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q-Gaussian integrable Hamiltonian reductions in anisentropic gasdynamics
1. | Australian Research Council Centre of Excellence for Mathematics & Statistics of Complex Systems, School of Mathematics, The University of New South Wales, Sydney, NSW2052, Australia |
2. | Department of Mathematics & Research Centre of Applied Mathematics (CIRAM), University of Bologna, Italy |
References:
[1] |
I. Bialynicki-Birula and J. Mycielski, Wave equations with logarithmic nonlinearities, Bull. Acad. Polon. Sci., 23 (1974), 461-466. |
[2] |
I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Phys., 100 (1976), 62-93.
doi: 10.1016/0003-4916(76)90057-9. |
[3] |
I. Bialynicki-Birula and J. Mycielski, Gaussons: solitons of the logarithmic Schrödinger equation, Physica Scripta, 20 (1979), 539-544.
doi: 10.1088/0031-8949/20/3-4/033. |
[4] |
D. Bohm, A suggested interpretation of the quantum theory in terms of ‘hidden' variables: I and II, Phys. Rev., 85 (1952), 166-193.
doi: 10.1103/PhysRev.85.166. |
[5] |
D. N. Christodoulidis, T. H. Coskun and R. I. Joseph, Incoherent spatial solitons in saturable nonlinear media, Opt. Lett., 22 (1997), 1080-1082.
doi: 10.1364/OL.22.001080. |
[6] |
S. Curilef, A. R. Plastino and A. Plastino, Tsallis' maximum entropy ansatz leading to exact analytic time dependent wave packet solutions of a nonlinear Schrödinger equation, Physica A, 392 (2013), 2631-2642.
doi: 10.1016/j.physa.2012.12.041. |
[7] |
L. de Broglie, The wave nature of the electron, Nobel Lectures, (1965), 244-246. |
[8] |
F. J. Dyson, Dynamics of a spinning gas cloud, J. Math. Mech., 18 (1968), 91-101.
doi: 10.1512/iumj.1969.18.18009. |
[9] |
B. Gaffet, Expanding gas clouds of ellipsoidal shape: New exact solutions, J. Fluid Mech., 325 (1996), 113-144.
doi: 10.1017/S0022112096008051. |
[10] |
B. Gaffet, Spinning gas clouds with precession: A new formulation, J. Phys. A: Math. Theor., 43 (2010), 165207, 11pp.
doi: 10.1088/1751-8113/43/16/165207. |
[11] |
M. Gell-Mann and C. Tsallis, Nonextensive Entropy: Interdisciplinary Applications, Oxford University Press, Oxford, 2004. |
[12] |
I. B. Gornushkin, S. V. Shabanov, N. Omenetto and J. D. Winefordner, Nonisothermal asymmetric expansion of laser induced plasmas into a vacuum, J. Appl. Phys., 100 (2006), 073304.
doi: 10.1063/1.2345460. |
[13] |
T. Hansson, D. Anderson and M. Lisak, Soliton interaction in logarithmically saturable media, Opt. Commun., 283 (2010), 318-322.
doi: 10.1016/j.optcom.2009.09.034. |
[14] |
K. Królikowski, D. Edmundson and O. Bang, Unified model for partially coherent solutions in logarithmically nonlinear media, Phys. Rev. E, 61 (2000), 3122-3126. |
[15] |
J. H. Lee, O. K. Pashaev, C. Rogers and W. K. Schief, The resonant nonlinear Schrödinger equation in cold plasma physics: application of Bäcklund transformations and superposition principles, J. Plasma Phys., 73 (2007), 257-272.
doi: 10.1017/S0022377806004648. |
[16] |
E. Madelung, Quartentheorie in Hydrodynamischen form, Zeit für Phys., 40 (1926), 322-326. |
[17] |
B. A. Malomed, Soliton Management in Periodic Systems, Springer, New York, 2006. |
[18] |
J. Naudts, Generalised Thermostatistics, Springer-Verlag London, Ltd., London, 2011.
doi: 10.1007/978-0-85729-355-8. |
[19] |
F. D. Nobre, M. A. Rego-Monteiro and C. Tsallis, Nonlinear relativistic and quantum equations with a common type of solution, Phys. Rev. Lett., 106 (2011), 140601.
