Gene alized a eling-wa e me hod, a ia ional app oach, and modi ied conse ed quan i ies
o he pe u bed nonlinea Sch ödinge equa ion
Niu ka R. Quin e o*
Depa amen o de Física Aplicada I, E.U.P., Uni e sidad de Se illa, c/Vi gen de Á ica 7, 41011 Se illa, Spain
F anz G. Me ens†
Physikalisches Ins i u , Uni e si ä Bay eu h, D-95440 Bay eu h, Ge many
A. R. Bishop
Theo 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
共Recei ed 16 Ma ch 2010; e ised manusc ip ecei ed 8 June 2010; published 29 July 2010兲
The gene alized a eling wa e me hod 共GTWM兲is de eloped o he nonlinea Sch ödinge equa ion
共NLSE兲wi h gene al pe u ba ions in o de o ob ain he equa ions o mo ion o an a bi a y numbe o
collec i e coo dina es. Rega dless o he pa icula ansa z ha is used, i is shown ha his al e na i e app oach
is equi alen o he Lag angian o malism, bu has he ad an age ha only he Hamil onian o he unpe u bed
sys em is equi ed, ins ead o he Lag angian o he pe u bed sys em. As an explici example, we ake 4
collec i e coo dina es, namely he posi ion, eloci y, ampli ude and phase o he soli on, and show ha he
GTWM yields he same equa ions o mo ion as he pe u ba ion heo y based on he In e se Sca e ing
T ans o m and as he ime a ia ion o he no m, i s momen o he no m, momen um, and ene gy o he
pe u bed NLSE.
DOI: 10.1103/PhysRe E.82.016606 PACS numbe 共s兲: 05.45.Y
I. INTRODUCTION
Wi h g ea gene ali y, and in many phenomena, such as
sca e ing and di usion, soli ons, and o he nonlinea cohe -
en exci a ions in ex ended sys ems beha e like pa icles
关1,2兴. This ac allows one o educe he in ini e numbe o
deg ees o eedom o he ex ended sys ems o only a ew,
when we a e in e es ed in he beha io o he soli on exci a-
ions. E.g., he dynamics o opological soli ons o he non-
linea Klein-Go don equa ions 共NLKGEs兲in he simples
case can be desc ibed in e ms o one so-called collec i e
coo dina e 共CC兲关3,4兴, ypically he cen e o he soli on.
Mo e sophis ica ed Ansä ze ake in o accoun o he deg ees
o eedom, using wo CCs 关5,6兴, h ee CCs 关7兴, o e en
mo e 关8兴.
The dynamics o non opological soli ons o he nonlinea
Sch ödinge equa ion 共NLSE兲is mo e complica ed: The
b igh one-soli on solu ion, which has an in e nal oscilla ion,
depends on ou pa ame e s 关2兴. The e o e, ypically ou
CCs, namely posi ion, ampli ude, eloci y and phase, ha e
been used.
The e a e se e al me hods o de e mine he equa ions o
mo ion ha he CCs sa is y. Fo ins ance, o modi ied
NLSEs se e al pe u ba ion heo ies can be applied 关9–12兴,
o ce ain sys ems one can use he pe u bed in e se sca e -
ing ans o m 共IST兲关13,14兴, and i a Lag angian densi y ex-
is s one can de i e Lag ange equa ions, which a e he e olu-
ion equa ions o he CCs 关15–17兴. Mo eo e , he ime
a ia ion o Mconse ed quan i ies o he unpe u bed sys-
em gene ally is ela ed wi h he e olu ion o MCCs 关18,19兴.
The e is ano he me hod, he so-called gene alized a el-
ing wa e me hod 共GTWM兲, in which a ce ain p ojec ion
echnique is used o ob ain he equa ions o mo ion o he
CCs. The GTWM was in oduced in a gene al way in Re .
关20兴: only he Hamil on equa ions o he unpe u bed sys em
mus be known and he unpe u bed sys em need no be in-
eg able. The me hod was applied o he ze o- empe a u e
dynamics 关20兴and he he mal di usion 关21,22兴o magne ic
o ices in he wo-dimensional 共2D兲aniso opic Heisenbe g
model. The GTWM was also applied o he dynamics o
opological soli ons in NLKGEs 关23兴. Toge he wi h he Rice
ansa z 关5,6兴, which in oduces he posi ion and he wid h o
he soli on as CCs, he me hod explained esonances due o
he ac ion o an ac o ce 关23–25兴and desc ibed anspo
phenomena o a che ype in sine-Go don and
␾
4models
wi h biha monic d i ing and damping 关26,27兴. The sine-
Go don model was applied, e.g., o long Josephson junc ions,
whe e he luxons play he ole o he soli ons, and he
a che e ec was con i med by expe imen s 关28兴.
