Compactons versus Solitons

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by  Paulo E. G. Assis

Compactons versus Solitons
Paulo E.G. Assis and Andreas Fring
(Dated: January 9, 2009)
WeinvestigatewhethertherecentlyproposedPT-symmetricextensionsofgeneralizedKorteweg-
de Vries equations admit genuine soliton solutions besides compacton solitary waves. For models
which admit stable compactons having a width which is independent of their amplitude and those
which possess unstable compacton solutions the Painlev´e test fails, such that no soliton solutions
canbefound. ThePainlev´etestispassedformodelsallowingforcompactonsolutionswhosewidth
isdeterminedbytheiramplitude. Consequentlythesemodelsadmitsoliton solutionsinadditionto
compactons and are integrable.
I. INTRODUCTION for the Hamiltonian density in (1) is
l,m,p
9 H
0 u +ul−2u +gimup−2um−3 p(p 1)u4 (3)
0 In a recent investigation Bender, Cooper, Khare, Mi- t x x − x
2 hailaandSaxena[1]havefoundcompactonsolutions,i.e. +2pmuu2u +m(m 2)u2u(cid:2)2 +mu2u u =0.
n solitary wave solutions with compact support, for - x xx − xx x xxx(cid:3)
PT
a symmetric extensions of generalized Korteweg-de Vries The main aim of this manuscript is to investigate
J (KdV) equations. The proposed models generalize vari- whether this equation admits soliton solutions and is

9 ous systems previously studied and are described by the therefore integrable for some specific choices of the pa-
Hamiltonian density rameters l,m,p. We will also address the question of

] whether it is possible to find solitons and compactons
h
t ul g in the same model or whether only one type of solu-
p- Hl,m,p =−l(l 1) − m 1up(iux)m. (1) tions may exist. To answer these questions one could
e − − of course construct explicitly the soliton solutions, con-
h served charges, Lax pairs, Dunkl operators, etc., which
The density reduces to a modification of a Hamil-
[ tonian descrHiplt,i2o,pn [2, 3] of generalized KdV-equations is usually a formidable task. Instead we will carry out
the Painlev´e test following a proposal originally made
1 [4], which are known to admit compacton solutions. For
v by Weiss, Tabor and Carnevale [14]. The test provides
l = 3, p = 0 and m = ε+1 one obtains a re-scaled ver-
7 anindicationabouttheexistenceofsolitonsolutionsand
sionofthe -symmetricextensionoftheKdV-equation
6 PT discriminatesbetweenmodels,whichareintegrablethose
2 (ε=1)introducedin[5]. ThefirstPT-symmetricexten- which are not.
1 sionsofthe KdV-equationproposedin[6]cannotbe ob-
. tained from (1) as they correspond to non-Hamiltonian
1
0 systems. II. THE PAINLEVE´ TEST
9 Thevirtueof -symmetry,i.e. invarianceunderasi-
0 multaneous pariPtyTtransformation : x x and time
: P → − The basic assumptionfor the existence ofa solitonso-
v reversalT :t→−t,i→−i,foraclassicalHamiltonianis lution is that it acquires the general form of a so-called
i thatitguaranteestherealityoftheenergyduetoitsanti-
X Painlev´e expansion [14]
linear nature [5]. When quantizing one also needs to
r ensure -symmetryofthecorrespoHndingwavefunctions ∞
a PT
inordertoobtainrealspectra[7,8,9,10]. Themostnat- u(x,t)= λ (x,t)φ(x,t)k+α. (4)
k
uralwaytoimplement -symmetryin(1)istokeepthe Xk=0
PT
interpretationfromthestandardKdV-equationandview
thefielduasavelocity,suchthatittransformsasu u. One further demands that in the limit φ(x,t) 0, the
Then is -symmetric for real coupling cons→tant functionu(x,t)ismeromorphic,suchthatthele→adingor-
l,m,p
g andHall possibPleTrealvalues of l,m,p. Alternatively, we dersingularityαisanegativeintegerandthe λk(x,t)are
could also allow a purely complex coupling constant, i.e. analyticfunctions. ThegeneralprocedureofthePainlev´e
g iR, by transforming the field as u u, such that test consists in substituting the expansion (4) into the
∈ is -symmetric when l is even→an−d p+m odd. equation of motion, (3) for the case at hand, and deter-
l,m,p
HForgeneraPlTreviewson -symmetryandnon-Hermitian miningthefunctionsλk(x,t)recursively. Apartialdiffer-
Hamiltonian systems sePeT[11, 12, 13]. ential equation is said to pass the Painlev´e test when all
λ (x,t) can be computed, including enough free param-
The equation of motion resulting from the variational k
eters to match the order of the differential equation. In
principle
[15] we recently applied this method to -symmetric
PT
∞ extensions of Burgers and the standard KdV-equation,
δ dx dn ∂
u = H = ( 1)n H (2) wheremoredetailsonthegeneralitiesandliteraturemay
t (cid:18) Rδu (cid:19)x nX=0 − (cid:18)dxn∂unx(cid:19)x be found.
