A family of nonlinear Schrödinger equations admitting \(q\)-plane wave solutions. (English) Zbl 1375.35507
Summary: Nonlinear Schrödinger equations with power-law nonlinearities have attracted considerable attention recently. Two previous proposals for these types of equations, corresponding respectively to the Gross-Pitaievsky equation and to the one associated with nonextensive statistical mechanics, are here unified into a single, parameterized family of nonlinear Schrödinger equations. Power-law nonlinear terms characterized by exponents depending on a real index \(q\), typical of nonextensive statistical mechanics, are considered in such a way that the Gross-Pitaievsky equation is recovered in the limit \(q \rightarrow 1\). A classical field theory shows that, due to these nonlinearities, an extra field \(\Phi (\overrightarrow{x}, t)\) (besides the usual one \(\Psi (\overrightarrow{x}, t)\)) must be introduced for consistency. The new field can be identified with \(\Psi^\ast (\overrightarrow{x}, t)\) only when \(q \to 1\). For \(q \neq 1\) one has a pair of coupled nonlinear wave equations governing the joint evolution of the complex valued fields \(\Psi (\overrightarrow{x}, t)\) and \(\Phi (\overrightarrow{x}, t)\). These equations reduce to the usual pair of complex-conjugate ones only in the \(q \to 1\) limit. Interestingly, the nonlinear equations obeyed by \(\Psi (\overrightarrow{x}, t)\) and \(\Phi (\overrightarrow{x}, t)\) exhibit a common, soliton-like, traveling solution, which is expressible in terms of the \(q\)-exponential function that naturally emerges within nonextensive statistical mechanics.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35C08 | Soliton solutions |
| 05A30 | \(q\)-calculus and related topics |
Keywords:
nonlinear Schrödinger equations; classical field theory; nonextensive thermostatistics; nonadditive entropiesReferences:
| [1] | Scott, A. C., The Nonlinear Universe (2007), Springer: Springer Berlin |
| [2] | Frank, T. D., Nonlinear Fokker-Planck Equations: Fundamentals and Applications (2005), Springer: Springer Berlin · Zbl 1071.82001 |
| [3] | Pitaevskii, L. P.; Stringari, S., Bose-Einstein Condensation (2003), Clarendon Press: Clarendon Press Oxford · Zbl 1110.82002 |
| [4] | Sulem, C.; Sulem, P.-L., The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (1999), Springer: Springer New York · Zbl 0928.35157 |
| [5] | Plastino, A. R.; Plastino, A., Non-extensive statistical mechanics and generalized Fokker-Planck equation, Physica A, 222, 347 (1995) |
| [6] | Tsallis, C.; Bukman, D. J., Anomalous diffusion in the presence of external forces: exact time-dependent solutions and their thermostatistical basis, Phys. Rev. E, 54, Article R2197 pp. (1996) |
| [7] | Andrade, J. S.; da Silva, G. F.T.; Moreira, A. A.; Nobre, F. D.; Curado, E. M.F., Thermostatistics of overdamped motion of interacting particles, Phys. Rev. Lett., 105, Article 260601 pp. (2010) |
| [8] | Tsallis, C., Introduction to Nonextensive Statistical Mechanics (2009), Springer: Springer New York · Zbl 1172.82004 |
| [9] | Nobre, F. D.; Rego-Monteiro, M. A.; Tsallis, C., Nonlinear relativistic and quantum equations with a common type of solution, Phys. Rev. Lett., 106, Article 140601 pp. (2011) |
| [10] | Tsallis, C., Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys., 52, 479 (1988) · Zbl 1082.82501 |
| [11] | Tirnakli, U.; Ozeren, S. F.; Buyukkilic, F.; Demirhan, D., The effect of nonextensivity on the time development of quantum systems, Z. Phys. B, 104, 341 (1997) |
| [12] | Czachor, M.; Naudts, J., Microscopic foundation of nonextensive statistics, Phys. Rev. E, 59, Article R2497 pp. (1999) |
| [13] | Rogers, C.; Ruggeri, T., Q-Gaussian integrable Hamiltonian reductions in anisentropic gas dynamics, Discrete Contin. Dyn. Syst., Ser. B, 19, 2297 (2014) · Zbl 1303.37026 |
| [14] | Curilef, S.; Plastino, A. R.; Plastino, A., Tsallis’ maximum entropy ansatz leading to exact analytical time dependent wave packet solutions of a nonlinear Schrödinger equation, Physica A, 392, 2631 (2013) · Zbl 1395.35172 |
