PHYSICAL REVIEW E 86, 046602 (2012)
Nonlinea Di ac equa ion soli a y wa es in ex e nal ields
F anz G. Me ens,1,*Niu ka R. Quin e o,2,†F ed Coope ,3,4,‡A inash Kha e,5,§and A adh Saxena4,|
1Physikalisches Ins i u , Uni e si ¨
a Bay eu h, D-95440 Bay eu h, Ge many
2IMUS and Depa amen o de Fisica Aplicada I, E.P.S. Uni e sidad de Se illa, 41011 Se illa, Spain
3San a Fe Ins i u e, San a Fe, New Mexico 87501, USA
4Theo e ical Di ision and Cen e o Nonlinea S udies, Los Alamos Na ional Labo a o y, Los Alamos, New Mexico 87545, USA
5Indian Ins i u e o Science Educa ion and Resea ch, Pune 411021, India
(Recei ed 13 Augus 2012; published 24 Oc obe 2012)
We conside nonlinea Di ac equa ions (NLDE’s) in he 1 +1 dimension wi h scala -scala sel -in e ac ion
g2
κ+1(¯
)κ+1in he p esence o a ious ex e nal elec omagne ic ields. We ind exac solu ions o special ex e nal
ields and we s udy he beha io o soli a y-wa e solu ions o he NLDE in he p esence o a wide a ie y o ields
in a a ia ional app oxima ion depending on collec i e coo dina es which allows he posi ion, wid h, and phase o
hese wa es o a y in ime. We ind ha in his app oxima ion he posi ion q( ) o he cen e o he soli a y wa e
obeys he usual beha io o a ela i is ic poin pa icle in an ex e nal ield. Fo ime-independen ex e nal ields,
we ind ha he ene gy o he soli a y wa e is conse ed bu no he momen um, which becomes a unc ion o
ime. We pos ula e ha , simila ly o he nonlinea Sch ¨
odinge equa ion (NLSE), a su icien dynamical condi ion
o ins abili y o a ise is ha dP( )/d ˙
q( )<0. He e P( ) is he momen um o he soli a y wa e, and ˙
qis he
eloci y o he cen e o he wa e in he collec i e coo dina e app oxima ion. We ound o ou choices o ex e nal
po en ials ha we always ha e dP( )/d ˙
q( )>0, so, when ins abili ies do occu , hey a e due o a di e en sou ce.
We in es iga e he accu acy o ou a ia ional app oxima ion using nume ical simula ions o he NLDE and ind
ha , when he o cing e m is small and we a e in a egime whe e he soli a y wa e is s able, ha he beha io o
he solu ions o he collec i e coo dina e equa ions ag ees e y well wi h he nume ical simula ions. We ound
ha he ime e olu ion o he collec i e coo dina es o he soli a y wa e in ou nume ical simula ions, namely he
posi ion o he a e age cha ge densi y and he momen um o he soli a y wa e, p o ide good indica o s o when
he soli a y wa e i s becomes uns able. When hese a iables s op being smoo h unc ions o ime ( ), hen he
soli a y wa e s a s o dis o in shape.
DOI: 10.1103/PhysRe E.86.046602 PACS numbe (s): 05.45.Y , 03.70.+k, 11.10.Lm
I. INTRODUCTION
Classical solu ions o nonlinea ield equa ions ha e a long
his o y as a model o ex ended pa icles [1–3]. In 1970,
Sole [3] p oposed ha he sel -in e ac ing 4-Fe mi heo y
was an in e es ing model o ex ended e mions. La e , S auss
and Vasquez [4] we e able o s udy he s abili y o his
model unde dila a ion and ound he domain o s abili y o
he Sole solu ions. Soli a y wa es in he 1 +1 dimensional
nonlinea Di ac equa ion (NLDE) ha e been s udied [5,6]in
he pas in he case o massi e G oss-Ne eu [7] (wi h N=1,
i.e., jus one localized e mion) and massi e Thi ing [8]
models. In hose s udies, i was ound ha hese equa ions
ha e soli a y-wa e solu ions o bo h scala -scala (S-S) and
ec o - ec o (V-V) in e ac ions. The in e ac ion be ween
soli a y wa es o di e en ini ial cha ge was s udied in de ail
o he S-S case in he wo k o Al a ez and Ca e as [9]
by Lo en z boos ing he s a ic solu ions and allowing hem
o sca e . Recen ly, we ex ended he solu ions p e iously
ound o a mo e gene al in e ac ion o he o m g2
κ+1(¯
)κ+1
[10]. Fo he non ela i is ic limi o he NLDE, namely he
nonlinea Sch ¨
odinge equa ion (NLSE), he e ha e been
*[email p o ec ed]
†[email p o ec ed]
‡[email p o ec ed]
§[email p o ec ed]
|a[email p o ec ed]
ecen s udies o he beha io o he o ced NLSE. Using a
collec i e coo dina e (CC) heo y, he au ho s ound [11–14]
ha a su icien dynamical condi ion o ins abili y o a ise
is ha dp( )/d < 0.He e p( ) is he no malized canonical
momen um p( )=1
M( )
∂L
∂˙
q,M( )=dx(x, )(x, )is he
mass, and ˙
q( )= ( ) is he eloci y o he soli a y wa e.
One o he poin s we will in es iga e in he pape is
whe he his dynamical s abili y c i e ion is also alid o
he NLDE. The e has been ecen in e es in he s abili y o
NLDE wi h highe -o de nonlinea i y [15]. Comech (p i a e
communica ion) has been able o p o e ha o κ=1, he
Vakhi o -Kolokolo [16] c i e ion gua an ees linea s abili y
in he non ela i is ic egime o he NLDE equa ion o
solu ions o he o m (in he es ame) (x, )=ψ(x)e−iω ,
whe e ωis less han bu app oxima ely equal o he mass
pa ame e min he Di ac equa ion. He was also able o show
linea ins abili y in he same non ela i is ic egime o κ⩾3.
This is he i s igo ous esul o he Di ac equa ion, bu
i only applies in he non ela i is ic egime. He e we wan
o unde s and i we can de e mine in he ela i is ic egime
o wha alues o ωdo he soli a y wa es become uns able,
wi h and wi hou o cing e ms, e en when hey a e s able
in he non ela i is ic egime. Wha we ind is ha when he
soli a y wa es a e only me as able o he un o ced p oblem,
he c i ical ime o he soli a y wa e o become uns able in he
o ced p oblem o weak o cing is simila o he c i ical ime
in he un o ced p oblem. When he soli a y wa e main ains
i s basic shape, he CC equa ions gi e a good desc ip ion o
he ac ual ime e olu ion a all imes. This is ue o weak
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MERTENS, QUINTERO, COOPER, KHARE, AND SAXENA PHYSICAL REVIEW E 86, 046602 (2012)
amp po en ials, ha monic po en ials, and spa ially pe iodic
po en ials, when ω>ω
c, and ωcis he c i ical alue abo e
which he un o ced soli a y wa e is s able. The collec i e
coo dina es q( ) and P( ), he posi ion and momen um o
he soli a y wa e, a e “smoo h” unc ions o o he ex e nal
po en ials we ha e chosen. Thei coun e pa s in he nume ical
simula ion a e he i s momen o he cha ge densi y and
he o al momen um o he nume ical solu ion. When he
nume ical e olu ion o hese coun e pa s o he collec i e
coo dina es s a de ia ing om hei CC alues, his is a signal
ha he shape o he soli a y wa e is beginning o change. This
usually apidly de elops in o non-smoo h beha io o q( ) and
P( ) in he nume ical solu ion. This is how we de e mine he
onse o he ins abili y ime c o he o ced NLDE soli a y
wa e. Un o una ely, o he po en ials we s udy, we always
ob ain dp/d ˙
q>0, which ul ills a necessa y condi ion o
s abili y. Thus, his c i e ion does no yield a p edic ion o he
ins abili ies.
