl. (SIGMA) 7, 001 (2011). (13 pages)CrossRefzbMATHGoogle Scholar
  21. 21.
    Rogers, C., Schief, W.K.: Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)CrossRefGoogle Scholar
  22. 22.
    Rogers, C.: Multi-component Ermakov and non-autonomous many-body system connections. Ricerche mat (2018). MathSciNetCrossRefGoogle Scholar
  23. 23.
    Rogers, C.: On hybrid Ermakov-Painlevé systems. Integrable reduction. J. Nonlinear Math. Phys. 24(2), 239–249 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Rogers, C., Schief, W.K.: On Ermakov-Painlevé II systems. Integrable reduction. Meccanica 51(12), 2967–2974 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Torres, P.: Modulated amplitude waves with non-trivial phase in quasi-1D inhomogeneous Bose–Einstein condensates. Phys. Lett. A 278(45), 3285–3288 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Torrontegui, E., Ibáñez, S., Chen, X., Ruschhaupt, A., Guéry-Odelin, D., Muga, J.G.: Fast atomic transport without vibrational heating. Phys. Rev. A 83, 013415 (2011). CrossRefGoogle Scholar
  27. 27.
    Vergel, D.G., Villasenor, E.J.S.: The time-dependent quantum harmonic oscillator revisited: applications to quantum field theory. Ann. Phys. 324, 1360–1385 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wu, Y., Xie, Q., Zhong, H., Wen, L., Hai, W.: Algebraic bright and vortex solitons in self-defocusing media with spatially inhomogeneous nonlinearity. Phys. Rev. A 87, 055801 (2013). CrossRefGoogle Scholar
  29. 29.
    Zullo, F.: Bäcklund transformations and Hamiltonian flows. J. Phys. A 46(14), 145203 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Zullo, F.: Bäcklund transformations for the elliptic Gaudin model and a Clebsch system. J. Math. Phys. 52, 073507 (2011). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2018

Authors and Affiliations

  1. 1.Dipartimento Scienze di Base e Applicate per l’IngegneriaSapienza Università di RomaRomeItaly
  2. 2.I.N.F.N. - Sezione Roma1 Gr. IV - Mathematical Methods in NonLinear PhysicsRomeItaly
  3. 3.DICATAM, Universitá di BresciaBresciaItaly

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