Using stationary solutions of the linearized two-dimensional Gross-Pitaevskii
equation, we describe the ``ship wave'' pattern occurring in the supersonic
flow of a Bose-Einstein condensate past an obstacle. It is shown that these
``ship waves'' are generated outside the Mach cone. The developed analytical
theory is confirmed by numerical simulations of the flow past body problem in
the frame of the full non-stationary Gross-Pitaevskii equation.Comment: 5 pages, 4 figure

A. Gammal

A. M. Kamchatnov

G. A. El

G. B. Whitham

L. P. Pitaevskii

R. S. Johnson

Yu. G. Gladush

English

arXiv

Crossref

Radiation of linear waves in the stationary flow of a Bose-Einstein condensate past an obstacle

Using stationary solutions of the linearized two-dimensional Gross-Pitaevskii equation, we describe the wave pattern occurring in the supersonic flow of a Bose-Einstein condensate past an obstacle. It is shown that these waves are generated outside the Mach cone. The developed analytical theory is confirmed by numerical simulations of the flow past body problem in the frame of the full nonstationary Gross-Pitaevskii equation. Relation of the developed theory with recent experiments is discussed

Gladush, Yu. G.

El, Gennady

Gammal, A.

Kamchatnov, A. M.

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Northumbria Research LinkCitation: Gladush, Yu. G., El, Gennady, Gammal, A. and Kamchatnov, A. M. (2007) Radiation of linear  waves in the stationary flow of a Bose-Einstein condensate past an obstacle. Physical Review A, 75 (3). 033619. ISSN 1050-2947 Published by: American Physical SocietyURL: https://doi.org/10.1103/PhysRevA.75.033619 <https://doi.org/10.1103/PhysRevA.75.033619>This version was downloaded from Northumbria Research Link: http://nrl.northumbria.ac.uk/42792/Northumbria University has developed Northumbria Research Link (NRL) to enable users to access the University’s research output. Copyright © and moral rights for items on NRL are retained by the individual author(s) and/or other copyright owners.  Single copies of full items can be reproduced, displayed or performed, and given to third parties in any format or medium for personal research or study, educational, or not-for-profit purposes without prior permission or charge, provided the authors, title and full bibliographic details are given, as well as a hyperlink and/or URL to the original metadata page. The content must not be changed in any way. Full items must not be sold commercially in any  format or medium without formal permission of the copyright holder.  The full policy is available online: http://nrl.northumbria.ac.uk/pol  i cies.html  This  document  may differ  from the  final,  published version of  the research  and has been made available online in accordance with publisher policies. To read and/or cite from the published version of the research, please visit the publisher’s website (a subscription may be required.)                        arXiv:cond-mat/0611149v1  [cond-mat.other]  6 Nov 2006Linear “ship waves” generated in stationary flow of a Bose-Einstein condensate pastan obstacleYu.G. Gladush1, G.A. El2, A. Gammal3, and A.M. Kamchatnov11Institute of Spectroscopy, Russian Academy of Sciences, Troitsk 142190, Moscow Region, Russia2 Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK3 Instituto de F´ısica, Universidade de Sa˜o Paulo, 05315-970, C.P. 66318 Sa˜o Paulo, BrazilUsing stationary solutions of the linearized two-dimensional Gross-Pitaevskii equation, we describethe “ship wave” pattern occurring in the supersonic flow of a Bose-Einstein condensate past anobstacle. It is shown that these “ship waves” are generated outside the Mach cone. The developedanalytical theory is confirmed by numerical simulations of the flow past body problem in the frameof the full non-stationary Gross-Pitaevskii equation.PACS numbers: 03.75.KkINTRODUCTIONExperimental creation of Bose-Einstein condensate(BEC) has led to emergence of a new field of nonlinearwave dynamics owing to a remarkable richness of non-linear wave patterns supported by this medium. First,vortices and bright and dark solitons were observed andtheir dynamics was studied theoretically in frameworkof the mean-field approach (see, e.g. [1] and referencestherein). Then, dispersive shocks generated by a largeand sharp disturbance of BEC were found in experiment[2, 3] and explained theoretically [3, 4] in framework ofthe