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On a Novel Resonant Ermakov-NLS System: Painlevé Reduction

  • Colin Rogers
  • Wolfgang K. SchiefEmail author
Chapter

Abstract

A novel resonant Ermakov-NLS system is introduced which admits symmetry reduction to a hybrid Ermakov-Painlevé II system. If the latter is Hamiltonian then combination with a characteristic Ermakov invariant provides an algorithmic integration procedure. The latter involves the isolation of positive solutions of a concomitant integrable Painlevé XXXIV equation. Explicit expressions for a multi-parameter class of wave packet representations for the original Ermakov-NLS system are obtained via the iterated application of a Bäcklund transformation admitted by the canonical Painlevé II equation.

References

  1. 1.
    Abdullaev, Y., Desyatnikov, A.S., Ostravoskaya, E.A.: Suppression of collapse for matter waves with orbital angular momentum. J. Opt. 13, 064023 (2011)CrossRefGoogle Scholar
  2. 2.
    Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: Nonlinear evolution equations of physical significance. Phys. Rev. Lett. 31, 125–127 (1973)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Abramov, A.A., Yukhno, L.F.: A method for the numerical solution of the Painlevé equations. Comput. Math. Math. Phys. 53, 540–563 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Amster, P., Rogers, C.: On a Ermakov-Painlevé II reduction in three-ion electrodiffusion. A Dirichlet boundary value problem. Discrete Contin. Dyn. Syst. 35, 3277–3292 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bass, L.K.: Electrical structures of interfaces in steady electrolysis. Trans. Faraday Soc. 60, 1656–1669 (1964)CrossRefGoogle Scholar
  6. 6.
    Bass, L.K.: Irreversible interactions between metals and electrolytes. Proc. R. Soc. Lond. A 277, 125–136 (1964)CrossRefGoogle Scholar
  7. 7.
    Bass, L., Nimmo, J.J.C., Rogers, C., Schief, W.K.: Electrical structures of interfaces: a Painlevé II model. Proc. R. Soc. Lond. A 466, 2117–2136 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables. I and II. Phys. Rev. 85, 166–193 (1952)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bracken, A.J., Bass, L., Rogers, C.: Bäcklund flux-quantization in a model of electrodiffusion based on Painlevé II. J. Phys. A Math. Theor. 45, 105204 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Clarkson, P.A.: Painlevé equations. Nonlinear special functions. J. Comput. Appl. Math. 153, 127–140 (2003)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Conte, R. (ed.): The Painlevé Property: One Century Later. Springer, New York (1999)zbMATHGoogle Scholar
  12. 12.
    Cornolti, F., Lucchesi, M., Zambon, B.: Elliptic Gaussian beam self-focussing in nonlinear media. Opt. Commun. 75, 129–135 (1990)CrossRefGoogle Scholar
  13. 13.
    de Broglie, L.: La mécanique ondulatoire et la structure atomique de la matiére et du rayonnement. J. Phys. Radium 8, 225–241 (1927)CrossRefGoogle Scholar
  14. 14.
    Desyatnikov, A.S., Buccoliero, D., Dennis, M.R., Kivshar, Y.S.: Suppression of collapse for spiralling elliptic solitons. Phys. Rev. Lett. 104, 053902-1–053902-4 (2010)Google Scholar
  15. 15.
    Ermakov, V.P.: Second-order differential equations: conditions of complete integrability. Univ. Izy. Kiev 20, 1–25 (1880)Google Scholar
  16. 16.
    Fornberg, B., Weideman, J.A.C.: A numerical methodology for the Painlevé equations. Oxford Centre for Collaborative Applied Mathematics Report 11/06 (2011)Google Scholar
  17. 17.
