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Gauge Equivalence among Quantum Nonlinear Many Body Systems

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Abstract

Transformations performing on the dependent and/or the independent variables are an useful method used to classify PDE in class of equivalence. In this paper we consider a large class of U(1)-invariant nonlinear Schrödinger equations containing complex nonlinearities. The U(1) symmetry implies the existence of a continuity equation for the particle density ρ≡|ψ|2 where the current j ψ has, in general, a nonlinear structure. We introduce a nonlinear gauge transformation on the dependent variables ρ and j ψ which changes the evolution equation in another one containing only a real nonlinearity and transforms the particle current j ψ in the standard bilinear form. We extend the method to U(1)-invariant coupled nonlinear Schrödinger equations where the most general nonlinearity is taken into account through the sum of an Hermitian matrix and an anti-Hermitian matrix. By means of the nonlinear gauge transformation we change the nonlinear system in another one containing only a purely Hermitian nonlinearity. Finally, we consider nonlinear Schrödinger equations minimally coupled with an Abelian gauge field whose dynamics is governed, in the most general fashion, through the Maxwell-Chern-Simons equation. It is shown that the nonlinear transformation we are introducing can be applied, in this case, separately to the gauge field or to the matter field with the same final result. In conclusion, some relevant examples are presented to show the applicability of the method.

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References

  1. Ablowitz, M.J., Benney, D.J.: Evolution of multi-phase modes for nonlinear dispersive waves. Stud. Appl. Math. 49, 225–238 (1979)

    Google Scholar 

  2. Aglietti, U., Griguolo, L., Jackiw, R., Pi, S.-Y., Seminara, D.: Anyons and chiral solitons on a line. Phys. Rev. Lett. 77, 4406–4409 (1996)

    Article  Google Scholar 

  3. Agrawal, G.P.: Modulation instability induced by cross-phase modulation. Phys. Rev. Lett. 59, 880–883 (1987)

    Article  Google Scholar 

  4. Barashenkov, I., Harin, A.: Nonrelativistic Chern-Simons theory for the repulsive Bose gas. Phys. Rev. Lett. 72, 1575–1579 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. Berkhoer, A.L., Zakharov, V.E.: Self excitation of waves with different polarizations in nonlinear media. Z. Eksp. Teor. Fiz. 58, 903–911 (1970) [Sov. Phys. JETP 31, 486–490 (1970)]

    Google Scholar 

  6. Bialynicki-Birula, I., Mycielski, J.: Nonlinear wave mechanics. Ann. Phys. (NY) 100, 62–93 (1976)

    Article  MathSciNet  Google Scholar 

  7. Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables. Phys. Rev. 85, 166–193 (1951)

    Article  MathSciNet  Google Scholar 

  8. Calogero, F., Degasperis, A., De Lillo, S.: The multicomponent Eckhaus equation. J. Phys. A: Math. Gen. 30, 5805–5814 (1997)

    Article  MATH  Google Scholar 

  9. Calogero, F.: Universal C-integrable nonlinear partial-differential equation in n+1 dimensions. J. Math. Phys. 34, 3197–3209 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  10. Calogero, F.: C-integrable nonlinear partial-differential equations in n+1 dimensions. J. Math. Phys. 33, 1257–1271 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  11. Calogero, F., Xiaoda, J.: C-integrable nonlinear PDES. 2. J. Math. Phys. 32, 875–887 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  12. Calogero, F., Xiaoda, J.: C-integrable nonlinear PDES. 2. J. Math. Phys. 32, 2703–2717 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  13. Calogero, F., De Lillo, S.: The Eckhaus PDE i ψ t +ψ xx +2(|ψ|2) x ψ+|ψ|4=0. Inverse Probl. 3, 633–681 (1987). Corrigendum: Inverse Probl. 4, 571 (1988)

    Article  Google Scholar 

  14. Chen, H.H., Lee, Y.C., Liu, C.S.: Integrability of non-linear Hamiltonian-systems by inverse scattering method. Phys. Scr. 20, 490–492 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  15. Dodonov, V.V., Mizrahi, S.S.: Generalized nonlinear Doebner-Goldin Schrödinger equation and the relaxation of quantum-systems. Physica A 214, 619–628 (1995)

    Article  Google Scholar 

  16. Doebner, H.-D., Zhdanov, R.: Nonlinear Dirac equations and nonlinear gauge transformations (2003). arXiv:quant-ph/0304167

