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September  2014, 19(7): 2297-2312. doi: 10.3934/dcdsb.2014.19.2297

q-Gaussian integrable Hamiltonian reductions in anisentropic gasdynamics

Colin Rogers 1, and  Tommaso Ruggeri 2,

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

Received  March 2013 Revised  March 2014 Published  August 2014

Integrable reductions in non-isothermal spatial gasdynamics are isolated corresponding to q-Gaussian density distributions. The availability of a Tsallis parameter q in the reductions permits the construction via a Madelung transformation of wave packet solutions of a class of associated q-logarithmic nonlinear Schrödinger equations involving a de Broglie-Bohm quantum potential term.
Keywords: gasdynamics., Q-Gaussian integrability, Tsallis statistical mechanics.
Mathematics Subject Classification: 37K10, 76B4.
Citation: Colin Rogers, Tommaso Ruggeri. q-Gaussian integrable Hamiltonian reductions in anisentropic gasdynamics. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2297-2312. doi: 10.3934/dcdsb.2014.19.2297
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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