Whitham theory of modulations of nonlinear waves(see also numerical experiment in [5]). At last, the sta-tionary waves generated by supersonic flow of BEC pastobstacles have been recently observed [6]. They werestudied in [7, 8, 9] where the main focus was on the non-linear component representing a train of solitons (in thesimplest case of a single soliton [8]) or, more precisely,having a form of a modulated nonlinear periodic wave.The theory developed in [7, 8, 9] shows that there existstationary spatial solutions of the Gross-Pitaevskii (GP)equation which describe nonlinear waves supported by asupersonic BEC flow with Mach numberM =ucs> 1, (1)where u is velocity of the oncoming flow at x → −∞and cs is the sound speed of the long-wavelength linearwaves. The density n of the condensate, as well as thecomponents of the velocity field, depend on the variablew = x− ay (2)alone, where a is the slope of the phase lines with respectto the y axis and it is supposed that the velocity of theoncoming flow is directed along the x axis. Then, theMach cone for sound waves with infinitely large wave-length corresponds to the slopeaM =√M2 − 1, (3)and it was shown in [9] that the spatial (oblique) solitonshave a > aM , that is they are located inside the Machcone. In particular, it was shown that the shallow solitonsare formed close to the Mach cone, a − aM ≪ aM , andthey are asymptotically described by the Korteweg-deVries (KdV) equation; and deep solitons have a≫ 1 (i.e.they are formed at small angles with respect to directionof the oncoming flow) and are asymptotically describedby the nonlinear Schro¨dinger (NLS) equation. On thecontrary, the linear waves are generated outside the Machcone and they have a < aM . In fact, these linear waveshad been observed in numerical simulations some yearsago [10] but the theory of their generation has not beendeveloped so far. The aim of this paper is to developsuch a theory.LINEAR WAVES GENERATED IN A BEC FLOWPAST AN OBSTACLEOur analysis is based on the use of the mean-filed de-scription of BEC dynamics in the framework of the GPequationi~∂ψ∂t= − ~22m∆ψ + U(r)ψ + g|ψ|2ψ, (4)where ψ(r) is the order parameter (“condensate wavefunction”), U(r) is the potential which confines atomsof a Bose gas in a trap and/or describes interaction ofthe BEC with the obstacle, and g is an effective couplingconstant arising due to inter-atomic collisions with thes-wave scattering length as,g = 4pi~2as/m, (5)m being the atomic mass. Here we consider the Bose-Einstein condensate with repulsive interaction betweenparticles for which g > 0.As suggested by the experimental set-up [6], we con-sider a two-dimensional flow of a condensate, so that thecondensate wave function ψ depends on only two spatial2coordinates, r = (x, y). To simplify the theory, we as-sume that the characteristic size of the obstacle is muchless than its distance from the center of the trap, so thatthe oncoming flow can be considered as uniform withconstant density n0 of atoms and constant velocity u0directed parallel to x axis (see also estimates in [8]). Itis convenient to transform Eq. (5) to a “hydrodynamic”form by means of the substitutionψ(r, t) =√n(r, t) exp(i~∫ru(r′, t)dr′), (6)where n(r, t) is density of atoms in BEC and u(r, t) de-notes its velocity field, and to introduce dimensionlessvariablesx˜ = x/√2ξ, y˜ = y/√2ξ, t˜ = (cs/2√2ξ)t,n˜ = n/n0, u˜ = u/cs,(7)where ξ = ~/√2mn0g is the BEC healing length and nu-merical constants are introduced for future convenience.As a result of this transformation we obtain the system(we omit tildes for convenience of the notation)12nt +∇(nu) = 0,12ut + (u∇)u+∇n+∇[(∇n)28n2− ∆n4n]= 0,(8)where ∇ = (∂x, ∂y). Since we shall consider waves farenough from the obstacle, the potential is omitted in (8).We are interested in linear waves propagating on thebackground flow with n = 1, u = M , v = 0. Hence, weintroducen = 1 + n1, u =M + u1, v = v1, (9)and linearize the system (8) with respect to small devia-tions n1, u1, v1. As a result we obtain the system12n1,t + u1,x +Mn1,x + v1,y = 0,12u1,t +Mu1,x + n1,x − 14 (n1,xxx + n1,xyy) = 0,12v1,t +Mv1,x + n1,y − 14 (n1,xxy + n1,yyy) = 0,(10)which describes propagation of linear waves in BEC witha uniform flow. We obtain the applicability condition ofthese equations, if we notice that in the linear wave u1 ∼Mn1 and the nonlinear terms of the order of magnitudeu1∇u1 ∼ ∇(Mu1)2 can be neglected as long as they