This pape is o ganized as ollows: In Sec. II,we e iew
he known exac solu ions o he un o ced NLDE and discuss
he conse a ion laws ha go e n hei beha io . In Sec. III
we ex end Bogolubsky’s discussion [17] o he s abili y o
hese soli a y-wa e solu ions o changes in he equency ω
o a bi a y nonlinea i y pa ame e κ. In Sec. IV we conside
he NLDE in ex e nal elec omagne ic ields and ind pa icula
exac soli a y-wa e solu ions and discuss he s abili y o hese
solu ions. In Sec. Vwe in oduce ou a ia ional me hod based
on using o ou a ia ional wa e unc ions he exac wa e
unc ions o he soli a y wa es o he un o ced p oblem, wi h
he posi ion, wid h pa ame e , and phase o hese solu ions
being p omo ed o collec i e coo dina es depending on ime.
We w i e he ela i is ic equa ions o hese collec i e coo di-
na es which a e simila o poin pa icle ela i is ic dynamical
equa ions. The po en ial he a e age posi ion o he soli a y
wa e sees is a pa icula a e age o he ex e nal po en ial
weigh ed wi h he cha ge densi y. In Sec. VI we pos ula e
ou s abili y c i e ion o an a bi a y ex e nal po en ial based
on jus sol ing he CC equa ions. This condi ion is a su icien
condi ion o ins abili y. In Sec. VII we examine and sol e
he collec i e coo dina e (CC) equa ions o h ee ypes o
po en ials—a amp po en ial, a ha monic po en ial, and a
spa ially pe iodic po en ial. We also compa e he solu ion o
he CC equa ions o he nume ical simula ion o he NLDE
equa ion. We s a e ou conclusions in Sec. VIII. In he appendix
we discuss iden i ies ha a e obeyed by he solu ions in he
es ame.
II. REVIEW OF EXACT SOLUTIONS TO THE NLDE
In his sec ion, we e iew he exac solu ions o he NLDE,
using he no a ion o Re . [10]. We a e in e es ed in soli a y-
wa e solu ion o he NLDE gi en by
(iγμ∂μ−m)+g2(¯
)κ=0.(2.1)
These equa ions can be de i ed in a s anda d ashion om he
Lag angian densi y
L=i
2[¯
γμ∂μ−∂μ¯
γμ]−m¯
 +g2
κ+1(¯
)κ+1.
(2.2)
Fo soli a y-wa e solu ions, he ield goes o ze o a in ini y.
I is su icien o go in o he es ame, since he heo y is
Lo en z in a ian and he mo ing solu ion can be ob ained by
a Lo en z boos . In he es ame we conside solu ions o he
o m
(x, )=e−iω ψ(x).(2.3)
We a e in e es ed in bound-s a e solu ions ha co espond
o posi i e equency ω⩾0 and which ha e ene gies in
he es ame less han he mass pa ame e m, i.e., ω<m.
In ou p e ious pape [10], we chose he ep esen a ion
γ0=σ3,iγ1=σ1. He e, o make con ac wi h he nume ical
simula ions pape o Al a ez and Ca e as [9] we choose
ins ead γ0=σ3;γ1=iσ2.De ining A,B ia
ψ(x)=A(x)
iB(x)=R(x)cos θ
isin θ,(2.4)
we ob ain he ollowing equa ions o Aand B:
dA
dx +(m+ω)B−g2(A2−B2)κB=0,
(2.5)
dB
dx +(m−ω)A−g2(A2−B2)κA=0.
A i s in eg al o hese equa ions can be ob ained by ealizing
ha om ene gy-momen um conse a ion we ha e
∂μTμν =0; Tμν =i
2[¯
γμ∂ν−∂ν¯
γμ]−gμν L.(2.6)
Thus, o s a iona y solu ions,
T10 =cons ,T
11 =cons .(2.7)
Now, using (2.3), we ob ain
T11 =ωψ†ψ−m¯
ψψ +LI;LI=g2
κ+1(¯
ψψ)κ+1.(2.8)
Fo soli a y-wa e solu ions anishing a in ini y, he cons an
is ze o and we ge he use ul i s in eg al,
T11 =ωψ†ψ−m¯
ψψ +LI=0.(2.9)
Mul iplying he equa ion o mo ion on he le by ¯
and using
(2.3) we ha e ha
(κ+1)LI=−ωψ†ψ+m¯
ψψ −¯
ψiγ1∂1ψ. (2.10)
The e o e, we can ew i e T11 =0as
ωκψ†ψ−mκ ¯
ψψ −¯
ψiγ1∂1ψ=0.(2.11)
Fo he Hamil onian densi y we ob ain
H=T00 =i
2[¯
γ1∂x−∂x¯
γ1]+m¯
 −LI
≡h1+h2−h3.(2.12)
Each o he hia e posi i e de ini e. F om Eqs. (2.9) and (2.10),
one has he ela ionship
κLI=−¯
ψiγ1∂xψ. (2.13)
F om his, we ha e
h3=1
κh1(2.14)
and, in pa icula o κ=1, H=m¯
ψψ.
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In e ms o R,θ one has
¯
ψiγ1∂1ψ=ψ†ψdθ
dx.(2.15)
This leads o he simple di e en ial equa ion o θ o soli a y
wa es,
dθ
dx =−ωκ+mκcos 2θ;ωκ≡κω;mκ=κm. (2.16)
The solu ion is (in his sec ion and wha ollows we will choose
he posi ion o he soli a y wa e o be ini ially a x0=0)
θ(x)= an−1(α anh βκx),(2.17)
whe e
α=mκ−ωκ
mκ+ωκ1/2
=m−ω
m+ω1/2
,β
κ=m2
κ−ω2
κ1/2.
(2.18)
Thus, we ha e
an θ(x)=α anh βκx, sin2θ(x)=(m−ω)sinh
2βκx
mcosh 2βκx+ω;
cos2θ(x)=(m+ω) cosh2βκx
mcosh 2βκx+ω,(2.19)
whe e we ha e used he iden i ies
1+α2 anh2βkx=mcosh 2βkx+ω
m+ωsech2βkx,
(2.20)
1−α2 anh2βkx=ωcosh 2βkx+m
m+ωsech2βkx.
Sol ing Eq. (2.9) o R2we ob ain
R2=(κ+1)(mcos 2θ−ω)
g2(cos 2θ)κ+11/κ
.(2.21)
Now we ha e
dθ
dx =β2
κ
ωκ+mκcosh 2βκx=−ωκ+mκcos 2θ, (2.22)
whe e βκ=m2
κ−ω2
κ=κ√m2−ω2,so
cos 2θ=mκ+ωκcosh 2βκx
ωκ+mκcosh 2βκx=m+ωcosh 2βκx
ω+mcosh 2βκx.(2.23)
We can ew i e R2using he igh -hand side o Eq. (2.22) as
R2=ω+mcosh 2βκx
m+ωcosh 2βκx (κ+1)β2
κ
g2κ2(m+ωcosh 2βκx)1/κ
.