2
A. Leading order singularities oftherecurrencerelationbecomesanidentity,theλ be-
r
comesafreeparameter,otherwisethePainlev´etestfails.
The Painlev´e test stays and falls with the possibility The possible values for r can be found by substituting
that the initial condition λ0 can be determined, which u(x,t) λ0(x,t)φ(x,t)α+λrφ(x,t)r+α (7)
is essential to commence the iterative procedure to solve →
into (3) and computing all possible values of r for which
the recurrence relation. We compute λ by substitut-
0
λ becomes a free parameter. Considering the case iii)
ing the first term in the expansion (4), i.e. u(x,t) r
λ (x,t)φ(x,t)α, into (3) and evaluating the values f→or for integer values l,m,p the coefficients of the leading
al0l possible leading order singularities α. The individual order φr+α(l−1)−1 is proportional to
terms in (3) have the following leading order behaviour: λ gα(2−l)/m(r+1)(r+αl)[r+α(l 1)]φα(2−l)+1. (8)
u φα−1, ul−2u φα(l−1)−1 and all remaining terms r − x
t x
are∼proportionalto∼φα(m+p−1)−m−1. Therefore the lead- This means that besides the so-called fundamental res-
onance at r = 1, we also find two more resonances at
ing order terms may only be cancelled if any of the fol-
r = αl,α(1 l). Since thedifferentialequation(3)isof
lowing three conditions hold: − −
order three all these models fully pass the Painlev´e test
i) α 1=α(l 1) 1 α(m+p 1) m 1, provided λ−αl and λα(1−l) can indeed be chosen freely.
− − − ≤ − − − Thestandardproceduretoverifythiswouldbenowto
which results from assuming that ut and ul−2ux derive the recursive equation resulting from combining
constitute the leading order terms. In this case we (4) and (3). Clearly for generic values of l,m,p this will
obtainl=2andtheinequalityα(2 m p) m. be extremely lengthy, but even for specific choices it is
− − ≤−
Thus α remains undetermined. fairly complicated. It suffices, however, to compute the
λ up to k > αl. We will present these values for vari-
ii) α 1=α(m+p 1) m 1 α(l 1) 1, k −
− − − − ≤ − − ousexamplesforseveralchoicesofthe parametersl,m,p
which corresponds to the assumption that ul−2ux correspondingtoscenariosleadingtosolutionswithqual-
isthe leastsingulartermandmatchingthe leading itatively different kinds of behaviour.
ordersofalltheremainingones. Thenweconclude
that l 2 and α is fixed to α=m/(m+p 2).
≤ −
III. GENERALIZED KDV-EQUATION
iii) α(l 1) 1=α(m+p 1) m 1 α 1.
− − − − − ≤ −
which is the consequence of u being least singular Cooper, Khare and Saxena [16] found that in the gen-
t
termandthematchingoftheremainingones. This eralizedKdVequation,i.e. m=2,a necessarycondition
means the leading order singularity of u(x,t) is of for compactons to be stable is to consider models with
the order 2<l<p+6. Thismeansnoneoftheconditionsi)orii)
fortheleadingordersingularitytocancelcanbesatisfied.
m
α= Z− and l 2. (5) The special choice l = p+2,0 < p 2 guarantees that
p+m l ∈ ≥ ≤
− the compacton solutions have in addition a width which
Cancelling the leading order terms then yields isindependentoftheiramplitude[2]. Forthatparticular
case alsothe conditioniii)admits no solution, suchthat
λ(n) =e2πinα/m[gl(l 1)]−α/m(iαφ )−α, (6) the Painlev´e test fails.