aremuch less than the linear ones ∼ ∇n1. Thus, we get thecriterionn1 ≪ 1/M2. (11)Hence, if M is large enough, the linear theory is appli-cable to description of waves outside the Mach cone farenough from the obstacle, where the density amplituden1 of the wave satisfies the condition (11). For harmonicwaves n1, u1, v1 ∝ exp[i(kxx+kyy)−iωt] the system (10)yields at once the dispersion relationω2=Mkx ± k√1 +k24, (12)where k =√k2x + k2y.Now we consider the stationary wave patterns farenough from the obstacle where the condition (11) is sup-posed to be fulfilled. In fact, this problem is analogousto the Kelvin theory of ship waves generated by a shipmoving in a deep water, but with a different dispersionlaw (12). Hence, we shall follow Kelvin’s method in itsform presented in [11, 12].First, we notice that in a stationary wave pattern ω =0 and, hence, the components of the wavevector k =(kx, ky) are the functions of the space coordinates (x, y)connected with each other by the relationshipG(kx, ky) ≡Mkx + k√1 +k24= 0, (13)where we have taken into account that for chosen geom-etry of the BEC flow the wave must propagate upwindi.e. kx < 0.Next, the “ship wave” pattern corresponds to a mod-ulated two-dimensional wave where the wavevector k isa gradient of the phase [11],θ =∫r0k · dr. (14)Hence, the components (kx, ky) satisfy the condition∂kx∂y− ∂ky∂x= 0, (15)which, with an account of (13), yields the equation forky∂ky∂x− f ′(ky)∂ky∂y= 0, (16)where f ′(ky) is defined by the derivative of an implicitfunction (13):f ′ = −∂G/∂ky∂G/∂kx. (17)It follows from Eq. (16) that kx and ky are constant alongthe characteristics defined as solutions of the equationdydx= −f ′(ky). (18)At last, since at large distances from the obstacle, thelatter can be considered as a point-like source of waves,3χ ηµkyxAFIG. 1: Sketch of the wave front with wavevector k normalto it at the point A and coordinates χ and η defining theparameters of the wave pattern at this point.-2 2 4 6 8 10x-15-10-551015yFIG. 2: Wave pattern of stationary linear waves generated inthe flow of BEC past a point-like obstacle. Dashed line cor-responds to the Mach cone of linear waves in the long wave-length limit.the resulting stationary wave is centered, that is the char-acteristics intersect at the point (x, y) = (0, 0) of the lo-cation of the obstacle. Hence, we obtain the solutionyx= tanχ = −∂G/∂ky∂G/∂kx, (19)where χ is the angle between the x axis and the radius-vector r of the point A with wavevector (kx, ky); seeFig. 1 where other convenient parameters are also de-fined, namely, the angle η between the wavevector kand a negative direction of the x axis, and the angleµ = pi − χ− η between vectors k and r.Thus, we havekx = −k cos η, ky = k sin η, (20)G(k, η) = −Mk cos η + k√1 +k24= 0. (21)Then elementary calculation gives for (19) the expressiontanχ =(2M2 − 1− tan2 η) tan η(M2 + 1) tan2 η − (M2 − 1) (22)and Eq. (21) yieldsk = 2√M2 cos2 η − 1. (23)Thus, we have found that for a fixed value of η the compo-nents (kx, ky) are constant along the line χ = const withχ defined by Eq. (22) and the length of the wavevectorgiven by (23). Therefore the phase (14) can be conve-niently calculated by integration along the line χ = constwith constant vector k, so thatθ = (k cosµ)r. (24)It means that the lines of constant phase (e.g. the wavecrests) θ are determined in parametrical form by Eq. (22)andr =θk cosµ, (25)where k is given by Eq. (23) and µ can be calculatedfrom tanµ = − tan(χ+ η) which gives, after elementaryalgebra, the expressiontanµ =2M2k2sin 2η. (26)This expression permits one to express Eq. (25) asr =4θk3√M2(M2 − 2) cos2 η + 1 (27)and Eq. (22) astanχ =(1 + k2/2) tan ηM2 − (1 + k2/2) . (28)At last, the curves with constant phase θ are given inCartesian coordinates by the formulasx = r cosχ =4θk3cos η(1−M2 cos 2η),y = r sinχ =4θk3sin η(2M2 cos2 η − 1).