We consider a recently proposed nonlinear Schroedinger equation exhibiting soliton-like solutions of the power-law form

$e_q^{i(kx-wt)}$
eqi(kx−wt)⁠, involving the q-exponential function which naturally emerges within nonextensive thermostatistics [
$e_q^z \equiv [1+(1\break-q)z]^{1/(1-q)}$
eqz≡[1+(1−q)z]1/(1−q)⁠, with
$e_1^z=e^z$
e1z=ez]. Since these basic solutions behave like free particles, obeying p = ℏk, E = ℏω, and E = p2/2m (1 ⩽ q < 2), it is relevant to investigate how they change under the effect of uniform acceleration, thus providing the first steps towards the application of the aforementioned nonlinear equation to the study of physical scenarios beyond free particle dynamics. We investigate first the behaviour of the power-law solutions under Galilean transformation and discuss the ensuing Doppler-like effects. We consider then constant acceleration, obtaining new solutions that can be equivalently regarded as describing a free particle viewed from an uniformly accelerated reference frame (with acceleration a) or a particle moving under a constant force −ma. The latter interpretation naturally leads to the evolution equation
$i\hbar \frac{\partial }{\partial t}\left( \frac{\Phi }{\Phi _0} \right) \,\, = \,\, - \frac{1}{2-q}\frac{\hbar ^2}{2m} \frac{\partial ^2}{\partial x^2} \left[\left( \frac{\Phi }{\Phi _0} \right)^{2-q}\right] + V(x)\left( \frac{\Phi }{\Phi _0} \right)^{q}$
iℏ∂∂tΦΦ0=−12−qℏ22m∂2∂x2ΦΦ02−q+V(x)ΦΦ0q with V(x) = max. Remarkably enough, the potential V couples to Φq, instead of coupling to Φ, as happens in the familiar linear case (q = 1).

Topics
Entropy, General relativity, Special relativity, Cognitive science, Operator theory, Probability theory, Plasma physics, de Broglie wavelength, Schrodinger equations, Statistical mechanics
1.
C.
 
Sulem
 and
P.-L.
 
Sulem
,
The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse
(
Springer
,
New York
,
1999
).
Google Scholar
 
2.
T. D.
 
Frank
,
Nonlinear Fokker-Planck Equations: Fundamentals and Applications
(
Springer
,
Berlin
,
2005
).
Google Scholar
 
3.
A. C.
 
Scott
,
The Nonlinear Universe
(
Springer
,
Berlin
,
2007
).
Google Scholar
 
4.
A. D.
 
Polyanin
 and
V. F.
 
Zaitsev
,
Handbook of Nonlinear Partial Differential Equations
(
Chapman and Hall/CRC
,
Boca Raton
,
2004
).
Google Scholar
 
5.
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
,
140601
(
2011
).
https://doi.org/10.1103/PhysRevLett.106.140601
Google Scholar
Crossref
Search ADS
PubMed
 
6.
F. D.
 
Nobre
,
M. A.
 
Rego-Monteiro
, and
C.
 
Tsallis
, “
A generalized nonlinear Schroedinger equation: Classical field-theoretic approach
,”
EPL
 
97
,
41001
(
2012
).
https://doi.org/10.1209/0295-5075/97/41001
Google Scholar
Crossref
Search ADS
 
7.
J. S.
Andrade
Jr. ,
G. F. T.
da Silva
 ,
A. A.
Moreira
 ,
F. D.
Nobre
 , and
E. M. F.
Curado
 , “
Thermostatistics of overdamped motion of interacting particles
,”
Phys. Rev. Lett.
 
105
,
260601
(
2010
);
https://doi.org/10.1103/PhysRevLett.105.260601
Crossref
Search ADS
[PubMed]
Google Scholar
 
Y.
Levin
 and
R.
Pakter
 , “
Comment on: Thermostatistics of overdamped motion of interacting particles
,”
Phys. Rev. Lett.
 
107
,
088901
(
2011
);
https://doi.org/10.1103/PhysRevLett.107.088901
Crossref
Search ADS
[PubMed]
Google Scholar
 
J. S.
 
Andrade
 Jr.,
G. F. T.
 
da Silva
,
A. A.
 
Moreira
,
F. D.
 
Nobre
, and
E. M. F.
 
Curado
, “
Andrade et al. reply
,”
Phys. Rev. Lett.
 
107
,
088902
(
2011
).
https://doi.org/10.1103/PhysRevLett.107.088902
Crossref
Search ADS
[PubMed]
Google Scholar
 
8.
M. S.
 
Ribeiro
,
F. D.
 
Nobre
, and
E. M. F.
 
Curado
, “
Time evolution of interacting vortices under overdamped motion
,”
Phys. Rev. E
 
85
,
021146
(
2012
).
https://doi.org/10.1103/PhysRevE.85.021146
Google Scholar
Crossref
Search ADS
 
9.
C.
 
Tsallis
, “
Possible generalization of Boltzmann-Gibbs statistics
,”
J. Stat. Phys.
 
52
,
479
(
1988
).
https://doi.org/10.1007/BF01016429
Google Scholar
Crossref
Search ADS
 
10.
C.
 
Tsallis
,
Introduction to Nonextensive Statistical Mechanics
(
Springer
,
New York
,
2009
).
Google Scholar
 
11.
C.
 
Beck
, “
Generalised information and entropy measures in physics
,”
Contemp. Phys.
 