  17. Doebner, H.-D., Goldin, G.A., Nettermann, P.: Properties of nonlinear Schrödinger equations associated with diffeomorphism group-representations. J. Math. Phys. 40, 49 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Doebner, H.-D., Goldin, G.A.: Introducing nonlinear gauge transformations in a family of nonlinear Schrödinger equations. Phys. Rev. A 54, 3764–3771 (1996)

    Article  MathSciNet  Google Scholar 

  19. Doebner, H.-D., Goldin, G.A.: Properties of nonlinear Schrödinger-equations associated with diffeomorphism group-representations. J. Phys. A: Math. Gen. 27, 1771–1780 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  20. Doebner, H.-D., Goldin, G.A.: On a general nonlinear Schrödinger equation admitting diffusion currents. Phys. Lett. A 162, 397–401 (1992)

    Article  MathSciNet  Google Scholar 

  21. Fermi, E.: Rend. R. Accad. Naz. Lincei 5, 795 (1955)

    Google Scholar 

  22. Feynmann, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)

    Google Scholar 

  23. Florjańczyk, M., Gagnon, L.: Dispersive-type solutions for the Eckhaus equation. Phys. Rev. A 45, 6881–6883 (1992)

    Article  Google Scholar 

  24. Florjańczyk, M., Gagnon, L.: Exact-solutions for a higher-order nonlinear Schrödinger equation. Phys. Rev. A 41, 4478–4485 (1990)

    Article  Google Scholar 

  25. Fordy, A.P.: Derivative nonlinear Schrödinger equations and hermitian symmetric-spaces. J. Phys. A: Math. Gen. 17, 1235–1245 (1984)

    Article  MathSciNet  Google Scholar 

  26. Gedalin, M., Scott, T.C.: Optical solitary waves in the higher order nonlinear Schrödinger equation, Band Y.B. Phys. Rev. Lett. 78, 448–451 (1997)

    Article  Google Scholar 

  27. Ginzburg, V., Pitaevskii, L.: On the theory of superfluidity. Z. Eksp. Theor. Fiz. 34, 1240–1245 (1958) [Sov. Phys. JETP 7, 858–861 (1958)]

    MathSciNet  Google Scholar 

  28. Gisin, L.: Microscopic derivation of a class of non-linear dissipative Schrödinger-like equations. Physica A 111, 364–370 (1961)

    Article  MathSciNet  Google Scholar 

  29. Goldin, G.A.: The diffeomorphism group-approach to nonlinear quantum-systems. Int. J. Mod. Phys. B 6, 1905–1916 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  30. Goldin, G.A., Menikoff, R., Sharp, D.H.: Diffeomorphism-groups, gauge groups, and quantum-theory. Phys. Rev. Lett. 51, 2246–2249 (1983)

    Article  MathSciNet  Google Scholar 

  31. Grigorenko, A.N.: Measurement description by means of a nonlinear Schrödinger equation. J. Phys. A: Math. Gen. 28, 1459–1466 (1995)

    Article  MATH  Google Scholar 

  32. Gross, E.P.: Hydrodynamics of a superfluid condensate. J. Math. Phys. 4, 195–207 (1963)

    Article  Google Scholar 

  33. Gross, E.P.: Structure of a quantized vortex in boson systems. Nuovo Cimento 20, 454–477 (1961)

    Article  MATH  Google Scholar 

  34. Guerra, F., Pusterla, M.: A nonlinear Schrödinger equation and its relativistic generalization from basic principles. Lett. Nuovo Cimento 34, 351–356 (1982)

    Article  MathSciNet  Google Scholar 

  35. Hacinliyan, I., Erbay, S.: Coupled quintic nonlinear Schrödinger equations in a generalized elastic solid. J. Phys. A: Math. Gen. 37, 9387–9401 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  36. Hasegawa, A., Kodama, Y.: Solitons in optical communication. Oxford University Press, London (1995)

    Google Scholar 

  37. Hasegawa, A., Tappert, F.D.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Appl. Phys. Lett. 23, 142–144 (1973)

    Article  Google Scholar 

  38. Hisakado, M., Wadati, M.: Integrable multicomponent hybrid nonlinear Schrödinger equations. J. Phys. Soc. Jpn. 64, 408–413 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  39. Hisakado, M., Iizuka, T., Wadati, M.: Coupled hybrid nonlinear Schrödinger equation and optical solitons. J. Phys. Soc. Jpn. 63, 2887–2894 (1994)