(29)Thus, we have found the expressions describing the linearwave pattern in a parametric form where the parameterη changes in the interval− arccos 1M≤ η ≤ arccos 1M. (30)For η = 0 we have x < 0 and y = 0, that is small valuesof η correspond to the wave before the obstacle. The4boundary values η = ± arccos (1/M) correspond to thelinesxy= ±√M2 − 1 = ±aM , (31)that is the curves of constant phase become asymptoti-cally the straight lines parallel to the Mach cone. Thegeneral pattern is shown in Fig. 2.NUMERICAL SIMULATIONSWe have compared the above approximate analyticaltheory with the exact numerical solution of the GP equa-tion, which in non-dimensional units (7) takes the stan-dard formiψt = − 12 (ψxx + ψyy) + U(x, y)ψ + |ψ|2ψ, (32)which corresponds for U = 0 to the system (8) withψ =√n exp(i∫ru(r′, t)dr′).In our simulations the obstacle was modeled by an impen-etrable disk with radius r = 1. Such an obstacle intro-duces large enough perturbation into the flow to generatean oblique solitons pair behind it (see [8]). We assumethat at the initial moment t = 0 there is no disturbancein the condensate, so that it is described by the planewave functionψ(x, y)|t=0 = exp(iMx)corresponding to a uniform condensate flow. For largeenough evolution time, the wave pattern around the ob-stacle tends to a stationary structure. An example ofsuch a structure for M = 2 is shown in Fig. 2. A dashedline corresponds to the analytic theory developed in thepreceding Section; as we see it agrees very well with thenumerical results for M = 2.The condition (11) indicates that the nonlinear effectsgrow up with increase of M . To demonstrate this explic-itly, we have compared the wavelength λ at y = 0 calcu-lated using the developed linear analytic theory with thesame parameter obtained from our full numerical simu-lations. According to linear theory, the wavelength aty = 0 (i.e. η = 0) is constant and equal toλ =2pik=pi√M2 − 1 , (33)where we have used Eq. (23). In Fig. 4 we compare thisdependence of the wavelength λ on the Mach numberM with the results of numerical simulations at the pointwith n1 ≈ 0.1. As we see, Eq. (33) is very accuratefor values ofM satisfying the condition (11) and discrep-ancy between analytical and numerical results slightly in-creases with increase ofM . In general, this plot confirmsvalidity of a linear theory in the region of its applicability.FIG. 3: Numerically calculated wave pattern of stationarylinear waves generated in the flow of BEC past an obstacle.The Mach number is equal to M = 2 and the radius of animpenetrable obstacle to r = 1. Dashed line corresponds tothe linear analytical theory for the line of constant phase. Itcorresponds to Eqs. (29) but the curve is shifted to 1 unit oflength to the left from the center of the obstacle for betterfitting to numerics. It should be noted that this distance isnegligibly small in our theory corresponding to a point-likeobstacle. A pair of oblique dark lines behind the obstaclecorrespond to spatial solitons studied in [8].1 2 3 4 5 M0123anal  ytical numericalλFIG. 4: Dependence of the wavelength of the wave generatedbefore the obstacle on the Mach number calculated numeri-cally (full time-dependent GP equation)—dots, and analyti-cally (linear theory)—dashed line.CONCLUSIONWe have developed here the theory of linear waves gen-erated by the flow of BEC past an obstacle. The linearapproximation is correct for small enough amplitudes ofthe perturbation. This condition is satisfied in the caseof small disturbance introduced by the obstacle and nottoo high values of the Mach number. Our numerical sim-ulations confirm the analytical theory in the region of its5applicability.AcknowledgementsAMK and YGG thank RFBR (grant 05-02-17351) andAG and GAE thank FAPESP and CNPq for financialsupport.[1] L.P. Pitaevskii and S. Stringari, Bose-Einstein Conden-sation, Cambridge University Press, Cambridge, 2003.[2] T.P. Simula, P. Engels, I. Coddington, V. Schweikhard,E.A. Cornell, and R.J. Ballagh, Phys. Rev. Lett. 94,080404 (2005).[3] M.A. Hoefer, M.J. Ablowitz, I. Coddington, E.A. Cor-nell, P. Engels, and V. Schweikhard, Phys. Rev. A 74,023623 (2006).[4] A.M. Kamchatnov, A. Gammal, and R.A. Kraenkel,Phys. Rev. A 69, 063605 (2004).[5] B. Damski, Phys.Rev. A 69, 043610 (2004).[6] E.A. Cornell, in “Conference on NonlinearWaves, Integrable Systems and their Applica-tions”, (Colorado Springs, June 2005); available athttp://jilawww.colorado.edu/bec/papers.html.[7] G.A. El and A.M. Kamchatnov, Phys. Lett A 350, 192(2006); erratum: Phys. Lett. A 352, 554 (2006).[8] G.A. El, A. Gammal, and A.M. Kamchatnov, Phys. Rev.Lett. 97, 180405 (2006).[9] G.A. El, Yu.G. Gladush, and A.M. Kamchatnov, e-printnlin.PS/0609022.[10] T. Winiecki, J.F. McCann, and C.S. Adams, Phys. Rev.Lett. 82, 5186 (1999).[11] G.B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York (1974).[12] R.S. Johnson, A Modern Introduction to the Mathemati-cal Theory of Water Waves, Cambridge University Press,Cambridge, (1997).