(2.24)
Using he iden i ies in Eq. (2.20), we ob ain he al e na i e
exp ession
R2=1+α2 anh2βκx
1−α2 anh2βκx
×sech2βκx(κ+1)β2
κ
g2κ2(m+ω)(1 −α2 anh2βκx)1/κ
.(2.25)
In pa icula , o κ=1,
R2=2(m−ω)
g2
(1 +α2 anh2βx)
(1 −α2 anh2βx)2sech2βx (2.26)
and
A2=R2cos2θ=2
g2
(m2−ω2)(m+ω) cosh2βx
(m+ωcosh 2βx)2,
(2.27)
B2=R2sin2θ=2
g2
(m2−ω2)(m−ω)sinh
2βx
(m+ωcosh 2βx)2.
Fo a bi a y κwe ha e
A=(m+ω) cosh2(κβx)
m+ωcosh(2κβx)(κ+1)β2
g2(m+ωcosh(2κβx))1
2κ
,
B=(m−ω)sinh
2(κβx)
m+ωcosh(2κβx)(κ+1)β2
g2(m+ωcosh(2κβx))1
2κ
.
(2.28)
Because o Lo en z in a iance we can ind he solu ion in a
ame mo ing wi h eloci y wi h espec o he es ame.
The Lo en z boos is gi en in e ms o he apidi y a iable η
as ollows (he e c=1):
= anh η;γ=1
√1− 2=cosh η;sinhη=
√1− 2.
(2.29)
In he mo ing ame, he ans o ma ion law o spino s ells
us ha
(x, )=cosh(η/2) sinh(η/2)
sinh(η/2) cosh(η/2) 
×0
1[γ(x− ),γ ( − x)]
0
2[γ(x− ),γ ( − x)],(2.30)
since
cosh(η/2) =(1 +γ)/2; sinh(η/2) =(γ−1)/2.(2.31)
This in componen o m,
1(x, )=[cosh(η/2)A(x)+isinh(η/2)B(x)]e−iω ,
(2.32)
2(x, )=[sinh(η/2)A(x)+icosh(η/2)B(x)]e−iω ,
whe e
x=γ(x− ); =γ( − x).(2.33)
No e ha cosh2(η/2) +sinh2(η/2) =cosh η=γ.
A. Conse a ion laws o he NLDE
The Lag angian is in a ian unde he ans o ma ion o
phase →ei, which by Noe he ’s heo em leads o he
conse ed cu en ,
∂μjμ(x)=0; jμ=¯
γμ. (2.34)
This leads o cha ge conse a ion,
Q=dx†, (2.35)
which, o he soli a y-wa e solu ion, leads o
Q=dx(A2+B2)=1
κβ (κ+1)β2
g2(m+ω)1/κ
Iκ(α2),(2.36)
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MERTENS, QUINTERO, COOPER, KHARE, AND SAXENA PHYSICAL REVIEW E 86, 046602 (2012)
whe e
Iκ(α2)=1
−1
dy 1+α2y2
(1 −y2)(κ−1)/κ [1 −α2y2](κ+1)/κ
=B1
2,1
κ2F11+1
κ,1
2,1
2+1
κ;α2
+α2B3
2,1
κ2F11+1
κ,3
2,3
2+1
κ;α2,
(2.37)
and 2F1is a hype geome ic unc ion. He e B(x,k) deno es he
β unc ion.
We also ha e ene gy-momen um conse a ion Eq. (2.6)
leading o conse a ion o ene gy and momen um,
E=T00dx;P=T01dx. (2.38)
Because o Lo en z in a iance i is su icien o calcula e he
ene gy-momen um enso in he como ing ame =0. The
ene gy momen um enso in an a bi a y ame is hen gi en
by
Tμν =μ
αν
βTαβ ;μ
α=cosh ηsinh η
sinh ηcosh η.(2.39)
In he es ame o he soli a y wa e, o he unpe u bed
sys em, one has ha
T00 =h11−1
κ+h2,(2.40)
whe e
h1=R2(x)dθ
dx =κβ2
m+ωcosh(2κβx)
×(κ+1)β2
g2(m+ωcosh(2κβx))1/κ
,(2.41)
h2=m¯
ψψ =m(A2−B2)
=m(κ+1)β2
g2(m+ωcosh(2κβx))1/κ
.(2.42)
In eg a ing in he es ame, we ge , o he es - ame ene gy,
E0=H11−1
κ+H2,(2.43)
whe e
H1=dxh1=β
m+ω(κ+1)β2
g2(m+ω)1/κ
×B1
2,1+1
κ2F11+1
κ,1
2,3
2+1
κ;α2,(2.44)
H2=dxh2=1
κβ (κ+1)β2
g2(m+ω)1/κ
×B1
2,1
κ2F11
κ,1
2,1
2+1
κ;α2.(2.45)
Since in he es ame o s a iona y solu ions T11 =T01 =
0, he ene gy o he soli a y wa e in he mo ing ame is
jus
E=E0cosh η=γE
0;P=E0sinh η, (2.46)
so he no m E2−P2=E2
0=M2
0.
In pa icula , o κ=1and m=1, we ha e ha
M0=2
g2Qsinh−1g2Q
2;Q=2√1−ω2
g2ω.(2.47)
We also ha e
H1=−
2√1−ω2−2 anh−11−ω
1+ω
g2,
(2.48)
H2=
4 anh−11−ω
1+ω
g2.
III. STABILITY OF EXACT SOLUTIONS
A. S abili y o changes in he equency a ixed cha ge
Bogolubsky [17] sugges ed ha he s abili y could be
asce ained by looking a a ia ions o he wa e unc ion,
keeping he cha ge ixed and seeing i he solu ion was a
minimum (s able o ha a ia ion) o maximum (uns able
o ha a ia ion) o he Hamil onian as a unc ion o he
pa ame e ω. This p inciple has been e y use ul in he pas
in de e mining he s abili y o scala wa e equa ions ha a e
Hamil onian dynamical sys ems. I he a ia ion dec eased he
ene gy, i u ned ou ha he soli a y wa es we e uns able.
Since in highe dimensions he e a e many deg ees o eedom
o pe u bing he sys em, his c i e ion is a su icien condi ion
o ins abili y. Fo he Di ac case, we ha e ound om ou
nume ical simula ions ha his c i e ion does no de e mine he
c i ical ωexcep when κ=1[18], he case o iginally s udied
by Bogolubsky [17]. Assuming we know he wa e unc ion a
he alue o ωco esponding o a ixed cha ge Q, i we change
he pa ame ic dependence on ω, his also changes he cha ge.
This can be co ec ed by assuming ha he new wa e unc ion
has a new no maliza ion ha co ec s o his. Tha is, i we
pa ame ize a es - ame soli a y-wa e solu ion o he NLDE
which has a cha ge Q[ω]by
ψs(x, )=χs(x,ω)e−iω ,(3.1)
hen we choose ou sligh ly changed wa e unc ion o be
˜
ψ[x, ,ω,ω]=√Q[ω]
√Q[ω]χs(x,ω)e−iω
≡ (ω,ω)χs(x,ω)e−iω .(3.2)
Thewa e unc ion ˜
ψ[x, ,ω,ω] hen has he same cha ge as
ψ[x, ,ω]. Inse ing his wa e unc ion in o he Hamil onian,
we ge a new p obe Hamil onian Hpdepending on bo h ω,ω.