    Article  Google Scholar 

  40. Ho, T.-L.: Spinor Bose condensates in optical traps. Phys. Rev. Lett. 81, 742–745 (1998)

    Article  Google Scholar 

  41. Jackiw, R.: A nonrelativistic chiral soliton in one dimension. J. Nonlin. Math. Phys. 4, 261–270 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  42. Jackiw, R., Pi, S.-Y.: Self-dual Chern-Simons solitons. Prog. Theor. Phys. Suppl. 107, 1–40 (1992)

    Article  MathSciNet  Google Scholar 

  43. Jackiw, R., Pi, S.-Y.: Classical and quantal nonrelativistic Chern-Simons theory. Phys. Rev. D 42, 3500–3513 (1990). Corrigendum: Phys. Rev. D 42, 3929–3929 (1993)

    Article  MathSciNet  Google Scholar 

  44. Karpman, V.I., Rasmussen, J.J., Shagalov, A.G.: Dynamics of solitons and quasisolitons of the cubic third-order nonlinear Schrödinger equation. Phys. Rev. E 64, 026614–13 (2001)

    Article  Google Scholar 

  45. Karpman, V.I., Shagalov, A.G.: Evolution of solitons described by the higher-order nonlinear Schrödinger equation. II. Numerical investigation. Phys. Lett. A 254, 319–324 (1999)

    Article  Google Scholar 

  46. Karpman, V.I.: Evolution of solitons described by higher-order nonlinear Schrödinger equations. Phys. Lett. A 244, 397–400 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  47. Karpman, V.I.: Radiation by solitons due to higher-order dispersion. Phys. Rev. E 47, 2073–2082 (1993)

    Article  MathSciNet  Google Scholar 

  48. Kaniadakis, G., Scarfone, A.M.: Nonlinear Schrödinger equations within the Nelson quantization picture. Rep. Math. Phys. 51, 225–231 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  49. Kaniadakis, G., Miraldi, E., Scarfone, A.M.: Cole-Hopf like transformation for a class of coupled nonlinear Schrödinger equations. Rep. Math. Phys. 49, 203–209 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  50. Kaniadakis, G., Scarfone, A.M.: Cole-Hopf-like transformation for Schrödinger equations containing complex nonlinearities. J. Phys. A: Math. Gen. 35, 1943–1959 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  51. Kaniadakis, G., Scarfone, A.M.: Nonlinear transformation for a class of gauged Schrödinger equations with complex nonlinearities. Rep. Math. Phys. 48, 115–121 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  52. Kaniadakis, G., Scarfone, A.M.: Nonlinear gauge transformation for a class of Schrödinger equations containing complex nonlinearities. Rep. Math. Phys. 46, 113–118 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  53. Kaniadakis, G., Quarati, P., Scarfone, A.M.: Soliton-like behavior of a canonical quantum system obeying an exclusion-inclusion principle. Physica A 255, 474–482 (1998)

    Article  Google Scholar 

  54. Kaniadakis, G., Quarati, P., Scarfone, A.M.: Nonlinear canonical quantum system of collectively interacting particles via an exclusion-inclusion principle. Phys. Rev. E 58, 5574–5585 (1998)

    Article  MathSciNet  Google Scholar 

  55. Kaper, H.G., Takáč, P.: Ginzburg-Landau dynamics with a time-dependent magnetic field. Nonlinearity 11, 291–305 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  56. Kaup, D.J., Newell, A.C.: Exact solution for a derivative non-linear Schrödinger equation. J. Math. Phys. 19, 798–801 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  57. Kostin, M.D.: Friction and dissipative phenomena in quantum mechanics. J. Stat. Phys. 12, 145–151 (1975)

    Article  Google Scholar 

  58. Kostin, M.D.: On the Schrödinger-Langevin equation. J. Chem. Phys. 57, 3589–3591 (1973)

    Article  Google Scholar 

  59. Kundu, A.: Comments on the Eckhaus PDE i ψ t +ψ xx +2(|ψ|2) x ψ+|ψ|4=0. Inverse Probl. 4, 1143–1144 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  60. Kundu, A.: Landau-Lifshitz and higher-order nonlinear-systems gauge generated from nonlinear Schrödinger type equations. J. Math. Phys. 25, 3433–3438 (1984)