doi: 10.1103/PhysRevLett.106.140601. |
[20] |
F. D. Nobre, M. A. Rego-Monteiro and C. Tsallis, A generalised nonlinear Schrödinger equation: Classical field-theoretic approach, Europhysics Letters, 97 (2012), 41001. |
[21] |
O. K. Pashaev, J. H. Lee and C. Rogers, Soliton resonances in a generalized nonlinear Schrödinger equation, J. Phys. A: Math. Theor., 41 (2008), 452001, 9pp.
doi: 10.1088/1751-8113/41/45/452001. |
[22] |
R. C. Prim, Steady rotational flow of ideal gases, J. Rational Mech. Anal, 1 (1952), 425-497. |
[23] |
J. R. Ray, Nonlinear superposition law for generalised Ermakov systems, Phys. Lett. A, 78 (1980), 4-6.
doi: 10.1016/0375-9601(80)90789-6. |
[24] |
J. L. Reid and J. R. Ray, Ermakov systems, nonlinear superposition and solution of nonlinear equations of motion, J. Math. Phys., 21 (1980), 1583-1587.
doi: 10.1063/1.524625. |
[25] |
C. Rogers, C. Hoenselaers and J. R. Ray, On 2+1-dimensional Ermakov systems, J. Phys. A. Math. & Gen., 26 (1993), 2625-2633.
doi: 10.1088/0305-4470/26/11/012. |
[26] |
C. Rogers and W. K. Schief, Multi-component Ermakov systems: Structure and linearization, J. Math. Anal. Appl., 198 (1996), 194-220.
doi: 10.1006/jmaa.1996.0076. |
[27] |
C. Rogers and H. An, Ermakov-Ray-Reid systems in 2+1-dimensional rotating shallow water theory, Stud. Appl. Math., 125 (2010), 275-299.
doi: 10.1111/j.1467-9590.2010.00488.x. |
[28] |
C. Rogers, B. Malomed, K. Chow and H. An, Ermakov-Ray-Reid systems in nonlinear optics, J. Phys. A: Math. Theor., 43 (2010), 455214, 15pp.
doi: 10.1088/1751-8113/43/45/455214. |
[29] |
C. Rogers and W. K. Schief, The pulsrodon in 2+1-dimensional magnetogasdynamics. Hamiltonian structure and integrability theory, J. Math. Phys., 52 (2011), 083701, 20pp.
doi: 10.1063/1.3622595. |
[30] |
C. Rogers and W. K. Schief, On the integrability of a Hamiltonian reduction of a 2+1-dimensional non-isothermal rotating gas cloud system, Nonlinearity, 24 (2011), 3165-3178.
doi: 10.1088/0951-7715/24/11/009. |
[31] |
C. Rogers, B. Malomed and H. An, Ermakov-Ray-Reid reductions of variational approximations in nonlinear optics, Stud. Appl. Math., 129 (2012), 389-413.
doi: 10.1111/j.1467-9590.2012.00557.x. |
[32] |
C. Rogers and H. An, On a 2+1-dimensional Madelung system with logarithmic and de Broglie-Bohm quantum potentials. Ermakov reduction, Physica Scripta, 84 (2011), 045004. |
[33] |
C. Rogers, W. K. Schief and W. H. Hui, On complex-lamellar motion of a Prim gas, J. Math. Anal. Appl., 266 (2002), 55-69.
doi: 10.1006/jmaa.2001.7685. |
[34] |
W. K. Schief, C. Rogers and A. Bassom, Ermakov systems with arbitrary order and dimension: Structure and linearization, J. Phys. A: Math. Gen., 29 (1996), 903-911.
doi: 10.1088/0305-4470/29/4/017. |
[35] |
W. K. Schief, H. An and C. Rogers, Universal and integrable aspects of an elliptic vortex representation in 2+1-dimensional magnetogasdynamics, Stud. Appl. Math., 130 (2013), 49-79.
doi: 10.1111/j.1467-9590.2012.00559.x. |
[36] |
A. W. Snyder and J. D. Mitchell, Mighty morphing and spatial solitons and bullets, Opt. Lett., 22 (1997), 16-18.
doi: 10.1364/OL.22.000016. |
[37] |
C. Tsallis, Introduction to Nonextensive Statistical Mechanics - Approaching a Complex World, Springer, New York, 2009. |
[38] |
W. G. Wagner, H. A. Haus and J. H. Marburger, Large-scale self-trapping of optical beams in the paraxial ray approximation, Phys. Rev., 175 (1968), 256-266.
