Pseudo Spectral Transform for Schr\"odinger--Poisson Equations

0.18 MB·

by  Jesús Fuentes

PSEUDO SPECTRAL TRANSFORM FOR
SCHRO¨DINGER–POISSON EQUATIONS
7
1 J.FUENTES,P.GALAVIZ,ANDT.MATOS
0
2

b
e
F Abstract. Here we present exact, stationary, parametric solutions to the
Schro¨dinger–Poisson system. We confront two images: on one hand, we
7
draw on the homotopy analysis method which leads us to a nonlinear inte-
1
gral scheme. Indeed, this approach might be simplified by looking for suffi-

cientlysmoothsolutionsvanishingasymptotically. However,sinceoursystem
]
h possesses stiffness an additional analysis has to be considered. On the other
p hand, we seek for exact solutions over the inverse scattering method by in-
- troducing a pseudo spectral transform. In fact, this pseudo spectral method
t generalises Korteweg–de Vries family’s kernel and let us to circumvent some
n
technicaldifficultiesoriginallyariseninourfirstapproachalthough,again,we
a
u cometoanintegralrepresentation, whichwetestforconvergence.
q
[

2 1. Introduction
v
TheSchr¨odinger–Poissonsystemisrelevantforseveralfieldsinphysicsandmath-
4
0 ematics. There are cosmologicalmodels where the dark matter takes the form of a
1 scalar field. Such field dynamics for galactic halos is described by a Schr¨odinger–
5 Poissonsystem [1, 2,3, 4, 5]. Hypothetical dense scalarfield knownas Bosonstars
0
are the relativistic counterpart of a galactic halo [6, 7, 8, 9, 10, 11, 12]. In the
.
1 astrophysics realm, Schr¨odiger equation describes the dynamics of a scalar field
0 while Poisson equation dictates the dynamics of the gravitational field. On the
7
other hand, in quantum mechanics and semiconductor theories the Schr¨odinger–
1
Poissonsystemmodels the interactionof chargedparticlesin crystals[13, 14], e.g..
:
v Schr¨odingerequationdescribes aparticledynamics under the presenceofa electric
i
X field given by Poissonequation.
A convenient way to solve the Schr¨odinger–Poisson equations is by numerical
r
a computations [15]. Although there are several theoretical studies showing some of
the general properties of the system, in particular its solitonic nature [16, 17, 18].
The first techniques to solve some celebrated solitonic equations were laid soon
after Zabusky and Kruskal reported their observations on the Fermi–Pasta–Ulam
puzzle [19] (even Seeger et al. noticed quite the same over a decade before [20]).
Among these techniques, the spectral transform, originally introduced by Gardner
et al. [21],isconsideredasthecornerstoneofsolitontheorygivenitisanextension
of the Fourier transform to the nonlinear framework [22]. The theory became
augmented by relevant enhancements and additional developments to the spectral
transform[23, 24] as well as the introduction of alternative approachesfounded on
the Darboux and B¨acklund transformations [25, 26, 27, 28, 29]. Today, the length
of these methods is such that these reach applications of indubitable interest in
other areas of mathematical physics [30].
1
2 J.FUENTES,P.GALAVIZ,ANDT.MATOS
In this workwe concentrateonthe spectraltransformin orderto seekfor exact,
stationarysolutionstotheSchr¨odinger–Poissonequations. Foremost,ourprocedure
is entirely heuristical and lies substantially on integral equations.
The paper is organized as follows: In section 2 approximated solutions are ob-
tained through the homotopy analysis method [31]. This algorithm enable us to
solvenonlinearintegralequations,beingananalogouspathtotheinversescattering
method where the linear integral equation of Gelfand–Levitan–Marchenko (GLM)
raises (section 3). Finally, our results are summarized in section 4. The presented
solutions are single–solitonic, i.e. are uniquely related to an individual bounded
state.
1.1. Units and Notation. In order to simplify the writing, we use dimensionless