As a unc ion o ω his new Hamil onian is s a iona y a he
alue ω=ω. The c i e ion Bogolubsky p oposed [17]is
ha he soli a y wa e is s able (uns able) wi h espec o his
a ia ion in ωacco ding o whe he his new Hamil onian
has a minimum (maximum) a ω=ω. Wha we will ind
o κ=1 is ha he e is a c i ical alue o ω(de e mined
by he coupling g and Q) below which he soli a y wa e is
uns able, and his esul is bo ne ou by nume ical simula ions
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NONLINEAR DIRAC EQUATION SOLITARY WAVES IN ... PHYSICAL REVIEW E 86, 046602 (2012)
which we will p esen below. Howe e , we will p esen in
ano he pape nume ical simula ions a a bi a y κwhich
sugges ha his me hod p o ides only a su icien condi ion
o ins abili y [18]. The p obe Hamil onian has he o m
Hp[ω,ω]=H1[ω] (ω,ω)2−1
κ (ω,ω)2(κ+1)
+H2[ω] (ω,ω)2.(3.3)
Fo κ=1weha e ha (ω,ω)2=β[ω]ω
β[ω]ω,whe e
β[ω]=√1−ω2. We hen ind ha he i s de i a i e
o Hpwi h espec o ωe alua ed a ω=ωis indeed
ze o. The second de i a i e e alua ed a ω=ωleads o he
ollowing exp ession:
∂2Hp
∂ω2ω=ω=−
2√1−ω2(ω2−3) +4 anh−11−ω
1+ω
g2ω2(ω2−1)2.
(3.4)
This unc ion is ze o a ωc=0.697586 and he second
de i a i e is nega i e below his alue o ωshowing an
ins abili y. In ou nume ical simula ions o he un o ced
NLDE [18], we ind ha , below his alue, he soli a y wa es
a e me as able, wi h he ime o he ins abili y o se in
inc easing exponen ially as a unc ion o ω o ω<ω
c.
IV. NLDE IN EXTERNAL ELECTROMAGNETIC FIELDS
We add elec omagne ic in e ac ions h ough he gauge
co a ian de i a i e
i∂μ→(i∂μ−eAμ), (4.1)
and hen, unde he combined ans o ma ions
→ei(x);Aμ→Aμ−1
e∂μ;¯
→¯
e−i(x),(4.2)
he Lag angian is in a ian . Again, he conse ed cu en is
gi en by Eq. (2.34). The gauge-in a ian Lag angian o he
ex e nal ield p oblem is
L=i
2[¯
γμ∂μ−∂μ¯
γμ]−m¯

+g2
κ+1(¯
)κ+1−e¯
γμAμ. (4.3)
One again inds ha ene gy-momen um is conse ed, wi h E
and Pgi en by Eq. (2.38). Because o Lo en z in a iance, i
is su icien o calcula e he ene gy-momen um enso in he
como ing ame =0. The ene gy-momen um enso in an
a bi a y ame is hen gi en by Eq. (2.39). Using he eedom
o gauge in a iance, one can choose he axial gauge A1=0,
eA0=V(x). In he axial gauge he Di ac equa ion becomes
iγμ∂μ−m +g2(¯
)κ−γ0V(x)=0.(4.4)
Going in o he es ame and choosing o ψ0 he ep esen a-
ion o Eq. (2.4), we ind ha he Di ac equa ion becomes
∂xA+(m+ω)B−g2[A2−B2]κB−V(x)B=0,
(4.5)
∂xB+(m−ω)A−g2[A2−B2]κA+V(x)A=0.
As in he case o he s anda d NLDE equa ion, we can again
look o s a ic solu ions o he o m ψ0, whe e A=Rcos θ
and B=Rsin θ. The conse a ion o T11 o s a ic solu ions
ha anish a in ini y yields he equa ion
T11 =ωψ†ψ−m¯
ψψ +LI=0,(4.6)
whe e
LI=g2
κ+1(¯
ψψ)κ+1−V(x)ψ†ψ. (4.7)
Mul iplying he Di ac equa ion [Eq. (4.4)] on he le by ¯

and using Eq. (2.3), we ob ain
ωψ†ψ+i¯
ψγ1∂xψ−m¯
ψψ +g2(¯
ψψ)κ+1−V(x)ψ†ψ=0.
(4.8)
The ene gy densi y is gi en by
h=i
2[¯
γ1∂x−∂x¯
γ1]+m¯

−g2
κ+1(¯
)κ+1+V(x)†
≡h1+h2−h3+h4.(4.9)
F om Eq. (4.6),(4.7), and Eq. (4.8) we again ind ha
h3=1
κh1.(4.10)
Mul iplying he Di ac equa ion [Eq. (4.4)] on he le by ¯

and using Eq.(2.3), elimina ing he sel -in e ac ion e m using
Eq. (4.6), we ob ain
i¯
ψγ1∂xψ+κ[m¯
ψψ +V(x)ψ†ψ−ωψ†ψ]=0.(4.11)
Realizing ha he equa ions o he s anda d NLDE a e
modi ied by eplacing ωby ω−V(x), we ob ain, o θ(x),
dθ
dx =−κω +κmcos(2θ)+κV(x),(4.12)
and o R(x) we ob ain
R2=(κ+1)(mcos 2θ−ω+V(x))
g2(cos 2θ)κ+11/κ
.(4.13)
No ice, i we now choose V(x)=μcos 2θ,wea i ea he
solu ions ound ea lie wi h m→m+μ. No e ha o a bound
s a e wi hou he ex e nal po en ial we needed ω<m.Sonow
we need ω<m+μ o βk o be eal. The po en ial is hen
V1(x)=μ(m+μ)+ωcosh 2βκx
ω+(m+μ) cosh 2βκx;
(4.14)
βκ=κ(m+μ)2−ω2.
The eigen alue ωis se by ixing he cha ge Qo he soli a y
wa e. We can choose a nega i e μsa is ying ω<m+μ
so he po en ial a small xlooks much like a ha monic ap.
Namely, choosing κ=1, μ=−1/4, m=1, ω=1/2, hen
V1(x)=−2 cosh √5x
2+3
12 cosh √5x
2+8
.(4.15)
This is plo ed in Fig. 1. The cha ge densi y o he soli a y
wa e co esponding o his ex e nal po en ial is plo ed in
Fig. 2.
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MERTENS, QUINTERO, COOPER, KHARE, AND SAXENA PHYSICAL REVIEW E 86, 046602 (2012)
2 1 1 2
x
0.24
0.23
0.22
0.21
0.20
V x
FIG. 1. (Colo online) Po en ial V1(x) co esponding o Eq. (4.14)
wi h μ=−1/4, ω=1/2.
A small x,V(x) has he expansion
V1(x)≃−1
4+x2
32 −13x4
1536 +O(x6).(4.16)
Ano he solu ion is ound i we le
V2(x)=μsin(2θ(x)).(4.17)
4 2 2 4
x
0.1
0.2
0.3
0.4
0.5
x
FIG. 2. (Colo online) Cha ge densi y R2(x) co esponding
o he po en ial V1(x)gi enbyEq.(4.14) wi h μ=−1/4,
ω=1/2.