0 − x However, for models which admit stable compacton
where 1 n p+m l indicates the different solutionshavinga widthdepending onthe amplitude we
roots of t≤he de≤termining−equation. can find solutions to the condition iii) and proceed with
the Painlev´e test. For instance, m = 2,p = 1,l = 5 is
In principle we could alsoenvisagea scenarioinwhich such a choice. In this case we find from (5) that α= 1
−
u andul−2u aretheleastdominanttermsandthelead- and the leading order singularity of the corresponding
intg order sinxgularity is cancelled by all the remaining differential equation is φ−5. Computing now order by
terms. However, all these terms only differ by an overall order the functions λk we find the two solutions
numericalfactor,suchthatλ0 turnsouttobezerointhis λ± = 2i 5gφ , λ± = i 5gφxx,
case and we can therefore discard this case. 0 ± x 1 ∓ φ
p p x
λ± = i√5g 3φ2xx−2φxφxxx , (9)
B. Resonances 2 ∓ 6 (cid:0) φ3 (cid:1)
x
3φ φ2 4i 5g3 6φ3 6φ φ φ +φ2φ
A key feature of the Painlev´e test is the occurrence λ±3 = t x∓ p (cid:0) 4x8xg−φ5 x 3x xx x 4x(cid:1).
of so-called resonances, which arise whenever the coeffi- x
± ±
cient in front of a specific λ in the recurrence relations Cruciallyweobservenextthatλ andλ canbe chosen
r ± 4 5
becomes zero. This implies that λ can not be deter- arbitrarily. The remaining λ for k > 5 can all be com-
r k
mined recursively. When in this case the remaining part puted,buttheexpressionsareallextremelycumbersome
3
andwewillthereforenotreportthemhere. Making,how- the λ(n) with n=1,2,3, of which the first terms are
k
ever,thefurtherassumptiononφtobeatravellingwave,
i.e. φ(x,t) = x ωt, simplifies the expressions consider- λ(n) = ie2πin/3(42g)1/3φ ,
−± ± 0 − x
sacbelny.arCiohroeodsuincge tλo4 = λ5 = 0 the two solutions for that λ1(n) = ie2πin/232(/231φg)1/3φxx, (11)
x
ie2πin/3(7g)1/3 3φ2 2φ φ
λ± = λ± =0 for κ=0,1,2,... λ(n) = xx− x xxx ,
3κ+1 3κ+2 2 2(6)(cid:0)2/3φ3 (cid:1)
± ± ω x
λ0 = ±2ip5g, λ3 =−16g, λ(n) = ie2πin/3(7g)1/3 6φ3xx−6φxφxxxφxx+φ2xφxxxx .
λ± = 3iω2 , λ± = ω3 , (10) 3 (cid:0) 4(6)2/3φ5x (cid:1)
6 ∓3584√5g5/2 9 573440g4 From (8) we know that we should encounter resonances
± 33iω4 at the level 6 and 7, which is indeed the case as we find
λ = ,
12 ±1669857280√5g11/2 that λ(n)and λ(n) can be chosen freely. The remaining
6 7
± 3ω5 λk(n) for k > 7 can all be computed iteratively and the
λ = , ...
15 −66794291200g7 Painlev´e test is passed for this example.