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Citation: Gladush, Yu. G., El, Gennady, Gammal, A. and Kamchatnov, A. M. (2007) Radiation of linear  
waves in the stationary flow of a Bose-Einstein condensate past an obstacle. Physical Review A, 75 
(3). 033619. ISSN 1050-2947 
Published by: American Physical Society
URL: https://doi.org/10.1103/PhysRevA.75.033619 <https://doi.org/10.1103/PhysRevA.75.033619>
This version was downloaded from Northumbria Research Link: http://nrl.northumbria.ac.uk/42792/
Northumbria University has developed Northumbria Research Link (NRL) to enable users to access 
the University’s research output. Copyright © and moral rights for items on NRL are retained by the 
individual author(s) and/or other copyright owners.  Single copies of full items can be reproduced, 
displayed or performed, and given to third parties in any format or medium for personal research or 
study, educational, or not-for-profit purposes without prior permission or charge, provided the authors, 
title and full bibliographic details are given, as well as a hyperlink and/or URL to the original metadata 
page. The content must not be changed in any way. Full items must not be sold commercially in any  
format or medium without formal permission of the copyright holder.  The full policy is available online: 
http://nrl.northumbria.ac.uk/pol  i cies.html  
This  document  may differ  from the  final,  published version of  the research  and has been made 
available online in accordance with publisher policies. To read and/or cite from the published version 
of the research, please visit the publisher’s website (a subscription may be required.)
                        