The solu ion o he di e en ial equa ion,
θ(x)=−κω +κmcos(2θ(x)) +κμsin(2θ(x)),
(4.18)
hen is ound o be
θ(x)= an−1μ+μ2+m2−ω2 anh [κ(x−x0)μ2+m2−ω2]
m+ω.(4.19)
Le ing x0=0, we ha e ha
an θ=μ+μ2+m2−ω2 anh (κxμ2+m2−ω2)
m+ω.(4.20)
The po en ial is, hus, gi en by
V2(x)=μsin 2θ=2μ an θ
1+ an2θ=2μ[μ+μ2+m2−ω2 anh (κxμ2+m2−ω2)]
(m+ω)[μ+√μ2+m2−ω2 anh (κx√μ2+m2−ω2)]2
(m+ω)2+1
.(4.21)
Fo he alues κ=1, μ=−1/4, m=1, ω=1/2, we ind ha V2(x) has a kinklike o m shown in Fig. 3. The cha ge densi y
R2co esponding o hese alues o he pa ame e s in V2(x) is shown in Fig. 4.
Fo small x o hese pa ame e s
V2(x)≃3
37 −1365x
5476 −54249x2
810448 +O(x3),(4.22)
and i ins ead we choose μ=1/4 and lea e he o he pa ame e s he same, we ge he opposi e ype o kink shown in Fig. 5.
Now o small xwe ha e
V2(x)≃3
37 +1365x
5476 −54249x2
810448 +O(x3).(4.23)
We can also sol e o he mo e gene al case
V3(x)=μ1cos(2θ(x)) +μ2sin(2θ(x)),(4.24)
and hen he solu ion o he di e en ial equa ion is
θ(x)=−κω +κ(m+μ1) cos(2θ(x)) +κμ2sin(2θ(x)),(4.25)
so we jus need o change m→m+μ1and μ→μ2 o ob ain
an θ=
μ2+μ2
2+(m+μ1)2−ω2 anh κxμ2
2+(m+μ1)2−ω2
m+μ1+ω.(4.26)
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NONLINEAR DIRAC EQUATION SOLITARY WAVES IN ... PHYSICAL REVIEW E 86, 046602 (2012)
The po en ial is hen gi en by
V3(x)=μ1cos 2θ+μ2sin 2θ=μ1
1− an2θ
1+ an2θ+2μ2
an θ
1+ an2θ=N(x)
D(x),(4.27)
whe e
N(x)=
2μ2μ2+μ2
2+(μ1+m)2−ω2 anh κxμ2
2+(μ1+m)2−ω2
μ1+m+ω
+μ11−μ2+μ2
2+(μ1+m)2−ω2 anh κxμ2
2+(μ1+m)2−ω22
(μ1+m+ω)2(4.28)
and
D(x)=μ2+μ2
2+(μ1+m)2−ω2 anh κxμ2
2+(μ1+m)2−ω22
(μ1+m+ω)2+1.(4.29)
Fo ω=1/2, μ1=μ2=−1/4, κ=1,m =1, V3(x) has he o m o Fig. 6. The cha ge densi y co esponding o his po en ial
is gi en in Fig. 7.
One can also conside he po en ial V4=μ an2θ. In ha case, on subs i u ing = an θ, we ob ain he di e en ial equa ion
o ,
d
dx =κ[ 2(μ−m−ω)+m+μ 4−ω].(4.30)
Thus, we ob ain
κx =
√2μ an−1√2μ
μ−√(μ+ω)2+m2+2m(ω−3μ)−m−ω
(μ+ω)2+m2+2m(ω−3μ)μ−(μ+ω)2+m2+2m(ω−3μ)−m−ω
−
√2μ an−1√2μ
μ+√(μ+ω)2+m2+2m(ω−3μ)−m−ω
(μ+ω)2+m2+2m(ω−3μ)μ+(μ+ω)2+m2+2m(ω−3μ)−m−ω
.(4.31)
A. “ω” s abili y o exac solu ions wi h κ=1 in he p esence
o an ex e nal ield
He e we ollow he me hod o Bogolubsky [17] and
conside s abili y o changes in he equency ω, keeping
he cha ge Q ixed. Tha is, i we pa ame ize a es - ame
2 1 1 2
x
0.1
0.1
0.2
V x
FIG. 3. (Colo online) Po en ial V2(x) co esponding o he exac
solu ion Eq. (4.21) wi h ω=1/2, μ=−1/4.
soli a y-wa e solu ion by
ψs(x, )=χs(x,ω)e−iω ,(4.32)
4 2 2 4
x
0.5
1.0
1.5
x
FIG. 4. (Colo online) The cha ge densi y o he soli a y wa e in
he po en ial V2(x)gi enEq.(4.21) wi h ω=1/2, μ=−1/4.
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MERTENS, QUINTERO, COOPER, KHARE, AND SAXENA PHYSICAL REVIEW E 86, 046602 (2012)
2 1 1 2
x
0.1
0.1
0.2
V x
FIG. 5. (Colo online) Po en ial V2(x) co esponding o he exac
solu ion Eq. (4.21) wi h μ=1/4.
hen we choose ou pe u bed wa e unc ion o be
˜
ψ[x, ,ω,ω]=√Q[ω]
√Q[ω]χs(x,ω)e−iω
≡ (ω,ω)χs(x,ω)e−iω .(4.33)
Again inse ing his wa e unc ion in o he Hamil onian, we
ge a new p obe Hamil onian Hpdepending on bo h ω,ω.
We will conside he mos gene al solu ion we ha e ound
whe e he po en ial has he o m o V3(x) gi en by Eq. (4.24).
The ene gy densi y is gi en by he ou e ms in Eq. (4.9),
(h1+h2−h3+h4), whe e
h1=R2dθ
dx;h2=mR2cos 2θ,
h3=g
2(R2cos 2θ)2;h4=R2(μ1cos 2θ+μ2sin 2θ).
We also ha e ha h3=h1. Following ou p e ious discussion,
he p obe Hamil onian has he o m
Hp[ω,ω]=H1[ω][ (ω,ω)2− (ω,ω)4]
+(H2+H4)[ω] (ω,ω)2.(4.34)
Fo ou choice o po en ial, we ha e he ollowing ela ionship:
dθ
dx =(m+μ1) cos 2θ+μ2sin 2θ−ω. (4.35)
3 2 1 1 2 3
x
0.3
0.2
0.1
0.1
V x
FIG. 6. (Colo online) Po en ial V3(x) co esponding o Eq. (4.27)
wi h ω=1/2,μ1=μ2=−1/4.
10 5 5 10
x
0.2
0.4
0.6
0.8
x
FIG. 7. (Colo online) Cha ge densi y o he soli a y wa e in
he po en ial V3(x) co esponding o Eq. (4.27) wi h ω=1/2, μ1=
μ2=−1/4.
Sol ing his equa ion, we ob ain
an θ≡T=D+E anh Bx, (4.36)
whe e
B=(m+μ1)2+μ2
2−ω2;E=B
d1
;
(4.37)
D=μ2
d1
;d1=m+μ1+ω.
We also ind
R2=2
g2(m+μ1) cos 2θ+μ2sin 2θ−ω
cos22θ
=2B2
d1g2sech2Bx [1 +(D+E anh Bx)2]
[1 −(D+E anh Bx)2]2.(4.38)
The cha ge is gi en by
Q[ω]=∞
−∞
dxR2=
2ω(m+μ1)2+μ2
2−ω2
g2(ω2−μ2
2).(4.39)
Also,
H1=−ωQ +H2+H4;H2+H4=(m+μ1)T1+μ2T2,
(4.40)
whe e
T1=dxR2cos 2θ=2[ anh−1(E−D)+ anh−1(E+D)]
=2
g2 anh−1(μ1+m)2+μ2
2−ω
μ1+m,(4.41)
T2=dxR2sin 2θ
=2μ2(m+μ1)
g2ω2−μ2
2−2μ2
g2
anh−1√(μ1+m)2+μ2
2−ω2
m+μ1
(μ1+m)2+μ2
2−ω2
.