For a travelling wave ansatz φ(x,t)=x ωt with the
−
choice λ(n) = λ(n) =0 the expressions simplify to
We conclude that the Painlev´e test is passed for this 6 7
choices of parameters, which means that besides stable λ(n) = λ(n) =λ(n) =λ(n) =0, for κ=0,1,...
compacton solutions, whose width depends on their am- 5κ+1 5κ+2 5κ+3 5κ+4
plitude,wealsofindgenuinesolitonsinthesemodelsand, λ(n) = ie2πin/3(42g)1/3, λ(n) = e4πin/3ω , (12)
provided the series (4) converges, they are therefore in- 0 − 5 36(42)1/3g4/3
tegrable. 17iω2
λ(n) = ,
In the unstable compacton regime, i.e. l 2 or 10 598752g3
≤
l p+6, the condition iii) can not be satisfied. Con-
se≥quently we do not expect to find genuine soliton solu- λ(n) = 53e2i3nπω3 ,...
tions. We have also verified this type of behaviour for 15 −21555072(42)2/3g14/3
other representative examples which we do not present
Thus we observe no qualitative difference in the -
here. PT
symmetric extensions in comparison to the case m = 2
andfindthatalsointhisonemayhavestablecompacton
solutions, whose width depends on their amplitude and
genuine solitons at the same time.
IV. PT-SYMMETRIC GENERALIZED In the unstable compacton regime, that is l 2 or
≤
KDV-EQUATION l p+3m, the condition iii) can not be satisfied and
≥
the Painlev´e test fails. Once again we do not represent
hereotherrepresentativeexamplesforwhichweobtained
For the -symmetric extensions of the general-
PT the same type of behaviour.
ized KdV-equation (3) the necessary condition for com-
pactons to be stable was extended by Bender et al [1] to
2<l <p+3m. Thusalsoforgenericvaluesofmnoneof
V. DEFORMATIONS OF BURGERS EQUATION
theconditionsi)orii)fortheleadingordersingularityto
cancelcanbesatisfied. Furthermore,therequirementfor
Considering m = 1,p = 1,l = 3 in the equation of
stablecompactonsolutionstopossessalsoawidthwhich
motion(3)isaverysimpleexampleleadingtoaPainlev´e
is independent of their amplitude was generalized in [1]
expansion for u(x,t), which can even be truncated after
tol=p+m. Asforthespecialcasem=2thisvalueco-
thesecondterm. Asthistypeofbehaviourisreminiscent
incides with the leading order singularity resulting from
ofB¨acklundtransformationgeneratingsolutionsfoundin
the condition iii) tending to infinity and therefore the
other models [14], we present this case briefly. For this
Painlev´e test fails.
choice (3) simply reduces to
As in the previous case, for models which have stable
compacton solutions whose width is a function of their u +uu 2igu + igu u2 u u =0, (13)
amplitude the Painlev´etest has a chance to pass, as one t x− xx u2 xx− x xxx
x (cid:0) (cid:1)
canfind a value forthe leading ordersingularityandpo-
whichcanbeviewedasadeformationofBurgersequation
tentiallyhasthecorrectamountofresonances. Weverify
[15] corresponding to the first three terms. Proceeding
this for the example m = 3,p = 1,l = 7, for which we
obtain α = 1 and φ−7 as the leading order singularity as in the previous sections, we find the solution
−
in(3). Since α/m=1/3inthiscase,wefindnowthree 6igφ 6igφ 3φ
− u(x,t)= − x + xx− t (14)
non-equivalentsolutionsrelatedto the differentrootsfor φ 2φ
x
4
provided that φ satisfies the equation integrable. We found thatthe generalizedKdVequation
resultingfrom andtheir -symmetricextensions
l,2,p
φ2φ +φ2φ =2φ φ φ . (15) have theHsame qualitatiPveTbehaviour in the three
x tt t xx tx t x l,m,p
H
differentregimes. Forconveniencewesummarizethedif-
Atravellingwaveφ(x,t)=x ωtisforinstanceasolution
ferent qualitative behaviours in the following table:
−
of (15), such that we obtain the simple expression
compactons solitons
6igφ 3 l,m,p
u(x,t)= x + ω (16) H
l =p+m stable, dependent A,β no
ωt x 2
−
2<l<p+3m stable, independent A,β yes
for the solution of (13). Incidentally, the travelling wave
l 2 or l p+3m unstable no
solution for Burger’s equation [14] coincides with (16). ≤ ≥
Table 1: The models and their solutions.