The aim o his wo k is o de elop he GTWM o an
a bi a y ansa z wi h MCCs o he pe u bed NLSE, see
Sec. II. Fu he mo e, we show in Sec. III ha in cases whe e
he Lag angian densi y o he pe u bed sys em exis s, he
GTWM is equi alen o he a ia ional app oach. In pa icu-
la , by using only 4 CCs we show in Sec. IV ha he GTWM
is also equi alen o he esul s ob ained om he pe u ba-
ion heo y based on he in e se sca e ing ans o m 关14兴,
om he adiaba ic pe u ba ion heo y 关29兴and om he
a ia ion o he no m, he i s momen o he no m, he
momen um, and he ene gy o he pe u bed NLSE. Finally
in Sec. V, we summa ize ou main esul s.
*[email p o ec ed]
†[email p o ec ed]
PHYSICAL REVIEW E 82, 016606 共2010兲
1539-3755/2010/82共1兲/016606共6兲©2010 The Ame ican Physical Socie y016606-1
II. GENERALIZED TRAVELING WAVE METHOD
The pe u bed NLSE ac ually consis s o wo equa ions
o he eal and imagina y pa s, o o u共x, 兲and uⴱ共x, 兲.
Namely,
iu +uxx +2兩u兩2u=R关u共x, 兲;x, 兴,共1兲
and i s complex conjuga e equa ion, whe e R关u共x, 兲;x, 兴
ep esen s a gene al pe u ba ion, which may also depend on
uⴱand he spa ial de i a i es o uand uⴱ. This sys em has
e y many applica ions in p ac ically all ields o physics,
which a e lis ed and discussed in se e al e iews a icles
关14–16,30兴. In he case o op ical soli ons, and xa e e-
placed by he p opaga ion dis ance and he pulse du a ion,
espec i ely.
Fo ou pu pose, we ew i e Eq. 共1兲
iu =
␦
H0
␦
uⴱ+R关u共x, 兲;x, 兴,共2兲
whe e
H0=
冕
−⬁
+⬁
dxH0=
冕
−⬁
+⬁
dx共uxux
ⴱ−u2uⴱ2兲.共3兲
Fo he ollowing, only his Hamil onian o he unpe u bed
sys em mus be known. We now assume ha he ime depen-
dence o u共x, 兲and uⴱ共x, 兲in Eq. 共2兲and i s complex con-
juga e equa ion only appea s ia a se o M eal collec i e
coo dina es 兵Y1共 兲,Y2共 兲,...YM共 兲其ªY
ជ
共 兲
u共x, 兲=u关x,Y
ជ
共 兲兴,uⴱ共x, 兲=uⴱ关x,Y
ជ
共 兲兴.共4兲
Then, we mul iply Eq. 共2兲by
⳵
uⴱ/
⳵
Yn, and i s complex con-
juga e equa ion by
⳵
u/
⳵
Yn, add he esul ing equa ions and
in eg a e o e he sys em, yielding
兺
j=1
M
IYnYjY
˙j=Fn共Y
ជ
兲+Rn共Y
ជ
兲,n=1,2, ...M,共5兲
wi h
IYnYj=
冕
−⬁
+⬁
dx i
冋
⳵
u
⳵
Yn
⳵
uⴱ
⳵
Yj
−
⳵
uⴱ
⳵
Yn
⳵
u
⳵
Yj
册
,共6兲
Fn共Y
ជ
兲=−
冕
−⬁
+⬁
dx
冋
␦
H0
␦
uⴱ
⳵
uⴱ
⳵
Yn
+
␦
H0
␦
u
⳵
u
⳵
Yn
册
=−
冕
−⬁
+⬁
dx
⳵
H0
⳵
Yn
=−
⳵
⳵
Yn
H0,共7兲
Rn共Y
ជ
兲=−
冕
−⬁
+⬁
dx
冉
R
⳵
uⴱ
⳵
Yn
+Rⴱ
⳵
u
⳵
Yn
冊
,共8兲
whe e he o e do deno es he de i a i e wi h espec o
ime. Equa ions 共5兲–共8兲 ep esen a se o M i s -o de ODEs
o ou MCCs. We no e ha o ce ain sys ems i is neces-
sa y o assume ha he ime dependence o he ields can also
appea ia Y
˙1,Y
˙2,...,Y
˙M, which a e o mally ea ed in he
same way as he CCs. This yields Msecond-o de ODEs o
he MCCs. Examples a e magne ic o ices 关20兴and opo-
logical soli ons in NLKGEs 关23,31兴; in bo h cases he wid h
o he exci a ions depends on he eloci y o he exci a ions.