quantities along the document, unless it is stated otherwise. In particular, we use
geometricalunitsintheveryfirstpartofsection2tointroducetheEinstein–Klein–
Gordon equations. From section 3, Schr¨odinger equation appears in its canonical
form, where ~=1.
We try to avoidallfunctionaldependences unless these are absolutelynecessary
because of ambiguity or lack of context. Also, the following notation is employed:
i is the imaginary unit (√ 1),
−
x,y are real independent variables,
ι,κ,µ,ν are discrete (integer) indexes (quantities),
u=u(x) is an arbitrary test function,
u denotes the derivative of the function u with respect to x,
x
R is a tensor,
µν
K() is a kernel,
·
K () is an iterated kernel,
µ
·
ψ,Ψ are wave functions,
S[u] is the spectral transform of the function u,
∂ denotesapartialdifferentialoperatoractingontheindependentvariable
x
x,
stands for the Laplace operator,
△
H is the Schr¨odinger operator,
unless a different thing is specified on the text.
2. First approximations
The Schr¨odinger–Poissonequations
(1a) [ +φ]ψ =iψ ,
t
−△
(1b) φ= ψ 2,
△ | |
are the canonical representation of the Schr¨odinger–Newton equations originally
introducedin[32], which, constitute the weakfield limit ofEinstein–Klein–Gordon
equations
1
(2) R g R=8π ψ T ψ ,
µν µν µν
− 2 h | | i
here R , g and T stand for the Ricci tensor, the metric and the energy–
µν µν
momentum operator, all in geometric units.
From now on we restrict our study to one spatial coordinate, namely x. Addi-
tionally, we will assume stationary solutions of the form ψ(x,t) = u(x)exp( iωt),
±
PSEUDO SPECTRAL TRANSFORM FOR SCHRO¨DINGER–POISSON EQUATIONS 3
where ω is a suitable real constant, t represents the time and u : X Y is, in
general,a holomorficfunction such that X and Y are open subsets in→C. Thence,
the nonlinear system in (1) is reduced to
(3a) ∂2+bφ u= ωu,
− x ∓
(3b) (cid:2) φ(cid:3)xx = u2,
| |
having introduced a complex parameter b, for generality. The choice in the sign
relatedtoωisphysicallyirrelevant. Also,noticethatuandφaretimeindependent.
Broadly speaking, u can be computed in the neighbourhood x x by Pad´e
0
| − |
approximants of the form [µ/µ+1] (x), with U = u /u (furthermore details can
U x
befoundin[33,34]). Yet,firstapproximationsofucanbeobtainedbyun-coupling
(3).
Consider (3a), which differentiated two times with respect to x reads
(4) u +b(φu +φ u)+2bφ u =ωu ,
xxxx xx xx x x xx
−
by making the substitutions φ u2, (bφ ω)u u and solving for φ , (4)
xx xx x
→ | | − →
becomes
u u−1(u )2 bu u2
(5) φ = dx u2 = xxxx− xx − | | .
x
X | | 2bux
Z
Yet again, consider (3a), by differentiating it with respect to x, solving for φ
x
and equating with (5), we obtain an uncoupled scheme of the Schr¨odinger–Poisson
equations:
b u2u3+[(u )2+2u u ]u 2(u )2u
xx x xxx xx x
(6) u = | | − ,
xxxx u2
that possess just the same integral curves than (3).
In particular, if we consider u as a real function to be regular for all real values
of x and to vanish asymptotically rapid, viz.
(7) lim xǫ+1u(x) =0, 0<ǫ,
x→±∞ | |
(cid:2) (cid:3)
only the first term in the right–hand side of eq. (6) does contribute significantly if
compared to the others, consequently
(8) u bu3,
xxxx
≈
which general Urysohn form is
bx3 x bx2 x
u(x)=C dyu3(y) C dyyu3(y)
1 2
6 − 2
(9) Z Z
bx x b x
+C dyy2u3(y) C dyy3u3(y)+F(x),
3 4
2 − 6
Z Z
hereC areconstanstsdeterminedfromboundaryconditionsandthefunctionF(x)
i
equals to c +c x+c x2+c x3+f(x), where c stand for constants and f =f(x)
1 2 3 4 i
is a real valued function to be continuous in the sense of Lipschitz, to assure that
equation (9) has a unique solution.
Inordertoapproximatethesolutionofthenonlinearintegralequation(9),hither
we focus on the homotopy analysis method. That being said, assume that each
4 J.FUENTES,P.GALAVIZ,ANDT.MATOS
individual term of the Urysohn form (9) has a kernel K(x,y) R. Let Ξ (u3) R