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Linear “ship waves” generated in stationary flow of a Bose-Einstein condensate past
an obstacle
Yu.G. Gladush1, G.A. El2, A. Gammal3, and A.M. Kamchatnov1
1Institute of Spectroscopy, Russian Academy of Sciences, Troitsk 142190, Moscow Region, Russia
2 Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
3 Instituto de F́ısica, Universidade de São Paulo, 05315-970, C.P. 66318 São Paulo, Brazil
Using stationary solutions of the linearized two-dimensional Gross-Pitaevskii equation, we describe
the “ship wave” pattern occurring in the supersonic flow of a Bose-Einstein condensate past an
obstacle. It is shown that these “ship waves” are generated outside the Mach cone. The developed
analytical theory is confirmed by numerical simulations of the flow past body problem in the frame
of the full non-stationary Gross-Pitaevskii equation.
PACS numbers: 03.75.Kk
INTRODUCTION
Experimental creation of Bose-Einstein condensate
(BEC) has led to emergence of a new field of nonlinear
wave dynamics owing to a remarkable richness of non-
linear wave patterns supported by this medium. First,
vortices and bright and dark solitons were observed and
their dynamics was studied theoretically in framework
of the mean-field approach (see, e.g. [1] and references
therein). Then, dispersive shocks generated by a large
and sharp disturbance of BEC were found in experiment
[2, 3] and explained theoretically [3, 4] in framework of
the Whitham theory of modulations of nonlinear waves
(see also numerical experiment in [5]). At last, the sta-
tionary waves generated by supersonic flow of BEC past
obstacles have been recently observed [6]. They were
studied in [7, 8, 9] where the main focus was on the non-
linear component representing a train of solitons (in the
simplest case of a single soliton [8]) or, more precisely,
having a form of a modulated nonlinear periodic wave.
The theory developed in [7, 8, 9] shows that there exist
stationary spatial solutions of the Gross-Pitaevskii (GP)
equation which describe nonlinear waves supported by a
supersonic BEC flow with Mach number
M =
u
cs
> 1, (1)
where u is velocity of the oncoming flow at x → −∞
and cs is the sound speed of the long-wavelength linear
waves. The density n of the condensate, as well as the
components of the velocity field, depend on the variable
w = x− ay (2)
alone, where a is the slope of the phase lines with respect
to the y axis and it is supposed that the velocity of the
oncoming flow is directed along the x axis. Then, the
Mach cone for sound waves with infinitely large wave-
length corresponds to the slope
aM =
√
M2 − 1, (3)
and it was shown in [9] that the spatial (oblique) solitons
have a > aM , that is they are located inside the Mach
cone. In particular, it was shown that the shallow solitons
are formed close to the Mach cone, a − aM ≪ aM , and
they are asymptotically described by the Korteweg-de
Vries (KdV) equation; and deep solitons have a≫ 1 (i.e.
they are formed at small angles with respect to direction
of the oncoming flow) and are asymptotically described
by the nonlinear Schrödinger (NLS) equation. On the
contrary, the linear waves are generated outside the Mach
cone and they have a < aM . In fact, these linear waves
had been observed in numerical simulations some years
ago [10] but the theory of their generation has not been
developed so far. The aim of this paper is to develop
such a theory.
LINEAR WAVES GENERATED IN A BEC FLOW
PAST AN OBSTACLE
Our analysis is based on the use of the mean-filed de-
scription of BEC dynamics in the framework of the GP
equation
i~
∂ψ
∂t
= − ~
2
2m
∆ψ + U(r)ψ + g|ψ|2ψ, (4)
where ψ(r) is the order parameter (“condensate wave
function”), U(r) is the potential which confines atoms
of a Bose gas in a trap and/or describes interaction of
the BEC with the obstacle, and g is an effective coupling
constant arising due to inter-atomic collisions with the
s-wave scattering length as,
g = 4π~2as/m, (5)
m being the atomic mass. Here we consider the Bose-
Einstein condensate with repulsive interaction between
particles for which g > 0.
As suggested by the experimental set-up [6], we con-
sider a two-dimensional flow of a condensate, so that the
condensate wave function ψ depends on only two spatial
2
coordinates, r = (x, y). To simplify the theory, we as-
sume that the characteristic size of the obstacle is much
less than its distance from the center of the trap, so that
the oncoming flow can be considered as uniform with
constant density n0 of atoms and constant velocity u0
directed parallel to x axis (see also estimates in [8]). It
is convenient to transform Eq. (5) to a “hydrodynamic”