In o de o ha e an example which can be es ed by he
pe u ba ion heo y based on he IST 关14兴we now ake he
one-soli on solu ion o he unpe u bed NLSE and make he
ollowing speci ic ansa z o u共x,Y
ជ
共 兲兲:
u共x, 兲=2i
␩
sech关2
␩
共x−
␨
兲兴e−i关2
␰
共x−
␨
兲+⌽兴,共9兲
wi h he 4 CCs soli on posi ion
␨
共 兲, phase ⌽共 兲, ampli ude
␩
共 兲, and eloci y
␰
共 兲. The soli on ene gy eads
H0=16
3
␩
共3
␰
2−
␩
2兲.共10兲
Se ing Y1=
␨
,Y2=⌽,Y3=
␩
and Y4=
␰
in Eqs. 共5兲–共8兲,we
ob ain o n=1
8
␩
˙
␰
+8
␩
␰
˙=
冕
−⬁
+⬁
dx
冉
R
⳵
uⴱ
⳵␨
+Rⴱ
⳵
u
⳵␨
冊
,共11兲
whe e 2
␩
is he soli on mass, and −4
␰
共 兲 he soli on eloci y
关30兴. The e o e his equa ion has he o m o he New onian
equa ion o mo ion o a poin pa icle wi h a iable mass.
The hs o Eq. 共11兲 ep esen s he ime-dependen o ce
ob ained by means o he a e age o he pe u ba ion Ro e
space, whe e he “weigh unc ion” is jus he a ia ion o
he ansa z wi h espec o he i s CC, he soli on posi ion.
In addi ion o he poin -pa icle ep esen a ion, he e olu ion
o
␨
and ⌽ akes in o accoun o he deg ees o eedom.
Then, in a simila way, i.e., se ing Y1=
␨
,Y2=⌽,Y3=
␩
and
Y4=
␰
in Eqs. 共5兲–共8兲, we ob ain o n=2,3,4
4
␩
˙=−
冕
−⬁
+⬁
dx
冉
R
⳵
uⴱ
⳵
␾
+Rⴱ
⳵
u
⳵
␾
冊
,共12兲
4共⌽
˙−2
␰␨
˙兲⬅4共
␾
˙+2
␨␰
˙兲
=−16共
␩
2−
␰
2兲+
冕
−⬁
+⬁
dx
冉
R
⳵
uⴱ
⳵
␩
+Rⴱ
⳵
u
⳵
␩
冊
,
共13兲
8
␩
␨
˙=−32
␩
␰
−
冕
−⬁
+⬁
dx
冉
R
⳵
uⴱ
⳵␰
+Rⴱ
⳵
u
⳵␰
冊
,共14兲
espec i ely, whe e ⌽=
␾
+2
␨␰
. Hence, he e olu ion o
␩
and
␨
is gi en di ec ly by Eqs. 共12兲and 共14兲, espec i ely.
Subs i u ing
␩
˙ om Eq. 共12兲in Eq. 共11兲 he equa ion o
mo ion o
␰
is a ained. Finally, using he ob ained equa ion
o
␰
˙in Eq. 共13兲 he ime e olu ion o ⌽is ound. Equa ions
共11兲–共14兲a e equi alen o he esul s o he adiaba ic pe -
u ba ion heo y 关29兴. They a e also equi alen o he Eqs.
共3.17兲–共3.20兲o Re . 关14兴 o
␩
˙,
␰
˙,
␨
˙, and
␾
˙ob ained by he
IST wi h a bi a y pe u ba ions. Howe e , he GTWM has
he ollowing ad an ages:
共1兲I does no equi e ha he IST can be applied o he
unpe u bed sys em. In ac , he speci ic o m o H0in Eq.