doi: 10.1103/PhysRev.175.256. |
show all references
References:
[1] |
I. Bialynicki-Birula and J. Mycielski, Wave equations with logarithmic nonlinearities, Bull. Acad. Polon. Sci., 23 (1974), 461-466. |
[2] |
I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Phys., 100 (1976), 62-93.
doi: 10.1016/0003-4916(76)90057-9. |
[3] |
I. Bialynicki-Birula and J. Mycielski, Gaussons: solitons of the logarithmic Schrödinger equation, Physica Scripta, 20 (1979), 539-544.
doi: 10.1088/0031-8949/20/3-4/033. |
[4] |
D. Bohm, A suggested interpretation of the quantum theory in terms of ‘hidden' variables: I and II, Phys. Rev., 85 (1952), 166-193.
doi: 10.1103/PhysRev.85.166. |
[5] |
D. N. Christodoulidis, T. H. Coskun and R. I. Joseph, Incoherent spatial solitons in saturable nonlinear media, Opt. Lett., 22 (1997), 1080-1082.
doi: 10.1364/OL.22.001080. |
[6] |
S. Curilef, A. R. Plastino and A. Plastino, Tsallis' maximum entropy ansatz leading to exact analytic time dependent wave packet solutions of a nonlinear Schrödinger equation, Physica A, 392 (2013), 2631-2642.
doi: 10.1016/j.physa.2012.12.041. |
[7] |
L. de Broglie, The wave nature of the electron, Nobel Lectures, (1965), 244-246. |
[8] |
F. J. Dyson, Dynamics of a spinning gas cloud, J. Math. Mech., 18 (1968), 91-101.
doi: 10.1512/iumj.1969.18.18009. |
[9] |
B. Gaffet, Expanding gas clouds of ellipsoidal shape: New exact solutions, J. Fluid Mech., 325 (1996), 113-144.
doi: 10.1017/S0022112096008051. |
[10] |
B. Gaffet, Spinning gas clouds with precession: A new formulation, J. Phys. A: Math. Theor., 43 (2010), 165207, 11pp.
doi: 10.1088/1751-8113/43/16/165207. |
[11] |
M. Gell-Mann and C. Tsallis, Nonextensive Entropy: Interdisciplinary Applications, Oxford University Press, Oxford, 2004. |
[12] |
I. B. Gornushkin, S. V. Shabanov, N. Omenetto and J. D. Winefordner, Nonisothermal asymmetric expansion of laser induced plasmas into a vacuum, J. Appl. Phys., 100 (2006), 073304.
doi: 10.1063/1.2345460. |
[13] |
T. Hansson, D. Anderson and M. Lisak, Soliton interaction in logarithmically saturable media, Opt. Commun., 283 (2010), 318-322.
doi: 10.1016/j.optcom.2009.09.034. |
[14] |
K. Królikowski, D. Edmundson and O. Bang, Unified model for partially coherent solutions in logarithmically nonlinear media, Phys. Rev. E, 61 (2000), 3122-3126. |
[15] |
J. H. Lee, O. K. Pashaev, C. Rogers and W. K. Schief, The resonant nonlinear Schrödinger equation in cold plasma physics: application of Bäcklund transformations and superposition principles, J. Plasma Phys., 73 (2007), 257-272.
doi: 10.1017/S0022377806004648. |
[16] |
E. Madelung, Quartentheorie in Hydrodynamischen form, Zeit für Phys., 40 (1926), 322-326. |
[17] |
B. A. Malomed, Soliton Management in Periodic Systems, Springer, New York, 2006. |
[18] |
J. Naudts, Generalised Thermostatistics, Springer-Verlag London, Ltd., London, 2011.
doi: 10.1007/978-0-85729-355-8. |
[19] |
F. D. Nobre, M. A. Rego-Monteiro and C. Tsallis, Nonlinear relativistic and quantum equations with a common type of solution, Phys. Rev. Lett., 106 (2011), 140601.
doi: 10.1103/PhysRevLett.106.140601. |
[20] |
F. D. Nobre, M. A. Rego-Monteiro and C. Tsallis, A generalised nonlinear Schrödinger equation: Classical field-theoretic approach, Europhysics Letters, 97 (2012), 41001. |
[21] |
O. K. Pashaev, J. H. Lee and C. Rogers, Soliton resonances in a generalized nonlinear Schrödinger equation, J. Phys. A: Math. Theor., 41 (2008), 452001, 9pp.