(4.42)
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NONLINEAR DIRAC EQUATION SOLITARY WAVES IN ... PHYSICAL REVIEW E 86, 046602 (2012)
Thus,
Hp[ω]=−ωQ[ω] (ω,ω)2[1 − (ω,ω)2]
+ (ω,ω)2(H2[ω]+H4[ω])[2 − (ω,ω)2].
(4.43)
I we ake he i s de i a i e o his p obe Hamil onian wi h
espec o ωand se ω=ω, we ind ha his is ze o only
when μ2=0. Thus, he p obe Hamil onian is s a iona y o ω
a ia ions only when μ2=0.Howe e , we can use his me hod
o he ex e nal po en ial V1(x)=μ1cos 2θ[see Eq. (4.14)].
In ha case, he i s de i a i e is au oma ically ze o when
ω=ω. The second de i a i e changes sign a a pa icula
alue o ω=ω, which, o m=1, is he solu ion o he
equa ion
2(μ1+1)3co h−1μ1+1
√μ1(μ1+2)−ω2+1
μ2
1+2μ1−ω2+1
−3μ2
1−6μ1+ω2−3=0.(4.44)
Nowweha e ha o he e obeanallowed ealsolu ion
ω<m+μ1.Soi μ1is posi i e, ωis shi ed upwa d
compa ed o ωcand so is he egion in ωspace whe e he
solu ions a e eal. I μ1is nega i e ωis shi ed downwa d,
bu also he allowed egime o ωis dec eased. The ne esul
is ha he possible egime o s abili y is app oxima ely 30%
o he allowed egion o eal solu ions independen o he
alue o μ1. P elimina y simula ions show ha some o he
exac solu ions wi h he po en ial (4.14) a e me as able. Fo
he well po en ial, μ1<0, he soli a y wa e is me as able,
p obably due o he ac ha he ini ial soli a y wa e
ex ends beyond he in lec ion poin s o he po en ial well.
De ails o hese simula ions will be discussed in a u u e
pape .
V. VARIATIONAL ANSATZ FOR THE NLDE
IN EXTERNAL FIELDS
The gauge-in a ian Lag angian o he ex e nal ield
p oblem is gi en by Eq. (4.3). Using he eedom o gauge
in a iance, one can choose he axial gauge A1=0, eA0=
V(x). Ou ansa z o he ial a ia ional wa e unc ion is
o assume ha , because o he smallness o he pe u ba ion,
he main modi ica ion o ou exac solu ions o he NLDE
equa ion wi hou an ex e nal ield is ha he pa ame e s
desc ibing he posi ion, momen um, boos , and phase become
ime dependen . Tha is, we eplace
→q( ); η→η( ); γω →p( );
ω =γω( − x)→φ( )−p( )[x−q( )],(5.1)
whe e φ( )=ωγ −p( )q( ).
Thus, ou ial wa e unc ion in componen o m is gi en
by
1(x, )=cosh η
2A(x)+isinh η
2B(x)e−iφ+ip(x−q),
(5.2)
2(x, )=sinh η
2A(x)+icosh η
2B(x)e−iφ+ip(x−q),
whe e x=cosh η( )[x−q( )]. Using his ial wa e unc ion
we can de e mine he e ec i e Lag angian o he a ia ional
pa ame e s. W i ing he Lag angian densi y as
L=L1+L2+L3,(5.3)
whe e
L1=i
2(¯
γμ∂μ−∂μ¯
γμ),
L2=−m¯
 +g2
κ+1(¯
)κ+1,(5.4)
L3=−eA0¯
γ0≡−V(x)†.
In eg a ing o e xand changing in eg a ion a iables o z=
(x−q) cosh η, one ob ains
L1=dxL1
=Q(p˙
q+˙
φ−p anh η)−I0(cosh η−˙
qsinh η),(5.5)
whe e
Q=dz[A2(z)+B2(z)] (5.6)
is as gi en by Eq. (2.36). No e ha
I0=dz(BA−AB)=H1,(5.7)
whe e H1is he es - ame kine ic ene gy and is gi en by
Eq. (2.44).He eB(x)=dB(x)
dx, and
L2=L2dx =− m
cosh ηI1+g2
(κ+1) cosh ηI2,(5.8)
whe e
I1=dz[A2(z)−B2(z)]; I2=dz[A2(z)−B2(z)](κ+1),
(5.9)
and
L3=−dzρ(z)Vz
cosh η+q( )=−U[η( ),q( )].
(5.10)
Pu ing hese e ms oge he , we ob ain
L=Q(p˙
q+˙
φ−p anh η)−I0(cosh η−˙
qsinh η)
−m
cosh ηI1+g2
(κ+1) cosh ηI2−U[η( ),q( )].(5.11)
We now ge he ollowing Lag ange’s equa ions:
d
d
δL
δ˙
φ=0→dQ
d =0→Q=cons ,(5.12)
i.e., he cha ge is canonically conjuga ed o he phase φ.The
canonical soli a y-wa e momen um, which is conjuga ed o
he soli a y-wa e posi ion, is
Pq=δL
δ˙
q=Qp +I0sinh η,
dPq
d =Q˙
p+I0cosh η˙η=δL
δq =−∂U
∂q ,
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MERTENS, QUINTERO, COOPER, KHARE, AND SAXENA PHYSICAL REVIEW E 86, 046602 (2012)
-20 -10 0 10 20 30
x
0
0.5
1
1.5
2
2.5
3
|ψ1(x, *)|2+|ψ2(x, *)|2
-20 0 20 40
x
0
0.5
1
1.5
2
2.5
3
|ψ1(x, *)|2+|ψ2(x, *)|2
0 200 400 600 800
-5
0
5
10
15
q( )
0 200 400 600 800
0
2
4
6
P( )
FIG. 13. (Colo online) Ha monic po en ial, V(x)=(V2/2)x2, wi h ωin he uns able egime. (Le uppe panel) Cha ge densi y ρQa
∗=0; 133.3. (Righ uppe panel) Cha ge densi y a ∗=150. (Lowe panels) Soli a y-wa e posi ion q( ) and momen um P( ) om he
nume ical solu ions o he CC equa ions (black solid lines) and nume ical simula ions (dashed lines) o he o ced NLDE. The ene gy ( ed o
middle cu e) and cha ge (blue o uppe cu e) a e also plo ed. Pa ame e s: g=1, m=1, ω=0.3, and V2=10−4. Ini ial condi ion: exac
soli a y wa e o he unpe u bed NLDE wi h ini ial eloci y (0) =0.1.
and sn(x,l) is a Jacobi ellip ic unc ion.