| [15] | Plastino, A. R.; Souza, A. M.C.; Nobre, F. D.; Tsallis, C., Stationary and uniformly accelerated states in nonlinear quantum mechanics, Phys. Rev. A, 90, Article 062134 pp. (2014) |
| [16] | Bountis, T.; Nobre, F. D., Travelling-wave and separated variable solutions of a nonlinear Schrödinger equation, J. Math. Phys., 57, Article 082106 pp. (2016) · Zbl 1345.35097 |
| [17] | Toranzo, I. V.; Plastino, A. R.; Dehesa, J. S.; Plastino, A., Quasi-stationary states of the NRT nonlinear Schrödinger equation, Physica A, 392, 3945 (2013) · Zbl 1395.81113 |
| [18] | Alves, L. G.A.; Ribeiro, H. V.; Santos, M. A.F.; Mendes, R. S.; Lenzi, E. K., Solutions for a \(q\)-generalized Schrödinger equation of entangled interacting particles, Physica A, 429, 35 (2015) · Zbl 1400.81199 |
| [19] | Pennini, F.; Plastino, A. R.; Plastino, A., Pilot wave approach to the NRT nonlinear Schrödinger equation, Physica A, 403, 195 (2014) · Zbl 1395.35174 |
| [20] | Plastino, A.; Rocca, M. C., From the hypergeometric differential equation to a non-linear Schrödinger one, Phys. Lett. A, 379, 2690 (2015) · Zbl 1361.35168 |
| [21] | Nobre, F. D.; Rego-Monteiro, M. A.; Tsallis, C., A generalized nonlinear Schrödinger equation: classical field-theoretic approach, Europhys. Lett., 97, Article 41001 pp. (2012) |
| [22] | Plastino, A. R.; Tsallis, C., Dissipative effects in nonlinear Klein-Gordon dynamics, Europhys. Lett., 113, Article 50005 pp. (2016) |
| [23] | Nobre, F. D.; Plastino, A. R., Generalized nonlinear Proca equation and its free-particle solutions, Eur. Phys. J. C, 76, 343 (2016) |
| [24] | Santos, A. P.; Pereira, F. I.M.; Silva, R.; Alcaniz, J. S., Consistent nonadditive approach and nuclear equation of state, J. Phys. G, Nucl. Part. Phys., 41, Article 055105 pp. (2014) |
| [25] | Pereira, F. I.M.; Silva, R.; Alcaniz, J. S., Nonextensive effects on the relativistic nuclear equation of state, Phys. Rev. C, 76, Article 015201 pp. (2007) · Zbl 1234.81147 |
| [26] | Lavagno, A.; Pigato, D., Nonextensive statistical effects and strangeness production in hot and dense nuclear matter, J. Phys. G, Nucl. Part. Phys., 39, Article 125106 pp. (2012) |
| [27] | Conroy, J. M.; Miller, H. G., Color superconductivity and Tsallis statistics, Phys. Rev. D, 78, Article 054010 pp. (2008) |
| [28] | Ourabah, K.; Tribeche, M., Implication of Tsallis entropy in the Thomas-Fermi model for self-gravitating fermions, Ann. Phys., 342, 78 (2014) |
| [29] | Borges, E. P., On a q-generalization of circular and hyperbolic functions, J. Phys. A, 31, 5281 (1998) · Zbl 0923.33012 |
| [30] | Beraldo e. Silva, L.; Mamon, G. A.; Duarte, M.; Wojtak, R.; Peirani, S.; Bou, G., q-Gaussian 3D velocity distributions in CDM haloes, Mon. Not. R. Astron. Soc., 452, 944 (2015) |
| [31] | Cardone, V. F.; Leubner, M. P.; Del Popolo, A., Galaxy models as equilibrium configurations in nonextensive statistics, Mon. Not. R. Astron. Soc., 414, 2265 (2011) |
| [32] | Das, S.; Bhaduri, R. K., Dark matter and dark energy from Bose-Einstein condensate, Class. Quantum Gravity, 32, Article 105003 pp. (2015) · Zbl 1328.83193 |
| [33] | Atazadeh, K.; Darabi, F.; Mousavi, M., Dark matter as the Bose-Einstein condensation in loop quantum cosmology, Eur. Phys. J. C, 76, 327 (2016) |
| [34] | Sanders, R. H., The Dark Matter Problem: A Historical Perspective (2010), Cambridge University Press · Zbl 1204.85004 |
| [35] | Ablowitz, M. J.; Musslimani, Z. H., Integrable nonlocal nonlinear Schrödinger equation, Phys. Rev. Lett., 110, Article 064105 pp. (2013) |
| [36] | Nassar, A. B.; Miret-Artés, S., Dividing line between quantum and classical trajectories in a measurement problem: Bohmian time constant, Phys. Rev. Lett., 111, Article 150401 pp. (2013) |
| [37] | Yamano, T., Traveling wave solution for a logarithmic nonlinear Schrodinger equation, Wave Motion, 67, 116 (2016) · Zbl 1524.35610 |
| [38] | Lavagno, A.; Scarfone, A. M.; Narayana Swany, P., Classical and quantum q-deformed physical systems, Eur. Phys. J. C, 47, 253 (2006) · Zbl 1191.81137 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