l,m,p
H
Clearly our investigations do not constitute a full
VI. CONCLUSIONS
fletched mathematical proof as we based our findings on
various representative examples for the different classes
In previous investigations [2, 3, 16] various criteria
and it wouldbe very interesting to settle this issue more
have been found, which separate the models into
Hl,m,p rigorously with a generic argumentation not relying on
three distinct classes exhibiting qualitatively different
case-by-casestudies. At the same time such a treatment
types of compacton solutions, unstable compactons and
would probably provide a deeper understanding about
stablecompactons,whichhaveeitherdependentorfreely
the separation of the different models. Nonetheless, our
selectable width A and amplitude β. We have carried
findings provide enough evidence to make it worthwhile
out the Painlev´e test for various examples for each of
to investigate the models which pass the test with other
these classes and found that all models which allow sta-
techniques developed in the field of integrable models,
ble compactons for which the width can not be chosen
whereas models which do not pass the test may be ex-
independently from their amplitude pass the Painlev´e
cluded from such investigations.
test. AssumingthatthePainlev´eexpansion(4)converges
these models possess the Painlev´e property [17] and al- Acknowledgments: P.E.G.A. is supported by a City
low therefore for genuine soliton solutions and are thus University London research studentship.
[1] C. Bender, F. Cooper, A. Khare, B. Mihaila, and downexplainedbyanti-linearity,J.Phys.B5,S416–S419
A. Saxena, Compactons in PT-symmetric general- (2003).
ized Korteweg-de Vries Equations, arXiv.org:0810.3460 [10] C. M. Bender, D. C. Brody, and H. F. Jones, Must a
(2008). HamiltonianbeHermitian?, Am.J.Phys.71,1095–1102
[2] F.Cooper,H.Shepard,andP.Sodano, Solitarywavesin (2003).
aclassofgeneralizedKorteweg-deVriesequations, Phys. [11] C. Figueira de Morisson Faria and A. Fring, Non-
Rev.E48, 4027–4032 (1993). HermitianHamiltonianswithrealeigenvaluescoupledto
[3] A.KhareandF.Cooper,One-parameterfamilyofsoliton electricfields: fromthetime-independenttothetimede-
solutions with compact support in a class of generalized pendentquantummechanicalformulation, LaserPhysics
Korteweg˘deVriesequations, Phys.Rev.E48(6),4843– 17, 424–437 (2007).
4844 (1993). [12] C. M. Bender, Making sense of non-Hermitian Hamilto-
[4] P. Rosenau and J. M. Hyman, Compactons: Solitons nians, Rept.Prog. Phys. 70, 947–1018 (2007).
with finite wavelength, Phys. Rev. Lett. 70(5), 564–567 [13] A. Mostafazadeh, Pseudo-Hermitian Quantum Mechan-
(1993). ics, arXiv:0810.5643 (2008).
[5] A.Fring, PT-Symmetricdeformations of theKorteweg- [14] J. Weiss, M. Tabor, and G. Carnevale, The Painlev´e
deVries equation, J. Phys. A40, 4215–4224 (2007). propertyforpartialdifferentialequations,J.Math.Phys.
[6] C.M.Bender,D.C.Brody,J.Chen,andE.Furlan, PT- 24, 522–526 (1983).
symmetric extension of the Korteweg-de Vries equation, [15] P. Assis and A. Fring, Integrable models from PT-
J. Phys. A40, F153–F160 (2007). symmetric deformations, arXiv.org:0810.3628 (2008).
[7] E. Wigner, Normal form of antiunitary operators, J. [16] F. Cooper, A. Khare, and A. Saxena, Exact Ellip-
Math. Phys.1, 409–413 (1960). tic Compactons in Generalized Korteweg-DeVries Equa-
[8] C. M. Bender and S. Boettcher, Real Spectra in Non- tions, Complexity 11, 30–34 (2006).
Hermitian Hamiltonians Having PT Symmetry, Phys. [17] B.GrammaticosandA.Ramani, Integrability-andHow
Rev.Lett. 80, 5243–5246 (1998). to detect it, Lect. Notes Phys. 638, 31–94 (2004).
[9] S. Weigert, PT-symmetry and its spontaneous break-