    Giannini, J.A., Joseph, R.I.: The role of the second Painlevé transcendent in nonlinear optics. Phys. Lett. A 141, 417–419 (1989)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Goncharenko, A.M., Logvin, Y.A., Samson, A.M., Shapovalov, P.S., Turovets, S.I.: Ermakov Hamiltonian systems in nonlinear optics of elliptic Gaussian beams. Phys. Lett. A 160, 138–142 (1991)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Goncharenko, A.M., Logvin, Y.A., Samson, A.M., Shapovalov, P.S.: Rotating ellipsoidal Gaussian beams in nonlinear media. Opt. Commun. 81, 225–230 (1991)Google Scholar
  20. 20.
    Goncharenko, A.M., Kukushkin, V.G., Logvin, Y.A., Samson, A.M.: Self-focussing of two orthogonally polarised light beams in a nonlinear medium. Opt. Quant. Electron. 25, 97–104 (1999)CrossRefGoogle Scholar
  21. 21.
    Gradshteyn, I.S., Ryzhik, I.M.: In: Jeffrey, A., Zwillinger, D. (eds.) Table of Integrals, Series, and Products, 6th edn. Academic Press, San Diego (2000)Google Scholar
  22. 22.
    Guilano, C.R., Marburger, J.H., Yariv, A.: Enhancement of self-focussing threshold in sapphire with elliptical beams. Appl. Phys. Lett. 21, 58–60 (1972)CrossRefGoogle Scholar
  23. 23.
    Kang, J.U., Stegeman, G.I., Aitchison, J.S., Akhmediev, N.: Nonlinear pulse propagation in birefringent optical fibres. Phys. Rev. Lett. 76, 3699–3702 (1996)CrossRefGoogle Scholar
  24. 24.
    Kutuzov, V., Petnikova, V.M., Shuvalov, V.V., Vysloukh, V.A.: Cross-modulation coupling of incoherent soliton models in photorefractive crystals. Phys. Rev. E 57, 6056–6065 (1998)CrossRefGoogle Scholar
  25. 25.
    Lee, J.H., Pashaev, O.K., Rogers, C., Schief, W.K.: The resonant nonlinear Schrödinger equation in cold plasma physics: application of Bäcklund-Darboux transformations and superposition principles. J. Plasma Phys. 73, 257–272 (2007)CrossRefGoogle Scholar
  26. 26.
    Liang, Z.F., Tang, X.Y.: Painlevé analysis and exact solutions of the resonant Davey-Stewartson system. Phys. Lett. A 274, 110–115 (2009)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Makhan’kov, V.G., Pashaev, O.K.: Nonlinear Schrödinger equation with noncompact isogroup. Theor. Math. Phys. 53, 979–987 (1982)Google Scholar
  28. 28.
    Malomed, B.A.: Soliton Management in Periodic Systems. Springer, New York (2006)Google Scholar
  29. 29.
    Manakov, S.V.: On the theory of two-dimensional stationary self-focussing of electromagnetic waves. Sov. Phys. JETP 38, 248–553 (1974)Google Scholar
  30. 30.
    Mecozzi, A., Antonelli, C., Shtaif, M.: Nonlinear propagation in multi-mode fibers in the strong coupling regime (2012). arXiv: 1203.6275.v2 [physics optics]Google Scholar
  31. 31.
    Pashaev, O.K., Lee, J.H.: Resonance solitons as black holes in Madelung fluid. Mod. Phys. Lett. A 17, 1601–1619 (2002)CrossRefGoogle Scholar
  32. 32.
    Pashaev, O.K., Lee, J.H., Rogers, C.: Soliton resonances in a generalised nonlinear Schrödinger equation. J. Phys. A Math. Theor. 41, 452001 (9pp) (2008)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Ray, J.R.: Nonlinear superposition law for generalised Ermakov systems. Phys. Lett. A 78, 4–6 (1980)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Reid, J.L., Ray, J.R.: Ermakov systems, nonlinear superposition and solution of nonlinear equations of motion. J. Math. Phys. 21, 1583–1587 (1980)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Rogers, C.: Elliptic warm-core theory. Phys. Lett. A 138, 267–273 (1989)Google Scholar
  36. 36.
    Rogers, C.: A novel Ermakov-Painleve II system: N+1-dimensional coupled NLS and elastodynamic reductions. Stud. Appl. Math. 133, 214–231 (2014)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Rogers, C.: Gausson-type representations in nonlinear physics: Ermakov modulation. Phys. Scr. 89, 105208 (8pp) (2014)CrossRefGoogle Scholar
  38. 38.