    Article  MathSciNet  Google Scholar 

  61. Li, Z., Li, L., Tian, H., Zhou, G.: New types of solitary wave solutions for the higher order nonlinear Schrödinger equation. Phys. Rev. Lett. 84, 4096–4099 (2000)

    Article  Google Scholar 

  62. Madelung, E.: Quantum theory in hydrodynamical form. Z. Phys. 40, 332–336 (1926)

    Google Scholar 

  63. Mahalingam, A., Porsezian, K.: Propagation of dark solitons in a system of coupled higher-order nonlinear Schrödinger equations. J. Phys. A: Math. Gen. 35, 3099–3109 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  64. Malomed, B.A., Stenflo, L.: Modulational instabilities and soliton-solutions of a generalized nonlinear Schrödinger equation. J. Phys. A: Math. Gen. 24, L1149–1153 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  65. Malomed, B.A.: Bound solitons in the nonlinear Schrödinger-Ginzburg-Landau equation. Phys. Rev. A 44, 6954–6957 (1991)

    Article  MathSciNet  Google Scholar 

  66. Malomed, B.A., Nepomnyashchy, A.A.: Kinks and solitons in the generalized Ginzburg-Landau equation. Phys. Rev. A 42, 6009–6014 (1990)

    Article  Google Scholar 

  67. Malomed, B.A.: Evolution of nonsoliton and quasi-classical wavetrains in nonlinear Schrödinger and Korteweg-Devries equations with dissipative perturbations. Physica D 29, 155–172 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  68. Manakov, S.V.: On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Z. Eksp. Teor. Fiz. 65, 505–516 (1973) [Sov. Phys. JETP 38, 248–253 (1974)]

    Google Scholar 

  69. Martina, L., Soliani, G., Winternitz, P.: Partially invariant solutions of a class of nonlinear Schrödinger equations. J. Phys. A: Math. Gen. 25, 4425–4435 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  70. Matthews, M.R., Anderson, B.P., Haljan, P.C., Hall, D.S., Holland, M.J., Williams, J.E., Wieman, C.E., Cornell, E.A.: Watching a superfluid untwist itself: Recurrence of Rabi oscillations in a Bose-Einstein condensate. Phys. Rev. Lett. 83, 3358–3361 (1999)

    Article  Google Scholar 

  71. Mollenauer, L.F., Stolen, R.H., Gordon, J.P.: Experimental-observation of picosecond pulse narrowing and solitons in optical fibers. Phys. Rev. Lett. 45, 1095–1098 (1980)

    Article  Google Scholar 

  72. Nakkeeran, K.: Exact dark soliton solutions for a family of N coupled nonlinear Schrödinger equations in optical fiber media. Phys. Rev. E 64, 046611–7 (2001)

    Article  Google Scholar 

  73. Nakkeeran, K.: On the integrability of the extended nonlinear Schrödinger equation and the coupled extended nonlinear Schrödinger equations. J. Phys. A: Math. Gen. 33, 3947–3949 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  74. Nakkeeran, K.: Exact soliton solutions for a family of N coupled nonlinear Schrödinger equations in optical fiber media. Phys. Rev. E 62, 1313–1321 (2000)

    Article  MathSciNet  Google Scholar 

  75. Newboult, G.K., Parker, D.F., Faulkner, T.R.: Coupled nonlinear Schrödinger equations arising in the study of monomode step-index optical fibers. J. Math. Phys. 30, 930–936 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  76. Noether, E.: Invariante Variationsprobleme, Nachr. Ges. Wiss. Gött. Math. Phys. Kl. 235 (1918) (English translation from Travel, M.A.: Transp. Theory Stat. Phys. 1(3), 183 (1971)

  77. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1986)

    MATH  Google Scholar 

  78. Pitaevskii, L.P.: Vortex lines in an imperfect Bose gas. Z. Eksp. Teor. Fiz. 40, 646–651 (1961) [Sov. Phys. JETP 13, 451–454 (1961)]

    Google Scholar 

  79. Radhakrishnan, R., Kundu, A., Lakshmanan, M.: Coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity: Integrability and soliton interaction in non-Kerr media. Phys. Rev. E 60, 3314–3323 (1999)