1. Ene gy conse a ion
F om Eq. (5.33) we ha e ha he soli a y-wa e ene gy is
gi en by
E=γM
0−cos kqI4[˙
q]+kγ cos kq ˙
q2I5[˙
q].(7.36)
In he non ela i is ic limi we ob ain
E=M0−cos kqI0
4+M0
2+kI0
5cos kq˙
q2.(7.37)
In he case o a weak po en ial (excep o M0→0 when
ω→1)
E=1+˙
q2
2M0−cos kqI0
4.(7.38)
2. Soli a y-wa e momen um and dynamical s abili y
The soli a y-wa e momen um is gi en by Eq. (5.36) and
becomes
P=γ(M0+kI5[˙
q] cos kq)˙
q. (7.39)
In he non ela i is ic egime we ob ain
P=M0+kI0
5cos kq˙
q. (7.40)
The necessa y condi ion o s abili y,
dP
d˙
q=M0+kI0
5cos kq>0,(7.41)
is sa is ied excep in he egime whe e M0→0, which is when
ω→1. In ha egime, he soli a y wa e is e y b oad and he
condi ion 2π/k bis no ul illed.
3. Nume ical esul s o q( )and P( )
Fo he pendulum equa ion he e is a c i ical ini ial eloci y
a which he coo dina e q( ) makes a ansi ion om pe iodic
mo ion o unbounded mo ion. This occu s when he modulus
pa ame e l=1. This yields he condi ion
c=4I0
4
M0
.(7.42)
Depending on ou choice o pa ame e s, o small-enough ,
cwill be in he non ela i is ic egime. We choose ω o be
in he s abili y egion o he un o ced p oblem (see Sec. III).
Fo g=1, m=1, ωc=0.697586 and choosing ω=0.9, hen
he wid h o he soli a y wa e is 1/(2β)=1.15. I we choose
k=0.1, hen he cha ac e is ic wa eleng h 2π/k =62.8
1/(2β). F om Eqs. (2.47) and (2.48) we ha e ha
Q=0.968644; H1=0.0625108 =I0,(7.43)
M0=H1+ωQ =0.934291.(7.44)
046602-16

NONLINEAR DIRAC EQUATION SOLITARY WAVES IN ... PHYSICAL REVIEW E 86, 046602 (2012)
-20 -10 0 10 20
0
0.05
0.1
0.15
0.2
|ψ1(x, *)|2+|ψ2(x, *)|2
0 1000 2000 3000 4000
-4
-2
0
2
4
q( )
0 1000 2000 3000 4000
-0.01
-0.005
0
0.005
0.01
P( )
FIG. 14. Pe iodic po en ial, V(x)=−cos(kx), e y low ini ial
eloci y. (Uppe panel) Cha ge densi y ρQa ∗=0; 2666.6. (Middle
panel) Soli a y-wa e posi ion q( ) om he nume ical solu ion o
Eq. (7.30) (solid line), app oxima e analy ical exp ession (7.35)
(do ed line), and nume ical simula ions (dashed line) o he o ced
NLDE. The h ee cu es a e supe imposed. (Lowe panel) Momen-
um P( ) om he nume ical solu ions o Eq. (7.30) (solid line) and
nume ical simula ions (dashed line) o he o ced NLDE. The cu es
a e supe imposed. Pa ame e s: g=1, m=1, ω=0.9, =0.001,
and k=0.1. Ini ial condi ion: exac soli a y wa e o he unpe u bed
NLDE wi h ini ial eloci y (0) =0.01.
The o he cons an s o his ini ial condi ion om Eq. (7.31)
a e
I0
4=0.94632; I0
5=0.433477.(7.45)
We ha e, i s , compa ed he analy ical solu ion Eq. (7.35)
o he pendulum equa ion wi h he nume ical solu ion o
Eq. (7.30).Fo <0.1 he esul s a e p ac ically iden ical,
o ⩾1 de ia ions occu .
-20 0 20 40 60
x
0
0.05
0.1
0.15
0.2
|ψ1(x, *)|2+|ψ2(x, *)|2
0 1000 2000 3000 4000 5000
-30
-20
-10
0
10
20
30
q( )
0 1000 2000 3000 4000 5000
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
P( )
FIG. 15. Spa ially pe iodic po en ial, V(x)=−cos(kx), ini-
ial eloci y jus below c. (Uppe panel) Cha ge densi y ρQa
∗=0; 5000. (Middle panel) Soli a y-wa e posi ion q( ) om he
nume ical solu ion o Eq. (7.30) (solid line), app oxima e analy ical
exp ession (7.35) (do ed line), and nume ical simula ions (dashed
line) o he o ced NLDE. Solid and do ed lines a e supe imposed.
(Lowe panel) Momen um P( ) om he nume ical solu ion o
Eq. (7.30) (solid line) and nume ical simula ions (dashed line) o
he o ced NLDE. Pa ame e s: g=1, m=1, ω=0.9, =0.001,
and k=0.1. Ini ial condi ion: exac soli a y wa e o he unpe u bed
NLDE wi h ini ial eloci y (0) =0.0626619.
Speci ically we ha e chosen he ini ial condi ion q(0) =
0,˙
q= 0 o he h ee cases (1) 0 c1 and hen
0sligh ly below (2) and abo e (3) he c i ical alue c,
namely
0= c∓.001.(7.46)
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MERTENS, QUINTERO, COOPER, KHARE, AND SAXENA PHYSICAL REVIEW E 86, 046602 (2012)
0 30 60 90 120 150
0
0.05
0.1
0.15
0.2
|ψ1(x, *)|2+|ψ2(x, *)|2
0 1000 2000 3000 4000
0
50
100
150
q( )
0 1000 2000 3000 4000
0
0.02
0.04
0.06
P( )
FIG. 16. Spa ially pe iodic po en ial, V(x)=−cos(kx), ini-
ial eloci y jus abo e c. (Uppe panel) Cha ge densi y ρQa
∗=0; 4000. (Middle panel) Soli a y-wa e posi ion q( ) om he
nume ical solu ion o Eq. (7.30) (solid line), app oxima e analy ical
exp ession (7.35) (do ed line), and nume ical simula ions (dashed
line) o he o ced NLDE. Solid and do ed lines a e supe imposed.
(Lowe panel) Momen um P( ) om he nume ical solu ion o
Eq. (7.30) (solid line) and om nume ical simula ions (dashed line)
o he o ced NLDE. Pa ame e s: g=1, m=1, ω=0.9, =0.001,
and k=0.1. Ini ial condi ion: exac soli a y wa e o he unpe u bed
NLDE wi h ini ial eloci y (0) =0.0646619.
Choosing =0.001 yields c=0.0636619, which is in he
non ela i is ic egime, so we expec Eq. (7.35) o hold. In
Fig. 14 we show ha o 0=0.01 he analy ic non ela i is ic
esul and he nume ical solu ion o Eq. (7.30) gi e he same
esul s as he solu ion o he NLDE. We also see ha he
shape o he cha ge densi y does no change in ime. In
Figs. 15 and 16 we show ha jus below and abo e he c i ical
eloci y, espec i ely, he analy ical esul (7.35) ag ees wi h
he nume ical solu ion o Eq. (7.30), bu bo h esul s di e
e y sligh ly om he simula ion esul s.
A summa y o he esul o ou simula ions o soli a y wa es
in di e en ex e nal ields is p esen ed in Table I.
VIII. CONCLUSIONS
In his s udy we ha e ound exac solu ions o he NLDE
wi h scala -scala in e ac ions o he o m g2
κ+1(¯
)κ+1in
ce ain ex e nal ields. We ha e discussed hei s abili y
wi h espec o “ω” a ia ions. We ha e also in oduced a
collec i e coo dina e me hod o s udying he ime e olu ion
o soli a y wa es in ex e nal ields and de e mined simple
equa ions o he collec i e coo dina es ha pa allel hose o
a ela i is ic poin pa icle. We ound ha unless (o un il)
he soli a y wa es displayed an ins abili y, he collec i e
coo dina es desc ibing he posi ion and momen um in he
CC equa ions ga e ema kably good ag eemen wi h hei
coun e pa s om ou simula ions. We hen p esen ed a
gene aliza ion o a dynamical s abili y c i e ion, based only
on sol ing he CC equa ions, ha was use ul in s udying
he s abili y o soli a y wa es in he o ced NLSE p oblem.
