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On viscous quantum hydrodynamics associated with nonlinear Schrödinger-Doebner-Goldin models
1. | Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain, Spain |
References:
[1] |
M. G. Ancona, Density-gradient theory analysis of electron distributions in heterostructures, Superlattics and Microstructures, 7 (1990), 119-130.
doi: 10.1016/0749-6036(90)90124-P. |
[2] |
A. Arnold, J. L. López, P. Markowich and J. Soler, An analysis of quantum Fokker-Planck models: a Wigner function approach, Rev. Mat. Iberoamericana, 20 (2004), 771-814.
doi: 10.4171/RMI/407. |
[3] |
G. Auberson and P. C. Sabatier, On a class of homogeneous nonlinear Schrödinger equations, J. Math. Phys., 35 (1994), 4028-4040.
doi: 10.1063/1.530840. |
[4] |
I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Phys., 100 (1976), 62-93.
doi: 10.1016/0003-4916(76)90057-9. |
[5] |
S. Brull and F. Méhats, Derivation of viscous correction terms for the isothermal quantum euler model, Z. Angew. Math. Mech., 90 (2010), 219-230.
doi: 10.1002/zamm.200900297. |
[6] |
S. Burger, F. Cataliotti, C. Fort, F. Minardi, M. Inguscio, M. Chiofalo and M.Tosi, Superfluid and dissipative dynamics of a Bose-Einstein condensate in a periodic optimal potential, Phys. Rev. Lett., 86 (2001), 4447-4450.
doi: 10.1103/PhysRevLett.86.4447. |
[7] |
P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112 (2003), 587-628.
doi: 10.1023/A:1023824008525. |
[8] |
H. D. Doebner and G. A. Goldin, On a general nonlinear Schrödinger equation admitting diffusioncurrents, Phys. Lett., A, 162 (1992), 397-401.
doi: 10.1016/0375-9601(92)90061-P. |
[9] |
H. D. Doebner and G. A. Goldin, Properties of nonlinear Schrödinger equations associated with diffeomorphism group representations, J. Phys. A: Math. Gen., 27 (1994), 1771-1780.
doi: 10.1088/0305-4470/27/5/036. |
[10] |
H. D. Doebner and G. A. Goldin, Introducing nonlinear gauge transformations in a family of nonlinear Schrödinger equations, Phys. Rev. A (3), 54 (1996), 3764-3771.
doi: 10.1103/PhysRevA.54.3764. |
[11] |
H. D. Doebner, G. A. Goldin and P. Nattermann, A family of nonlinear Schrödinger equations: linearizing transformations and resulting structure, in "Quantization, Coherent States and Complex Structures" (eds. J.-P. Antoine et al.), Plenum, (1996), 27-31. |
[12] |
P. Frampton, "Gauge Field Theories," Wiley-VCH, Berlin, 2008.
doi: 10.1002/9783527623358. |
[13] |
L. Fritsche and M. Haugk, A new look at the derivation of the Schrödinger equation from Newtonian mechanics, Ann. Phys., 12 (2003), 371-403.
doi: 10.1002/andp.200310017. |
[14] |
P. Garbaczewski, Modular Schrödinger equation and dynamical duality, Phys. Rev. E (3), 78 (2008), 031101.
doi: 10.1103/PhysRevE.78.031101. |
[15] |
M. P. Gualdani and A. Jüngel, Analysis of the viscous quantum hydrodynamic equations for semiconductors, Europ. J. Appl. Math., 15 (2004), 577-595.
doi: 10.1017/S0956792504005686. |
[16] |
P. Guerrero, J. L. López and J. Nieto, Global $H^1$ solvability of the 3D logarithmic Schrödinger equation, Nonlinear Analysis R. W. A., 11 (2010), 79-87.