form by means of the substitution
ψ(r, t) =
√
n(r, t) exp
(
i
~
∫
r
u(r′, t)dr′
)
, (6)
where n(r, t) is density of atoms in BEC and u(r, t) de-
notes its velocity field, and to introduce dimensionless
variables
x̃ = x/
√
2ξ, ỹ = y/
√
2ξ, t̃ = (cs/2
√
2ξ)t,
ñ = n/n0, ũ = u/cs,
(7)
where ξ = ~/
√
2mn0g is the BEC healing length and nu-
merical constants are introduced for future convenience.
As a result of this transformation we obtain the system
(we omit tildes for convenience of the notation)
1
2
nt +∇(nu) = 0,
1
2
ut + (u∇)u+∇n+∇
[
(∇n)2
8n2
− ∆n
4n
]
= 0,
(8)
where ∇ = (∂x, ∂y). Since we shall consider waves far
enough from the obstacle, the potential is omitted in (8).
We are interested in linear waves propagating on the
background flow with n = 1, u = M , v = 0. Hence, we
introduce
n = 1 + n1, u =M + u1, v = v1, (9)
and linearize the system (8) with respect to small devia-
tions n1, u1, v1. As a result we obtain the system
1
2
n1,t + u1,x +Mn1,x + v1,y = 0,
1
2
u1,t +Mu1,x + n1,x − 14 (n1,xxx + n1,xyy) = 0,
1
2
v1,t +Mv1,x + n1,y − 14 (n1,xxy + n1,yyy) = 0,
(10)
which describes propagation of linear waves in BEC with
a uniform flow. We obtain the applicability condition of
these equations, if we notice that in the linear wave u1 ∼
Mn1 and the nonlinear terms of the order of magnitude
u1∇u1 ∼ ∇(Mu1)2 can be neglected as long as they are
much less than the linear ones ∼ ∇n1. Thus, we get the
criterion
n1 ≪ 1/M2. (11)
Hence, if M is large enough, the linear theory is appli-
cable to description of waves outside the Mach cone far
enough from the obstacle, where the density amplitude
n1 of the wave satisfies the condition (11). For harmonic
waves n1, u1, v1 ∝ exp[i(kxx+kyy)−iωt] the system (10)
yields at once the dispersion relation
ω
2
=Mkx ± k
√
1 +
k2
4
, (12)
where k =
√
k2x + k
2
y.
Now we consider the stationary wave patterns far
enough from the obstacle where the condition (11) is sup-
posed to be fulfilled. In fact, this problem is analogous
to the Kelvin theory of ship waves generated by a ship
moving in a deep water, but with a different dispersion
law (12). Hence, we shall follow Kelvin’s method in its
form presented in [11, 12].
First, we notice that in a stationary wave pattern ω =
0 and, hence, the components of the wavevector k =
(kx, ky) are the functions of the space coordinates (x, y)
connected with each other by the relationship
G(kx, ky) ≡Mkx + k
√
1 +
k2
4
= 0, (13)
where we have taken into account that for chosen geom-
etry of the BEC flow the wave must propagate upwind
i.e. kx < 0.
Next, the “ship wave” pattern corresponds to a mod-
ulated two-dimensional wave where the wavevector k is
a gradient of the phase [11],
θ =
∫
r
0
k · dr. (14)
Hence, the components (kx, ky) satisfy the condition
∂kx
∂y
− ∂ky
∂x
= 0, (15)
which, with an account of (13), yields the equation for
ky
∂ky
∂x
− f ′(ky)
∂ky
∂y
= 0, (16)
where f ′(ky) is defined by the derivative of an implicit
function (13):
f ′ = −∂G/∂ky
∂G/∂kx
. (17)
It follows from Eq. (16) that kx and ky are constant along
the characteristics defined as solutions of the equation
dy
dx
= −f ′(ky). (18)
At last, since at large distances from the obstacle, the
latter can be considered as a point-like source of waves,
3
χ η
µ
k
y
x
A
FIG. 1: Sketch of the wave front with wavevector k normal
to it at the point A and coordinates χ and η defining the
parameters of the wave pattern at this point.
-2 2 4 6 8 10
x
-15
-10
-5
5
10
15
y
FIG. 2: Wave pattern of stationary linear waves generated in
the flow of BEC past a point-like obstacle. Dashed line cor-
responds to the Mach cone of linear waves in the long wave-
length limit.
the resulting stationary wave is centered, that is the char-
acteristics intersect at the point (x, y) = (0, 0) of the lo-
cation of the obstacle. Hence, we obtain the solution
y
x
= tanχ = −∂G/∂ky
∂G/∂kx
, (19)
where χ is the angle between the x axis and the radius-
vector r of the point A with wavevector (kx, ky); see
Fig. 1 where other convenient parameters are also de-
fined, namely, the angle η between the wavevector k
and a negative direction of the x axis, and the angle
µ = π − χ− η between vectors k and r.
Thus, we have
kx = −k cos η, ky = k sin η, (20)
G(k, η) = −Mk cos η + k
√
1 +
k2
4
= 0. (21)
Then elementary calculation gives for (19) the expression
tanχ =
(2M2 − 1− tan2 η) tan η
(M2 + 1) tan2 η − (M2 − 1) (22)
and Eq. (21) yields
k = 2
√
M2 cos2 η − 1. (23)
Thus, we have found that for a fixed value of η the compo-
nents (kx, ky) are constant along the line χ = const with
χ defined by Eq. (22) and the length of the wavevector
given by (23). Therefore the phase (14) can be conve-