共3兲, which is ela ed o he unpe u bed NLSE, is no used in
he de i a ion o he e olu ion Eqs. 共5兲 o he CCs.
QUINTERO, MERTENS, AND BISHOP PHYSICAL REVIEW E 82, 016606 共2010兲
016606-2
共2兲The GTWM wo ks o an a bi a y numbe o CCs.
Fo example he ansa z o Re . 关37兴wi h 6 CCs can be used.
See also he ema ks a he end o he conclusions sec ion.
共3兲The p ocedu e which yields he e olu ion Eqs. 共5兲 o
he CCs is ex emely concise. A e ha ing chosen a speci ic
ansa z, he calcula ion o he in eg als Eq. 共6兲and he o ces
Eq. 共7兲is s aigh o wa d. The same holds o he pe u ba-
ion e ms Eq. 共8兲when he pe u ba ion Rhas been speci-
ied.
III. VARIATIONAL APPROACH AND GTWM
An ad an age o he GTWM is ela ed o he ac ha i
can be applied in cases whe e he Lag angian densi y
L共u,uⴱ,ux,ux
ⴱ,u ,u
ⴱ兲o he pe u bed sys em is unknown.
Howe e , i Lexis s and i , in addi ion, Land he ansa z
Eqs. 共4兲sa is y ce ain condi ions 关see Eqs. 共22兲and 共23兲
below兴, we show ha he a ia ional app oach and he
GTWM a e equi alen , i.e., bo h me hods yield he same
equa ions o mo ion o he CCs.
In o de o p o e his s a emen , we sepa a e he pe u ba-
ion R关u共x, 兲;x, 兴in wo pa s
R关u共x, 兲;x, 兴=−i
␤
u+B关u共x, 兲;x, 兴,共15兲
such ha he dissipa ion in he sys em appea s only in he
i s e m, −i
␤
u. We assume ha he pe u bed NLSE Eq. 共1兲
is equi alen o an Eule -Lag ange equa ion, gene alized by a
dissipa i e e m on he hs,
d
d
⳵
L
⳵
u
ⴱ+d
dx
⳵
L
⳵
ux
ⴱ−
⳵
L
⳵
uⴱ=
␦
F
␦
u
ⴱ,共16兲
wi h he dissipa ion unc ion
F=−i
␤
共uu
ⴱ−uⴱu 兲,共17兲
and simila ly o he complex conjuga e o Eq. 共1兲. We no e
ha mo e complica ed dissipa ion e ms han he simple e m
−i
␤
uin Eq. 共15兲can also be ea ed by gene alizing he
Eule -Lag ange o malism 关17兴.
Now, inse ing he ansa z Eq. 共4兲in Land F关32,33兴and
in eg a ing, we ob ain
L=
冕
−⬁
+⬁
dxL共u,uⴱ,ux,ux
ⴱ,u ,u
ⴱ兲=L共Y
ជ
;Y
ជ
˙兲,共18兲
F=
冕
−⬁
+⬁
dxF共u,uⴱ,ux,ux
ⴱ,u ,u
ⴱ兲=F共Y
ជ
;Y
ជ
˙兲.共19兲
Hence, he Mcollec i e coo dina es sa is y Mgene alized
Lag ange equa ions
d
d
⳵
L
⳵
Y
˙n
−
⳵
L
⳵
Yn
−
⳵
F
⳵
Y
˙n
=0, 共20兲
o equi alen ly
冕
−⬁
+⬁
dx
冋
d
d
⳵
L
⳵
Y
˙n
−
⳵
L
⳵
Yn
−
⳵
F
⳵
Y
˙n
册
=0, 共21兲
whe e n=1,...,M. F om now on we deno e Y=Ynand as-
sume 兵see Eq. 共4兲in 关32兴其 ha
冋
⳵
L
⳵
ux
⳵
u
⳵
Y
册
x→+⬁
−
冋
⳵
L
⳵
ux
⳵
u
⳵
Y
册
x→−⬁
=0, 共22兲
冋
⳵
L
⳵
ux
ⴱ
⳵
uⴱ
⳵
Y
册
x→+⬁
−
冋
⳵
L
⳵
ux
ⴱ
⳵
uⴱ
⳵
Y
册
x→−⬁
=0. 共23兲
As he ollowing ela ions hold
d
d
⳵
L
⳵
Y
˙=d
d
冉
⳵
L
⳵
u
⳵
u
⳵
Y
冊
+d
d
冉
⳵
L
⳵
u
ⴱ
⳵
uⴱ
⳵
Y
冊
,共24兲
⳵
L
⳵
Y=
⳵
L
⳵
u
⳵
u
⳵
Y+
⳵
L
⳵
uⴱ
⳵
uⴱ
⳵
Y+
⳵
L
⳵
ux
⳵
ux
⳵
Y
+
⳵
L
⳵
ux
ⴱ
⳵
ux
ⴱ
⳵
Y+
⳵
L
⳵
u
⳵
u
⳵
Y+
⳵
L
⳵
u
ⴱ
⳵
u
ⴱ
⳵
Y,共25兲
⳵
F
⳵
Y
˙=
⳵
F
⳵
u
⳵
u
⳵
Y+
⳵
F
⳵
u
ⴱ
⳵
uⴱ
⳵
Y,共26兲
and aking in o accoun he ela ions Eqs. 共22兲and 共23兲, Eq.