doi: 10.1016/j.nonrwa.2008.10.017. |
[17] |
P. Guerrero, J. L. López, J. Montejo-Gámez, J. Nieto, Wellposedness of a nonlinear, logarithmic Schrödinger equation of Doebner-Goldin type modeling quantum dissipation, preprint. |
[18] |
R. Harvey, Navier-Stokes analog of quantum mechanics, Phys. Rev., 152 (1966), 1115.
doi: 10.1103/PhysRev.152.1115. |
[19] |
A. Jüngel, "Transport Equations for Semiconductors," Lect. Notes Phys., 773, Springer, Berlin, 2009. |
[20] |
A. Jüngel, Effective velocity in compressible Navier-Stokes equations wirth third-order derivatives, Nonlinear Analysis, 74 (2011), 2813-2818.
doi: 10.1016/j.na.2011.01.002. |
[21] |
A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045.
doi: 10.1137/090776068. |
[22] |
A. Jüngel, J. L. López and J. Montejo-Gámez, A new derivation of the quantum Navier-Stokes equations in the Wigner-Fokker-Planck approach, To appear in J. Stat. Phys., 145 (2011), 1661-1673. |
[23] |
A. Jüngel and J.-P. Milisić, Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution equations, Kinetic and related models, 4 (2011), 785-807. |
[24] |
A. Jüngel, M. C. Mariani and D. Rial, Local existence of solutions to the transient quantum hydrodynamic equations, Math. Models Meth. Appl. Sci., 12 (2002), 485-495.
doi: 10.1142/S0218202502001751. |
[25] |
G. Kaniadakis and A. M. Scarfone, Nonlinear transformation for a class of gauged Schrödinger equations with complex nonlinearities, Reports on Math. Phys., 48 (2001), 115-121.
doi: 10.1016/S0034-4877(01)80070-7. |
[26] |
J. L. López, Nonlinear Ginzburg-Landau-type approach to quantum dissipation, Phys. Rev. E., 69 (2004), 026110.
doi: 10.1103/PhysRevE.69.026110. |
[27] |
J. L. López, J. Montejo-Gámez, A hydrodynamic approach to multidimensional dissipation-based Schrödinger models from quantum Fokker-Planck dynamics, Physica D, 238 (2009), 622-644. |
[28] |
R. Mosna, I. Hamilton and L. Delle Site, Variational approach to dequantization, J. Phys. A, 39 (2006), L229-L235.
doi: 10.1088/0305-4470/39/14/L03. |
[29] |
P. Nattermann and W. Scherer, Nonlinear gauge transformations and exact solutions of the Doebner-Goldin equation, in "Nonlinear, Deformed and Irreversible Quantum Systems" (eds. Doebner et al.), World Scientific, (1995), 188-199. |
[30] |
P. Nattermann and R. Zhdanov, On integrable Doebner-Goldin equations, J. Phys. A, 29 (1996), 2869-2886.
doi: 10.1088/0305-4470/29/11/021. |
[31] |
E. Nelson, Derivation of the Schrödinger equation from Newtonian Mechanics, Phys. Rev., 150 (1966), 1079-1085.
doi: 10.1103/PhysRev.150.1079. |
[32] |
C. Sabatier, Multidimensional nonlinear Schrödinger equations with exponentially confined solutions, Inverse Problems, 6 (1990), L47-L53.
doi: 10.1088/0266-5611/6/5/002. |
[33] |
A. Scarfone, Gauge equivalence among quantum nonlinear many body systems, Act. Appl. Math., 102 (2008), 179-217.
doi: 10.1007/s10440-008-9213-7. |
[34] |
A. G. Ushveridze, Dissipative quantum mechanics. A special Doebner-Goldin equation, its properties and exact solutions, Phys. Lett. A, 185 (1994), 123-127.
doi: 10.1016/0375-9601(94)90834-6. |
[35] |
A. G. Ushveridze, The special Doebner-Goldin equation as a fundamental equation of dissipative quantum mechanics, Phys. Lett. A, 185 (1994), 128-132.