niently calculated by integration along the line χ = const
with constant vector k, so that
θ = (k cosµ)r. (24)
It means that the lines of constant phase (e.g. the wave
crests) θ are determined in parametrical form by Eq. (22)
and
r =
θ
k cosµ
, (25)
where k is given by Eq. (23) and µ can be calculated
from tanµ = − tan(χ+ η) which gives, after elementary
algebra, the expression
tanµ =
2M2
k2
sin 2η. (26)
This expression permits one to express Eq. (25) as
r =
4θ
k3
√
M2(M2 − 2) cos2 η + 1 (27)
and Eq. (22) as
tanχ =
(1 + k2/2) tan η
M2 − (1 + k2/2) . (28)
At last, the curves with constant phase θ are given in
Cartesian coordinates by the formulas
x = r cosχ =
4θ
k3
cos η(1−M2 cos 2η),
y = r sinχ =
4θ
k3
sin η(2M2 cos2 η − 1).
(29)
Thus, we have found the expressions describing the linear
wave pattern in a parametric form where the parameter
η changes in the interval
− arccos 1
M
≤ η ≤ arccos 1
M
. (30)
For η = 0 we have x < 0 and y = 0, that is small values
of η correspond to the wave before the obstacle. The
4
boundary values η = ± arccos (1/M) correspond to the
lines
x
y
= ±
√
M2 − 1 = ±aM , (31)
that is the curves of constant phase become asymptoti-
cally the straight lines parallel to the Mach cone. The
general pattern is shown in Fig. 2.
NUMERICAL SIMULATIONS
We have compared the above approximate analytical
theory with the exact numerical solution of the GP equa-
tion, which in non-dimensional units (7) takes the stan-
dard form
iψt = − 12 (ψxx + ψyy) + U(x, y)ψ + |ψ|
2ψ, (32)
which corresponds for U = 0 to the system (8) with
ψ =
√
n exp
(
i
∫
r
u(r′, t)dr′
)
.
In our simulations the obstacle was modeled by an impen-
etrable disk with radius r = 1. Such an obstacle intro-
duces large enough perturbation into the flow to generate
an oblique solitons pair behind it (see [8]). We assume
that at the initial moment t = 0 there is no disturbance
in the condensate, so that it is described by the plane
wave function
ψ(x, y)|t=0 = exp(iMx)
corresponding to a uniform condensate flow. For large
enough evolution time, the wave pattern around the ob-
stacle tends to a stationary structure. An example of
such a structure for M = 2 is shown in Fig. 2. A dashed
line corresponds to the analytic theory developed in the
preceding Section; as we see it agrees very well with the
numerical results for M = 2.
The condition (11) indicates that the nonlinear effects
grow up with increase of M . To demonstrate this explic-
itly, we have compared the wavelength λ at y = 0 calcu-
lated using the developed linear analytic theory with the
same parameter obtained from our full numerical simu-
lations. According to linear theory, the wavelength at
y = 0 (i.e. η = 0) is constant and equal to
λ =
2π
k
=
π√
M2 − 1
, (33)
where we have used Eq. (23). In Fig. 4 we compare this
dependence of the wavelength λ on the Mach number
M with the results of numerical simulations at the point
with n1 ≈ 0.1. As we see, Eq. (33) is very accurate
for values ofM satisfying the condition (11) and discrep-
ancy between analytical and numerical results slightly in-
creases with increase ofM . In general, this plot confirms
validity of a linear theory in the region of its applicability.
FIG. 3: Numerically calculated wave pattern of stationary
linear waves generated in the flow of BEC past an obstacle.
The Mach number is equal to M = 2 and the radius of an
impenetrable obstacle to r = 1. Dashed line corresponds to
the linear analytical theory for the line of constant phase. It
corresponds to Eqs. (29) but the curve is shifted to 1 unit of
length to the left from the center of the obstacle for better
fitting to numerics. It should be noted that this distance is
negligibly small in our theory corresponding to a point-like
obstacle. A pair of oblique dark lines behind the obstacle
correspond to spatial solitons studied in [8].
1 2 3 4 5 
M
0
1
2
3

analytical
numerical
λ
FIG. 4: Dependence of the wavelength of the wave generated
before the obstacle on the Mach number calculated numeri-
cally (full time-dependent GP equation)—dots, and analyti-
cally (linear theory)—dashed line.
CONCLUSION
We have developed here the theory of linear waves gen-
erated by the flow of BEC past an obstacle. The linear
approximation is correct for small enough amplitudes of
the perturbation. This condition is satisfied in the case
of small disturbance introduced by the obstacle and not
too high values of the Mach number. Our numerical sim-
ulations confirm the analytical theory in the region of its
5
applicability.
Acknowledgements
AMK and YGG thank RFBR (grant 05-02-17351) and
AG and GAE thank FAPESP and CNPq for financial
support.
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