共21兲becomes
冕
−⬁
+⬁
dx
再
d
d
冉
⳵
L
⳵
u
冊
+d
dx
冉
⳵
L
⳵
ux
冊
−
⳵
L
⳵
u−
⳵
F
⳵
u
冎
⳵
u
⳵
Y
+
冕
−⬁
+⬁
dx
再
d
d
冉
⳵
L
⳵
u
ⴱ
冊
+d
dx
冉
⳵
L
⳵
ux
ⴱ
冊
−
⳵
L
⳵
uⴱ−
⳵
F
⳵
u
ⴱ
冎
⳵
uⴱ
⳵
Y=0.
共27兲
Inse ing he Lag angian densi y Land he dissipa ion unc-
ion Fin Eq. 共27兲, he i s and second cu ly b acke s become
he NLS Eq. 共1兲and i s co esponding complex conjuga e
equa ion, espec i ely 共up o a cons an ac o 兲, whe e Ris
gi en by Eq. 共15兲. Then, Eq. 共27兲 ep esen s he in eg a ion
o he sum o he i s NLSE Eq. 共1兲mul iplied by
⳵
u/
⳵
Yand
he second NLSE, i.e., he complex conjuga e o Eq. 共1兲,
mul iplied by
⳵
uⴱ/
⳵
Y. This p ojec ion p ocedu e is exac ly
he same as ha which has p oduced he MEqs. 共5兲in Sec.
II. Thus, he Lag angian app oach is equi alen o he
GTWM and ice e sa. Howe e , in p ac ice he GTWM
equi es mo e wo k i M⬎3, because M共M−1兲/2 in eg als
IYnYjand Min eg als Rn共Y
ជ
兲in Eqs. 共6兲and 共8兲, espec i ely,
mus be calcula ed. In he Lag angian me hod one has o
pe o m only M+3 in eg als o ob ain Land F. On he o he
hand, he GTWM is mo e gene al since he Lag angian den-
si y o he pe u bed sys em need no be known.
IV. GTWM AND MODIFIED CONSERVED QUANTITIES
A e y peculia p ope y o GTWMs is ela ed o i s
ela ionship wi h he so-called modi ied conse a ion laws
共MCL兲共 he ime e olu ion o he quan i ies which a e con-
se ed o he unpe u bed sys em兲关18兴. Such an equi alence
was al eady shown o NLKGEs in 关23兴since usually he
GENERALIZED TRAVELING-WAVE METHOD,…PHYSICAL REVIEW E 82, 016606 共2010兲
016606-3
equa ions o mo ion o M CCs a e ela ed wi h he i s M
MCLs.