doi: 10.1016/0375-9601(94)90835-4. |
show all references
References:
[1] |
M. G. Ancona, Density-gradient theory analysis of electron distributions in heterostructures, Superlattics and Microstructures, 7 (1990), 119-130.
doi: 10.1016/0749-6036(90)90124-P. |
[2] |
A. Arnold, J. L. López, P. Markowich and J. Soler, An analysis of quantum Fokker-Planck models: a Wigner function approach, Rev. Mat. Iberoamericana, 20 (2004), 771-814.
doi: 10.4171/RMI/407. |
[3] |
G. Auberson and P. C. Sabatier, On a class of homogeneous nonlinear Schrödinger equations, J. Math. Phys., 35 (1994), 4028-4040.
doi: 10.1063/1.530840. |
[4] |
I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Phys., 100 (1976), 62-93.
doi: 10.1016/0003-4916(76)90057-9. |
[5] |
S. Brull and F. Méhats, Derivation of viscous correction terms for the isothermal quantum euler model, Z. Angew. Math. Mech., 90 (2010), 219-230.
doi: 10.1002/zamm.200900297. |
[6] |
S. Burger, F. Cataliotti, C. Fort, F. Minardi, M. Inguscio, M. Chiofalo and M.Tosi, Superfluid and dissipative dynamics of a Bose-Einstein condensate in a periodic optimal potential, Phys. Rev. Lett., 86 (2001), 4447-4450.
doi: 10.1103/PhysRevLett.86.4447. |
[7] |
P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112 (2003), 587-628.
doi: 10.1023/A:1023824008525. |
[8] |
H. D. Doebner and G. A. Goldin, On a general nonlinear Schrödinger equation admitting diffusioncurrents, Phys. Lett., A, 162 (1992), 397-401.
doi: 10.1016/0375-9601(92)90061-P. |
[9] |
H. D. Doebner and G. A. Goldin, Properties of nonlinear Schrödinger equations associated with diffeomorphism group representations, J. Phys. A: Math. Gen., 27 (1994), 1771-1780.
doi: 10.1088/0305-4470/27/5/036. |
[10] |
H. D. Doebner and G. A. Goldin, Introducing nonlinear gauge transformations in a family of nonlinear Schrödinger equations, Phys. Rev. A (3), 54 (1996), 3764-3771.
doi: 10.1103/PhysRevA.54.3764. |
[11] |
H. D. Doebner, G. A. Goldin and P. Nattermann, A family of nonlinear Schrödinger equations: linearizing transformations and resulting structure, in "Quantization, Coherent States and Complex Structures" (eds. J.-P. Antoine et al.), Plenum, (1996), 27-31. |
[12] |
P. Frampton, "Gauge Field Theories," Wiley-VCH, Berlin, 2008.
doi: 10.1002/9783527623358. |
[13] |
L. Fritsche and M. Haugk, A new look at the derivation of the Schrödinger equation from Newtonian mechanics, Ann. Phys., 12 (2003), 371-403.
doi: 10.1002/andp.200310017. |
[14] |
P. Garbaczewski, Modular Schrödinger equation and dynamical duality, Phys. Rev. E (3), 78 (2008), 031101.
doi: 10.1103/PhysRevE.78.031101. |
[15] |
M. P. Gualdani and A. Jüngel, Analysis of the viscous quantum hydrodynamic equations for semiconductors, Europ. J. Appl. Math., 15 (2004), 577-595.
doi: 10.1017/S0956792504005686. |
[16] |
P. Guerrero, J. L. López and J. Nieto, Global $H^1$ solvability of the 3D logarithmic Schrödinger equation, Nonlinear Analysis R. W. A., 11 (2010), 79-87.