Fo ou simula ions o he exac e olu ion, as well as he
e olu ion o he collec i e coo dina es, we concen a ed on
κ=1. Fo he o cing e ms we used simple es po en ials
such as amp, ha monic, and pe iodic po en ials. In many
ins ances, we ound ha he ins abili y o he soli a y wa e
solu ion was ela ed o he me as abili y o he soli a y wa e
in he absence o ex e nal o ces and he c i ical ime o
b eakup was qui e close o he ime ound o he un o ced
p oblem.
We had hoped ha a gene aliza ion o he me hod used
o map ou domains o ins abili y in he NLSE using he
much simple solu ions o he collec i e coo dina e p oblem
would also wo k o he NLDE equa ion. Un o una ely o
he p oblems we s udied, we ob ained dp
d˙
q>0, which ul ills a
necessa y condi ion o s abili y, so his me hod did no gi e
any in o ma ion abou ins abili ies. Wha we did ind using
he collec i e coo dina e app oxima ion was ha , s a ing
wi h exac solu ions o he un o ced p oblem, hese soli a y
wa es main ained shape in he CC app oxima ion apa om
he pa ame e s becoming unc ions o ime. The collec i e
a iables in he simula ions, namely q( ) and P( ), we e
smoo h unc ions o a easonable pe iod o ime, e en in he
case when he soli a y wa es we e only me as able. When
hese collec i e a iables began o apidly oscilla e and/o
di e ge om hei alues ound in he collec i e coo dina e
calcula ion, hen ha “de ined” he onse o he ins abili y.
The c i e ion we used o he onse o ins abili y using he
collec i e coo dina es is a su icien condi ion and we ound
no cases whe e he condi ion o his dynamic ins abili y was
sa is ied. Possibly his is a esul o he ac ha ex e nal
ields di e om ex e nal sou ces. Fo he ex e nal sou ce
p oblem, we would expec in he non ela i is ic egime
ha we would eco e he known esul s o he o ced
NLSE wi h sou ce e ms due o he a gumen s o Comech
[15].
The simula ions in his pape a e con ined o he κ=1
case. The nume ical s abili y o soli a y wa es in he absence
046602-18
NONLINEAR DIRAC EQUATION SOLITARY WAVES IN ... PHYSICAL REVIEW E 86, 046602 (2012)
TABLE I. Simula ion esul s o h ee simple po en ials using di e en pa ame e se s; g=1andm=1.
Po en ial Cases Resul s
V(x)=−V1xω=0.9>ω
c=0.697586 S able soli on, wid h Lo en z con ac ed,
V1=10−2,10
−3,10
−4heigh inc eases
ω=0.3<ω
cAsymme ic shape, me as able o ⩽100,
V1=0.01 uns able o 110
ω=0.3<ω
cMe as able o ⩽100, spli s in o
V1=0.0001 wo soli ons and adia ion o 120
V(x)=1
2V2x2ω=0.9>ω
cS able soli on, ha monic oscilla ions
V2=0.0001, 0=0.1
ω=0.9>ω
c, Me as able o 350,
V2=0.0001, 0=0.9 uns able o 350
ω=0.3<ω
c, Me as able o 120, spli s in o
V2=0.0001, 0=0.1 wo soli ons and adia ion o 120
V(x)=−cos(kx)ω=0.9>ω
c,k=0.1, =0.0001 S able soli on, ha monic oscilla ions
0=0.01  c=0.0636619
ω=0.9>ω
c,k=0.1, =0.0001 S able soli on, e y anha monic
0= c−0.001 oscilla ions
ω=0.9>ω
c,k=0.1, =0.0001 S able soli on, ansla ional mo ion
0= c+0.001 plus oscilla ions
o ex e nal po en ials o gene al κwill be p esen ed in a
subsequen publica ion [18]. The semiclassical educ ion o
NLDE o NLSE and implica ions o soli a y-wa e s abili y
ha e been ecen ly discussed in a igo ous ashion by Comech
[15]. Ou nume ical indings [18] ag ee wi h his analysis in
he non ela i is ic egime.
ACKNOWLEDGMENTS
This wo k was suppo ed in pa by he US Depa men
o Ene gy. F.G.M. acknowledges he hospi ali y o he Ma he-
ma ical Ins i u e o he Uni e si y o Se ille (IMUS) and o he
Theo e ical Di ision and Cen e o Nonlinea S udies a Los
Alamos Na ional Labo a o y and inancial suppo by he Plan
P opio o he Uni e si y o Se ille and by Jun a de Andalucia.
N.R.Q. acknowledges inancial suppo om he Humbold
Founda ion h ough Resea ch Fellowship o Expe ienced
Resea che s SPA 1146358 STP and by he MICINN h ough
FIS2011-24540 and by Jun a de Andalucia unde P ojec s
No. FQM207, No. FQM-00481, No. P06-FQM-01735, and
No. P09-FQM-4643.
APPENDIX: REST-FRAME IDENTITIES
In he es ame, ene gy-momen um conse a ion leads
o iden i ies among he a ious in eg als ha a ise. The
Lag angian in he axial gauge is gi en by
L=i
2[¯
γμ∂μ−∂μ¯
γμ]−m¯
 +g2
κ+1(¯
)κ+1
−V(x)†. (A1)
In he es ame, he wa e unc ion is gi en by
0=ψe−iω =A(x)
iB(x)e−iω .(A2)
The ene gy-momen um conse a ion is gi en by Eq. (2.6) and
leads o wo independen equa ions. The i s is
∂0T01 +∂1T11 =0.(A3)
In he es ame T0x=0, so Txx =cons . I he solu ion goes
o ze o a in ini y, hen he cons an is ze o. We hen ha e he
ela ionship
T11 =i
2[¯
γ1∂1−∂1¯
γ1]+L
=ωψ†ψ−m¯
ψψ +g2
k+1(¯
ψψ)k+1−V(x)ψ†ψ=0.
(A4)
In eg a ing o e space we ge he ela ions
ωQ −mI1+g2
κ+1I2−dxρ(x)V(x)=0.(A5)
The second conse a ion law is
∂0T00 +∂1T10 =0,(A6)
which leads o he conse a ion o ene gy. The ene gy o he
soli a y wa e in he es ame de ines he es mass M0,
E=T00dx =M0.(A7)
We ha e ha
T00 =i
2[¯
ψγ1∂1ψ−∂1¯
ψγ1ψ]
+m¯
ψψ −g2
k+1(¯
ψψ)k+1+V(x)ψ†ψ
=(AB1−BA1)+m(A2−B2)
−g2
k+1(A2−B2)κ+1+(A2+B2)V(x).(A8)
046602-19
MERTENS, QUINTERO, COOPER, KHARE, AND SAXENA PHYSICAL REVIEW E 86, 046602 (2012)
In eg a ing, we ob ain
M0=I0+mI1−g2
k+1I2+ρ(x)V(x).(A9)
Using he iden i y o Eq. (A5), we hen ha e, e en in he
p esence o in e ac ions, ha
M0=I0+ωQ. (A10)
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