In his sec ion, we show ha in he case o he NLSE he
GTWM wi h he ansa z
u共x, 兲=u共x−
␨
,⌽,
␩
,
␰
兲,uⴱ共x, 兲=uⴱ共x−
␨
,⌽,
␩
,
␰
兲,
共28兲
yields a i s only 3 MCLs, namely he ime a ia ion o he
no m, he momen um, and he ene gy. In Eq. 共28兲
␨
deno es
he soli on posi ion and ⌽ ep esen s a phase, 共so ha
⳵
u/
⳵
⌽=−iu and
⳵
uⴱ/
⳵
⌽=iuⴱ兲. We s ess ha we need no
ye make a speci ic ansa z o u. Wi h M=4 and se ing n
=1 in Eqs. 共5兲–共8兲we ob ain
兺
j=1
4
I
␨
YjY
˙j=dP
d +
冕
−⬁
+⬁
dx
⳵
⳵
x
i
2关u uⴱ−uu
ⴱ兴,共29兲
F1共
␨
,⌽,
␩
,
␰
兲=
冕
−⬁
+⬁
dx
⳵
H0
⳵
x,共30兲
R1共
␨
,⌽,
␩
,
␰
兲=
冕
−⬁
+⬁
dx
冉
R
⳵
uⴱ
⳵
x+Rⴱ
⳵
u
⳵
x
冊
,共31兲
whe e Pis he momen um
P=
冕
−⬁
+⬁
dx i
2关uux
ⴱ−uⴱux兴.共32兲
Subs i u ing Eqs. 共29兲–共31兲in Eq. 共5兲and aking in o accoun
ha
冕
−⬁
+⬁
dx
⳵
⳵
x
再
i
2关u uⴱ−uu
ⴱ兴−H0
冎
=0, 共33兲
we ob ain
dP
d =
冕
−⬁
+⬁
dx
冉
R
⳵
uⴱ
⳵
x+Rⴱ
⳵
u
⳵
x
冊
,共34兲
which ep esen s he a ia ion o he momen um wi h espec
o ime o he pe u bed NLSE Eq. 共1兲and i s complex
conjuga e equa ion.
Taking n=2 in Eqs. 共5兲–共8兲, we ob ain
兺
j=1
4
I⌽YjY
˙j=−dN
d ,N=
冕
−⬁
+⬁
dx兩u兩2,共35兲
F2共
␨
,⌽,
␩
,
␰
兲=−
⳵
H0
⳵
⌽=−
⳵
⳵
⌽
冕
−⬁
+⬁
dx共兩ux兩2−兩u兩4兲=0,
共36兲
R2共
␨
,⌽,
␩
,
␰
兲=−
冕
−⬁
+⬁
dx
冉
R
⳵
uⴱ
⳵
⌽+Rⴱ
⳵
u
⳵
⌽
冊
=−
冕
−⬁
+⬁
dx i共Rⴱu−Ruⴱ兲.共37兲
Hence, subs i u ing Eqs. 共35兲–共37兲in Eq. 共5兲ou second
modi ied conse a ion law eads
dN
d =
冕
−⬁
+⬁
dx i共Rⴱu−Ruⴱ兲,共38兲
and ep esen s he a ia ion o he no m N.
Finally, we inse in Eqs. 共5兲–共8兲Y1=
␨
,Y2=⌽,Y3=
␩
, and
Y4=
␰
, wi h M=4. Then, we mul iply hese equa ion by
␨
˙,⌽
˙,
␩
˙, and
␰
˙, espec i ely; adding hem yields
兺
n=1
4
兺
j=1
4
IYnYjY
˙jY
˙n=0, 共39兲
F1
␨
˙+F2⌽
˙+F3
␩
˙+F4
␰
˙=−dH0
d ,共40兲
R1
␨
˙+R2⌽
˙+R3
␩
˙+R4
␰
˙=−
冕
−⬁
+⬁
dx共Ru
ⴱ+Rⴱu 兲,共41兲
so ha he e olu ion o he ene gy is gi en by
dH0
d =−
冕
−⬁
+⬁
dx共Ru
ⴱ+Rⴱu 兲,共42兲
which can be w i en as
dH0
d =−兺
j=1
4
Y
˙j
冕
−⬁
+⬁
dx
冉
R
⳵
uⴱ
⳵
Yj
+Rⴱ
⳵
u
⳵
Yj
冊
.共43兲
Equa ion 共43兲 ep esen s he hi d MCL o NLSE.
We would like o ind a ou h MCL, since we a e using
ou CC equa ions om ou GTWM. I is na u al o expec
ha hese equa ions a e ela ed no only o he i s h ee
conse a ion laws bu also o he ou h one. The in ini e
sequence o conse a ion laws o he unpe u bed NLSE can
be ob ained ia he ollowing ecu ence ela ion o densi-
ies, which was ob ained in he amewo k o he IST 关34兴,
bm+1 =uⴱd
dx
冉
bm
uⴱ
冊
+兺
k+j=m−1
bkbj,共44兲
wi h b0=兩u兩2,m=0,1,....Asimila ela ion, wi h bm+1
ⴱ, holds
o he complex conjuga e o Eq. 共1兲wi h R=0. In his way,
o m=0 he momen um densi y P=−共i/2兲共b1−b1
ⴱ兲is ob-
ained 关34兴, which is iden ical wi h ha in Eq. 共32兲. Fo m
=1, we ob ain he ene gy densi y and o m=2 a new densi y,
␳
1=−共i/2兲共b3−b3
ⴱ兲=i
2兵3兩u兩2共uux
ⴱ−uxuⴱ兲+ux
ⴱuxx −uxuxx
ⴱ其.