doi: 10.1016/j.nonrwa.2008.10.017. |
[17] |
P. Guerrero, J. L. López, J. Montejo-Gámez, J. Nieto, Wellposedness of a nonlinear, logarithmic Schrödinger equation of Doebner-Goldin type modeling quantum dissipation, preprint. |
[18] |
R. Harvey, Navier-Stokes analog of quantum mechanics, Phys. Rev., 152 (1966), 1115.
doi: 10.1103/PhysRev.152.1115. |
[19] |
A. Jüngel, "Transport Equations for Semiconductors," Lect. Notes Phys., 773, Springer, Berlin, 2009. |
[20] |
A. Jüngel, Effective velocity in compressible Navier-Stokes equations wirth third-order derivatives, Nonlinear Analysis, 74 (2011), 2813-2818.
doi: 10.1016/j.na.2011.01.002. |
[21] |
A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045.
doi: 10.1137/090776068. |
[22] |
A. Jüngel, J. L. López and J. Montejo-Gámez, A new derivation of the quantum Navier-Stokes equations in the Wigner-Fokker-Planck approach, To appear in J. Stat. Phys., 145 (2011), 1661-1673. |
[23] |
A. Jüngel and J.-P. Milisić, Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution equations, Kinetic and related models, 4 (2011), 785-807. |
[24] |
A. Jüngel, M. C. Mariani and D. Rial, Local existence of solutions to the transient quantum hydrodynamic equations, Math. Models Meth. Appl. Sci., 12 (2002), 485-495.
doi: 10.1142/S0218202502001751. |
[25] |
G. Kaniadakis and A. M. Scarfone, Nonlinear transformation for a class of gauged Schrödinger equations with complex nonlinearities, Reports on Math. Phys., 48 (2001), 115-121.
doi: 10.1016/S0034-4877(01)80070-7. |
[26] |
J. L. López, Nonlinear Ginzburg-Landau-type approach to quantum dissipation, Phys. Rev. E., 69 (2004), 026110.
doi: 10.1103/PhysRevE.69.026110. |
[27] |
J. L. López, J. Montejo-Gámez, A hydrodynamic approach to multidimensional dissipation-based Schrödinger models from quantum Fokker-Planck dynamics, Physica D, 238 (2009), 622-644. |
[28] |
R. Mosna, I. Hamilton and L. Delle Site, Variational approach to dequantization, J. Phys. A, 39 (2006), L229-L235.
doi: 10.1088/0305-4470/39/14/L03. |
[29] |
P. Nattermann and W. Scherer, Nonlinear gauge transformations and exact solutions of the Doebner-Goldin equation, in "Nonlinear, Deformed and Irreversible Quantum Systems" (eds. Doebner et al.), World Scientific, (1995), 188-199. |
[30] |
P. Nattermann and R. Zhdanov, On integrable Doebner-Goldin equations, J. Phys. A, 29 (1996), 2869-2886.
doi: 10.1088/0305-4470/29/11/021. |
[31] |
E. Nelson, Derivation of the Schrödinger equation from Newtonian Mechanics, Phys. Rev., 150 (1966), 1079-1085.
doi: 10.1103/PhysRev.150.1079. |
[32] |
C. Sabatier, Multidimensional nonlinear Schrödinger equations with exponentially confined solutions, Inverse Problems, 6 (1990), L47-L53.
doi: 10.1088/0266-5611/6/5/002. |
[33] |
A. Scarfone, Gauge equivalence among quantum nonlinear many body systems, Act. Appl. Math., 102 (2008), 179-217.
doi: 10.1007/s10440-008-9213-7. |
[34] |
A. G. Ushveridze, Dissipative quantum mechanics. A special Doebner-Goldin equation, its properties and exact solutions, Phys. Lett. A, 185 (1994), 123-127.
doi: 10.1016/0375-9601(94)90834-6. |
[35] |
A. G. Ushveridze, The special Doebner-Goldin equation as a fundamental equation of dissipative quantum mechanics, Phys. Lett. A, 185 (1994), 128-132.
doi: 10.1016/0375-9601(94)90835-4. |
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