50
,
495
(
2009
).
https://doi.org/10.1080/00107510902823517
Google Scholar
Crossref
Search ADS
 
12.
A.
 
Carati
,
S. L.
 
Cacciatori
, and
L.
 
Galgani
, “
Discrete matter, far fields, and dark matter
,”
EPL
 
83
,
59002
(
2008
).
https://doi.org/10.1209/0295-5075/83/59002
Google Scholar
Crossref
Search ADS
 
13.
S.
Umarov
 ,
C.
Tsallis
 , and
S.
Steinberg
 ,
Milan J. Math.
 
76
,
307
(
2008
);
https://doi.org/10.1007/s00032-008-0087-y
Crossref
Search ADS
Google Scholar
 
S.
 
Umarov
,
C.
 
Tsallis
,
M.
 
Gell-Mann
, and
S.
 
Steinberg
, “
Generalization of symmetric α-stable Lévy distributions for q > 1
,”
J. Math. Phys.
 
51
,
033502
(
2010
).
https://doi.org/10.1063/1.3305292
Crossref
Search ADS
[PubMed]
Google Scholar
 
14.
A. R.
Plastino
 and
A.
Plastino
 , “
Non-extensive statistical mechanics and generalized Fokker-Planck equation
,”
Physica A
 
222
,
347
(
1995
);
https://doi.org/10.1016/0378-4371(95)00211-1
Crossref
Search ADS
Google Scholar
 
C.
 
Tsallis
 and
D. J.
 
Bukman
, “
Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis
,”
Phys. Rev. E
 
54
,
R2197
(
1996
).
https://doi.org/10.1103/PhysRevE.54.R2197
Crossref
Search ADS
Google Scholar
 
15.
P.
 
Douglas
,
S.
 
Bergamini
, and
F.
 
Renzoni
, “
Tunable Tsallis distributions in dissipative optical lattices
,”
Phys. Rev. Lett.
 
96
,
110601
(
2006
).
https://doi.org/10.1103/PhysRevLett.96.110601
Google Scholar
Crossref
Search ADS
PubMed
 
16.
B.
 
Liu
 and
J.
 
Goree
, “
Superdiffusion and non-Gaussian statistics in a driven-dissipative 2D dusty plasma
,”
Phys. Rev. Lett.
 
100
,
055003
(
2008
).
https://doi.org/10.1103/PhysRevLett.100.055003
Google Scholar
Crossref
Search ADS
PubMed
 
17.
R. G.
 
DeVoe
, “
Power-law distributions for a trapped ion interacting with a classical buffer gas
,”
Phys. Rev. Lett.
 
102
,
063001
(
2009
).
https://doi.org/10.1103/PhysRevLett.102.063001
Google Scholar
Crossref
Search ADS
PubMed
 
18.
R. M.
 
Pickup
,
R.
 
Cywinski
,
C.
 
Pappas
,
B.
 
Farago
, and
P.
 
Fouquet
, “
Generalized spin-glass relaxation
,”
Phys. Rev. Lett.
 
102
,
097202
(
2009
).
https://doi.org/10.1103/PhysRevLett.102.097202
Google Scholar
Crossref
Search ADS
PubMed
 
19.
V.
 
Khachatryan
 et al (
CMS Collaboration
), “
Transverse-momentum and pseudorapidity distributions of charged hadrons in pp collisions at
$\sqrt{s} = 7\ {\rm TeV}$
s=7 TeV
,”
Phys. Rev. Lett.
 
105
,
022002
(
2010
).
https://doi.org/10.1103/PhysRevLett.105.022002
Google Scholar
Crossref
Search ADS
PubMed
 
20.
E. P.
 
Borges
, “
On a generalization of circular and hyperbolic functions
,”
J. Phys. A
 
31
,
5281
(
1998
).
https://doi.org/10.1088/0305-4470/31/23/011
Google Scholar
Crossref
Search ADS
 
21.
M.
Jauregui
 and
C.
Tsallis
 , “
New representations of π and Dirac delta using the nonextensive-statistical-mechanics q-exponential function
,”
J. Math. Phys.
 
51
,
063304
(
2010
);
https://doi.org/10.1063/1.3431981
Crossref
Search ADS
Google Scholar
 
A.
 
Chevreuil
,
A.
 
Plastino
, and
C.
 
Vignat
, “
On a conjecture about Dirac's delta representation using q-exponentials
,”
J. Math. Phys.
 
51
,
093502
(
2010
).
https://doi.org/10.1063/1.3478886
Crossref
Search ADS
Google Scholar
 
22.
G.
 
Kaniadakis
,
A.
 
Lavagno
, and
P.
 
Quarati
, “
Wigner equation for particles obeying an exclusion-inclusion principle
,”
Phys. Rev. E
 
57
,
1395
(
1998
).
https://doi.org/10.1103/PhysRevE.57.1395
Google Scholar
Crossref
Search ADS
 
23.
A.
 
Peres
,
Quantum Theory: Concepts and Methods
(
Kluwer
,
Dordrecht
,
1993
).
Google Scholar
 
© 2013 American Institute of Physics.
2013
American Institute of Physics
You do not currently have access to this content.