共45兲
Using he one-soli on solu ion o he unpe u bed NLSE,
i.e., Equa ion 共9兲wi h cons an eloci y and ampli ude and
QUINTERO, MERTENS, AND BISHOP PHYSICAL REVIEW E 82, 016606 共2010兲
016606-4
he linea unc ions in ime
␨
=
␨
0−4
␰
and ⌽=⌽0
−4共
␰
2+
␩
2兲 , his yields a new conse ed quan i y H1
=兰−⬁
+⬁dx
␳
1共x, 兲=32
␩
␰
共
␰
2−
␩
2兲. Due o he pe u ba ions R
and Rⴱin Eq. 共1兲and i s complex conjuga e equa ion, espec-
i ely, he e olu ion o H1is de e mined by 关18兴
dH1
d =
冕
−⬁
+⬁
dx关Rⴱ共6兩u兩2ux+uxxx兲+R共6兩u兩2ux
ⴱ+uxxx
ⴱ兲兴,
共46兲
whe e H1is he Hamil onian o he i s in eg able hie a chy
o he NLSE 关35兴. Compa ing he hs o Eq. 共8兲wi h he hs
o Eq. 共46兲we ealize ha in he o me equa ion he pe u -
ba ion Ris mul iplied by he i s de i a i e o uⴱwi h e-
spec o a gi en CC, whe eas in he la e one he hi d spa ial
de i a i e o uⴱappea s. So, he MCL Eq. 共46兲is no equi a-
len o any equa ion o combina ion o equa ions o he sys-
em Eqs. 共5兲–共8兲ob ained by means o GTWM.
Howe e , we can achie e ou goal by conside ing he
e olu ion o he i s momen o he no m 关19兴,
dN1
d =2P+
冕
−⬁
+⬁
dx x共Rⴱu−Ruⴱ兲,N1=
冕
−⬁
+⬁
dx x兩u兩2.
共47兲
We now show ha Eq. 共47兲 oge he wi h he i s h ee
MCLs yields he ou CC Eqs. 共11兲–共14兲. He e we use he
speci ic ansa z Eq. 共9兲 o which he no m
N=4
␩
,共48兲
he i s momen o he no m,
N1=4
␩
␨
,共49兲
he momen um
P=−8
␩
␰
,共50兲
and he ene gy is gi en by Eq. 共10兲. Subs i u ing Eqs.
共48兲–共50兲and 共10兲in he Eqs. 共38兲,共34兲,共47兲, and 共43兲,
espec i ely, we ob ain
4
␩
˙=
冕
−⬁
+⬁
dx i共Rⴱu−Ruⴱ兲,共51兲
−8
␩
␰
˙−8
␰
␩
˙=
冕
−⬁
+⬁
dx
冉
R
⳵
uⴱ
⳵
x+Rⴱ
⳵
u
⳵
x
冊
,共52兲
4
␩
␨
˙+4
␨
␩
˙=−16
␩
␰
+i
冕
−⬁
+⬁
dx x共Rⴱu−Ruⴱ兲,共53兲
−16共
␩
2−
␰
2兲
␩
˙+32
␩
␰␰
˙=−兺
j=1
4
Y
˙j
冕
−⬁
+⬁
dx
冉
R
⳵
uⴱ
⳵
Yj
+Rⴱ
⳵
u
⳵
Yj
冊
.