    Rogers, C.: Integrable substructure in a Korteweg capillarity model. A Kármán-Tsien type constitutive relation. J. Nonlinear Math. Phys. 21, 74–88 (2014)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Rogers, C., An, H.: Ermakov-Ray-Reid systems in 2+1-dimensional rotating shallow water theory. Stud. Appl. Math. 125, 275–299 (2010)Google Scholar
  40. 40.
    Rogers, C., An, H.: On a 2+1-dimensional Madelung system with logarithmic and with Bohm quantum potentials: Ermakov reduction. Phys. Scr. 84, 045004 (7pp) (2011)CrossRefGoogle Scholar
  41. 41.
    Rogers, C., Pashaev, O.K.: On a 2+1-dimensional Whitham-Broer-Kaup system: a resonant NLS connection. Stud. Appl. Math. 127, 114–152 (2011)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Rogers, C., Schief, W.K.: Multi-component Ermakov systems: structure and linearisation. J. Math. Anal. Appl. 198, 194–220 (1996)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Rogers, C., Schief, W.K.: Intrinsic geometry of the NLS equation and its auto-Bäcklund transformation. Stud. Appl. Math. 101, 267–287 (1998)MathSciNetCrossRefGoogle Scholar
  44. 44.
    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)Google Scholar
  45. 45.
    Rogers, C., Schief, W.K.: On the integrability of a Hamiltonian reduction of a 2+1-dimensional non-isothermal rotating gas cloud system. Nonlinearity 24, 3165–3178 (2011)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Rogers, C., Schief, W.K.: The pulsrodon in 2+1-dimensional magneto-gasdynamics. Hamiltonian structure and integrability. J. Math. Phys. 52, 083701 (2011)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Rogers, C., Schief, W.K.: On Ermakov-Painlevé II systems. Integrable reduction. Meccanica 51, 2967–2974 (2016)CrossRefGoogle Scholar
  48. 48.
    Rogers, C., Hoenselaers, C., Ray, J.R.: On 2+1-dimensional Ermakov systems. J. Phys. A Math. Gen. 26, 2625–2633 (1993)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Rogers, C., Bassom, A.P., Schief, W.K.: On a Painlevé II model in steady electrolysis: application of a Bäcklund transformation. J. Math. Anal. Appl. 240, 367–381 (1999)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Rogers, C., Malomed, B., Chow, K., An, H.: Ermakov-Ray-Reid systems in nonlinear optics. J. Phys. A Math. Theor. 43, 455214 (2010)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Rogers, C., Malomed, B., An, H.: Ermakov-Ray-Reid reductions of variational approximations in nonlinear optics. Stud. Appl. Math. 129, 389–413 (2012)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Rogers, C., Yip, L.P., Chow, K.W.: A resonant Davey-Stewartson capillarity model system. Soliton generation. Int. J. Nonlinear Sci. Numer. Simul. 10, 397–405 (2009)zbMATHGoogle Scholar
  53. 53.
    Schief, W.K., Rogers, C., Bassom, A.: Ermakov systems of arbitrary order and dimension. Structure and linearisation. J. Phys. A Math. Gen. 29, 903–911 (1996)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Wagner, W.G., Haus, H.A., Marburger, J.H.: Large scale self-trapping of optical beams in the paraxial ray approximation. Phys. Rev. 175, 256–266 (1968)CrossRefGoogle Scholar
  55. 55.
    Wai, P.K.A., Menyuk, C.R., Chen, H.H.: Stability of solitons in randomly varying birefringent fibers. Opt. Lett. 16, 1231–1233 (1991)CrossRefGoogle Scholar
  56. 56.
    Zhang, J.F., Li, Y.S., Meng, J., Wo, L., Malomed, B.A.: Matter-wave solitons and finite amplitude Bloch waves in optical lattices with a spatially modulated linearity. Phys. Rev. A 82, 033614 (2010)CrossRefGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.School of Mathematics and Statistics and Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex SystemsThe University of New South WalesSydneyAustralia

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