    Article  Google Scholar 

  80. Ryskin, N.M.: Schrödinger bound nonlinear equations for the description of multifrequency wave packages distribution in nonlinear medium with dispersion. Z. Eksp. Teor. Fiz. 106, 1542–1546 (1994) [Sov. Phys. JETP 79, 833–834 (1994)]

    Google Scholar 

  81. Sakovich, S.Y., Tsuchida, T.: Symmetrically coupled higher-order nonlinear Schrödinger equations: singularity analysis and integrability. J. Phys. A: Math. Gen. 33, 7217–7226 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  82. Scarfone, A.M.: Stochastic quantization of an interacting classical particle system, J. Stat. Mech.: Theory Exp. P03012+16 (2007)

  83. Scarfone, A.M.: Canonical quantization of classical systems with generalized entropies. Rep. Math. Phys. 55, 169–177 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  84. Scarfone, A.M.: Canonical quantization of nonlinear many-body systems. Phys. Rev. E 71, 051103–15 (2005)

    Article  MathSciNet  Google Scholar 

  85. Scarfone, A.M.: Gauge transformation of the third kind for U(1)-invariant coupled Schrödinger equations. J. Phys. A: Math. Gen. 38, 7037–7050 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  86. Schuch, D.: Nonunitary connection between explicitly time-dependent and nonlinear approaches for the description of dissipative quantum systems. Phys. Rev. A 53, 945–940 (1997)

    MathSciNet  Google Scholar 

  87. Schuch, D., Chung, K.-M., Hartmann, H.: Nonlinear Schrödinger-type field equation for the description of dissipative systems 3. Frictionally damped free motion as an example for an aperiodic motion. J. Math. Phys. 25, 3086–3092 (1984)

    Article  MathSciNet  Google Scholar 

  88. Shchesnovich, V.S., Doktorov, E.V.: Perturbation theory for the modified nonlinear Schrödinger solitons. Physica D 129, 115–129 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  89. Shi, H., Zheng, W.-M.: Bose-Einstein condensation in an atomic gas with attractive interactions. Phys. Rev. A 55, 2930–2934 (1997)

    Article  Google Scholar 

  90. Stratopoulos, G.N., Tomaras, T.N.: Vortex pairs in charged fluids. Phys. Rev. B 54, 12493–12504 (1996)

    Article  Google Scholar 

  91. Stringari, S.: Collective excitations of a trapped Bose condensed gas. Phys. Rev. Lett. 77, 2360–2363 (1996)

    Article  Google Scholar 

  92. Tsuchida, T., Wadati, M.: Complete integrability of derivative nonlinear Schrödinger-type equations. Inverse Probl. 15, 1363–1373 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  93. Tsuchida, T., Wadati, M.: New integrable systems of derivative nonlinear Schrödinger equations with multiple components. Phys. Lett. A 257, 53–64 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  94. Vinoj, M.N., Kuriakose, V.C.: Multisoliton solutions and integrability aspects of coupled higher-order nonlinear Schrödinger equations. Phys. Rev. E 62, 8719–8725 (2000)

    Article  MathSciNet  Google Scholar 

  95. Weinberg, S.: Precision tests of quantum mechanics. Phys. Rev. Lett. 62, 485–488 (1989)

    Article  Google Scholar 

  96. Weinberg, S.: Testing quantum mechanics. Ann. Phys. 194, 336–386 (1989)

    Article  MathSciNet  Google Scholar 

  97. Weinberg, S.: Understanding the Fundamental Constitutents of Matter. Plenum, New York (1978). A. Zichichi (ed.)

    Google Scholar 

  98. Wilczek, F.: Fractional Statistics and Anyon Superconductivity. World Scientific, Singapore (1990)

    Google Scholar 

  99. Wilczek, F.: Magnetic flux, angular momentum, and statistics. Phys. Rev. Lett. 48, 1144–1146 (1982)

    Article  MathSciNet  Google Scholar 

  100. Yip, S.-K.: Internal vortex structure of a trapped spinor Bose-Einstein condensate. Phys. Rev. Lett. 83, 4677–4681 (1999)

    Article  Google Scholar 

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  1. Istituto Nazionale di Fisica della Materia (CNR-INFM) and Dipartimento di Fisica, Unitá del Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

    Antonio M. Scarfone

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Scarfone, A.M. Gauge Equivalence among Quantum Nonlinear Many Body Systems. Acta Appl Math 102, 179–217 (2008). https://doi.org/10.1007/s10440-008-9213-7

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