共54兲
F om Eq. 共51兲and he iden i y
⳵
u/
⳵␰
=−2i共x−
␨
兲u, Eq. 共53兲
becomes
8
␩
␨
˙=−32
␩
␰
−
冕
−⬁
+⬁
dx
冉
R
⳵
uⴱ
⳵␰
+Rⴱ
⳵
u
⳵␰
冊
,共55兲
which p o ides he e olu ion equa ion o
␨
˙. Taking in o ac-
coun Eqs. 共51兲,共52兲, and 共55兲, Eq. 共54兲becomes
␩
˙
再
4共⌽
˙−2
␰␨
˙兲−16共
␰
2−
␩
2兲−
冕
−⬁
+⬁
dx
冉
R
⳵
uⴱ
⳵
␩
+Rⴱ
⳵
u
⳵
␩
冊冎
=0,
共56兲
yielding he e olu ion equa ion o ⌽since
␩
e ol es in
ime. F om Eqs. 共51兲and 共52兲we ob ain he equa ion o
mo ion o
␩
and
␰
. F om Eqs. 共55兲and 共56兲, he equa ions
o
␨
and ⌽can be ob ained. In ac , hese ou equa ions
ag ee wi h Eqs. 共11兲–共14兲ob ained ia GTWM.
V. CONCLUSIONS
We ha e de eloped he GTWM o he pe u bed NLS
Eqs. 共1兲and he complex conjuga e o Eq. 共1兲共 his is di e -
en om ea lie applica ions o he me hod which s a ed
om he Hamil on equa ions o magne ic o ices and o
soli ons in NLKGEs兲. The GTWM is based on a p ojec ion
echnique: The NLSEs Eq. 共2兲and he complex conjuga e o
Eq. 共2兲a e mul iplied by de i a i es o uⴱand uwi h espec
o one o he CCs, espec i ely. Then he esul ing equa ions
a e added and in eg a ed o e he sys em. This yields he
e olu ion equa ions o an a bi a y numbe o CCs, Eqs.
共5兲–共8兲, wi hou using a speci ic ansa z. This me hod is mo e
gene al han he a ia ional app oach because i is no nec-
essa y o know he Lag angian densi y Lo he pe u bed
sys em. E.g., see Eq. 共1兲o 关36兴, whe e he Lag angian den-
si y does no exis , bu he Hamil onian H0o he unpe -
u bed sys em is known. In cases whe e Lexis s and condi-
ions Eqs. 共22兲and 共23兲a e sa is ied, we ha e shown ha he
GTWM and he a ia ional o malism a e equi alen , i.e.,
om bo h heo ies he same e olu ion equa ions o he CCs
a e ob ained.
Using 4 collec i e coo dina es wi h he ansa z Eq. 共9兲,
namely, he posi ion
␨
, eloci y
␰
, ampli ude
␩
and phase ⌽
o he soli on, we ha e explici ly shown ha he GTWM is
equi alen o he esul s ob ained om he IST, om he
adiaba ic pe u ba ion heo y and om he ime a ia ion o
he no m, he i s momen o he no m, he momen um, and
he ene gy o he sys em. Howe e , we s ess ha he
GTWM is e y concise, and mo e gene al han he IST and
he ime- a ia ion o he conse ed quan i ies since i can be
applied when we ha e an ansa z wi h mo e han 4 CCs 共see
he ansä ze wi h 5 CCs in Re s. 关36,38兴 o soli on p opaga-
ion in op ical ibe s and wi h 6 CCs used in 关37兴兲. Mo eo e ,
he pe u ba ions conside ed in Re s. 关36,38兴include se e al
dissipa i e e ms which p esen no addi ional di icul y o
he GTWM, in con as o he a ia ional me hod. Wo k
along his line is in p og ess.
ACKNOWLEDGMENTS
We hank Yu i Gaididei 共Kie 兲and Edwa d A e alo
共D esden兲 o e y use ul discussions on his wo k. N.R.Q.
GENERALIZED TRAVELING-WAVE METHOD,…PHYSICAL REVIEW E 82, 016606 共2010兲
016606-5

acknowledges inancial suppo by he Minis e io de Edu-
cación y Ciencia 共MEC, Spain兲 h ough G an No. FIS2008-
02380/FIS, and by he Jun a de Andalucía unde P ojec Nos.
FQM207, FQM-00481, P06-FQM-01735, and P09-FQM-
4643. F.G.M. acknowledges he hospi ali y o he Uni e si y
o Se ille and o he Theo e ical Di ision and Cen e o
Nonlinea S udies a Los Alamos Labo a o y. Wo k a Los
Alamos was suppo ed by U.S. DOE. F.G.M. acknowledges
inancial suppo by IMUS and by he Plan P opio o he
Uni e si y o Se ille.
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