doi: 10.1088/1751-8113/41/45/452001. |
[22] |
R. C. Prim, Steady rotational flow of ideal gases, J. Rational Mech. Anal, 1 (1952), 425-497. |
[23] |
J. R. Ray, Nonlinear superposition law for generalised Ermakov systems, Phys. Lett. A, 78 (1980), 4-6.
doi: 10.1016/0375-9601(80)90789-6. |
[24] |
J. L. Reid and J. R. Ray, Ermakov systems, nonlinear superposition and solution of nonlinear equations of motion, J. Math. Phys., 21 (1980), 1583-1587.
doi: 10.1063/1.524625. |
[25] |
C. Rogers, C. Hoenselaers and J. R. Ray, On 2+1-dimensional Ermakov systems, J. Phys. A. Math. & Gen., 26 (1993), 2625-2633.
doi: 10.1088/0305-4470/26/11/012. |
[26] |
C. Rogers and W. K. Schief, Multi-component Ermakov systems: Structure and linearization, J. Math. Anal. Appl., 198 (1996), 194-220.
doi: 10.1006/jmaa.1996.0076. |
[27] |
C. Rogers and H. An, Ermakov-Ray-Reid systems in 2+1-dimensional rotating shallow water theory, Stud. Appl. Math., 125 (2010), 275-299.
doi: 10.1111/j.1467-9590.2010.00488.x. |
[28] |
C. Rogers, B. Malomed, K. Chow and H. An, Ermakov-Ray-Reid systems in nonlinear optics, J. Phys. A: Math. Theor., 43 (2010), 455214, 15pp.
doi: 10.1088/1751-8113/43/45/455214. |
[29] |
C. Rogers and W. K. Schief, The pulsrodon in 2+1-dimensional magnetogasdynamics. Hamiltonian structure and integrability theory, J. Math. Phys., 52 (2011), 083701, 20pp.
doi: 10.1063/1.3622595. |
[30] |
C. Rogers and W. K. Schief, On the integrability of a Hamiltonian reduction of a 2+1-dimensional non-isothermal rotating gas cloud system, Nonlinearity, 24 (2011), 3165-3178.
doi: 10.1088/0951-7715/24/11/009. |
[31] |
C. Rogers, B. Malomed and H. An, Ermakov-Ray-Reid reductions of variational approximations in nonlinear optics, Stud. Appl. Math., 129 (2012), 389-413.
doi: 10.1111/j.1467-9590.2012.00557.x. |
[32] |
C. Rogers and H. An, On a 2+1-dimensional Madelung system with logarithmic and de Broglie-Bohm quantum potentials. Ermakov reduction, Physica Scripta, 84 (2011), 045004. |
[33] |
C. Rogers, W. K. Schief and W. H. Hui, On complex-lamellar motion of a Prim gas, J. Math. Anal. Appl., 266 (2002), 55-69.
doi: 10.1006/jmaa.2001.7685. |
[34] |
W. K. Schief, C. Rogers and A. Bassom, Ermakov systems with arbitrary order and dimension: Structure and linearization, J. Phys. A: Math. Gen., 29 (1996), 903-911.
doi: 10.1088/0305-4470/29/4/017. |
[35] |
W. K. Schief, H. An and C. Rogers, Universal and integrable aspects of an elliptic vortex representation in 2+1-dimensional magnetogasdynamics, Stud. Appl. Math., 130 (2013), 49-79.
doi: 10.1111/j.1467-9590.2012.00559.x. |
[36] |
A. W. Snyder and J. D. Mitchell, Mighty morphing and spatial solitons and bullets, Opt. Lett., 22 (1997), 16-18.
doi: 10.1364/OL.22.000016. |
[37] |
C. Tsallis, Introduction to Nonextensive Statistical Mechanics - Approaching a Complex World, Springer, New York, 2009. |
[38] |
W. G. Wagner, H. A. Haus and J. H. Marburger, Large-scale self-trapping of optical beams in the paraxial ray approximation, Phys. Rev., 175 (1968), 256-266.
doi: 10.1103/PhysRev.175.256. |
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