µ
∈ ∈
be the µ–th order deformation constraint given by
µ ν
(10) Ξµ(u3)= uµ−ν uβuν−β,
ν=0 β=0
X X
then, the homotopy–series solution to (9) is computed through
∞ ∞ x
(11) u(x)= uµ(x)= dyK(x,y)Ξµ−1(u3) ,
µ=0 µ=0(cid:26)Z (cid:27)
X X
subject to the initial guess u = f. The last expression is a pivotal trace in our
0
quest for exact solutions. A way to obtain general exact solutions to (3) is by the
inversescatteringmethod, whichis the spannedversionof (11). See ourdiscussion
insection3. Inourcurrentanalysis,while the kernelin(11)is separable,giventhe
interval x [χ ,χ ], convergence of series (11) is assured iff
1 2
∈
χ2
(12) 1 dyK(y,y) 1,
− ≤
(cid:12) Zχ1 (cid:12)
(cid:12) (cid:12)
is undeniably satisfied1. Thu(cid:12)s, at the hand of all(cid:12)owing f = sech(ax), with a some
(cid:12) (cid:12)
real constant, it does imply that uniquely the C and C terms in (9) will behave
2 4
as required by (7), so the constants c = c = c = c = 0 and the solution to (9)
1 2 3 4
is approximated in virtue of
∞ C x2 x C x
(13) u(x)=sech(ax) b 2 dyyΞµ−1(u3)+ 4 dyy3Ξµ−1(u3) ;
− 2 6
µ=1(cid:26) Z Z (cid:27)
X
indeed,theprogramme(13)leadtotheexactsolutionto(9),ifftheseries(11)does
converge. Unfortunately, these integrals may become complicated, hence, in most
cases,(13)shouldbeenoughfornumericalcomputationofthefirstroughsolutions.
Moreover, the fact that we have restricted our study to those solutions com-
pelled to suffice (7), is only a feasible gimmick to work for solutions of accesible
interpretation in the physics environment. Nonetheless, a different class might be
considered for another purposes.
In the next section we study the generalizationof the former blueprint in accor-
dance with a parallel essay from soliton theory.
3. Devising exact solutions
Straightaway,lookingatthestationarySchr¨odingerequationdefinedinallx R
∈
(14) HΨ=k2Ψ,
the discrete part of the operator H= ∂ +u, consists of A negative eigenvalues
xx
−
k2 = p2, with p > 0, α = 1,2,...,A. To each of these eigenvalues corresponds
− α α
a unique solution Ψ impelled to satisfy the boundary condition:
α
(15) lim [exp(p x)Ψ (x)]=1, α=1,2,...,A.
α α
x→∞
1Cf. [35,36]foracomprehensivediscussion.
PSEUDO SPECTRAL TRANSFORM FOR SCHRO¨DINGER–POISSON EQUATIONS 5
Whereasthecontinuumpart,featuredbyallpositiverealvaluesoftheeigenvalue
k2, does typify the solution Ψ to accomplish the asymptotic boundary conditions:
Ψ T(k)exp( ikx), x ,
(16) → − →−∞
Ψ exp( ikx)+R(k)exp(ikx), x + ,
→ − → ∞
where T(k) and R(k) stand for the transmission and reflection coefficients2.
Having said all that, the extended spectral transform S of the function u is
defined as
(17) S[u]:= R(k), <k< ; p , q , α=1,2,...,A, β Z ,
α α
{ −∞ ∞ ∈ }
whereq are the normalizationfactors andβ is a discrete parameterrelatedto the
α
GLM equation
∞
(18) W(x,y)+K(x+y)+ dzW(x,z)K(z+y)=0, y >x,
Zx
through the complex valued function
α
1 1
(19) K(x)= dk exp(ikx)R(k)+ q Γ ,p ( x)β+1 ,
α α
2π β+1 −
Z α=1 (cid:20) (cid:21)
X
where
∞
Γ(a,z):= ds exp( s)sa−1,
−
Zz
is the so called incomplete gamma function.
We request the extended spectral transform of (3) as the specified set
(20) S[u]= R(k)=0, <k < ; p =p, q =q, A=1,β =1 ,
1 1
{ −∞ ∞ }
and, correspondingly, the GLM equation becomes
(21)
∞
1 1
W(x,y)+qΓ , p(x+y)2 +ζq dz W(x,z)Γ , p(z+y)2 =0, y >x,
2 2
(cid:20) (cid:21) Zx (cid:20) (cid:21)
with ζ =0 a parameter in pursuance of convenience.
6
Since the kernel in (21) is not separable, it is a matter of appositeness to solve
(18) for W(x,y) in terms of von Neumann series.
In that event, consider the initial definition,
∞
′ ′ ′
(22) K (z+y):= dz K(z+z )K(z +y).
1
Zz
hence, after the GLM equation has been iterated µ>1 times, there results
(23) W(x,y)=K(x+y) ζσ (x,y) ρ (x,y),
µ µ
− − −
where
µ ∞
(24a) σ (x,y)= ζν−1 dzK (z+y)K(x+z),
µ ν
ν=1 Zx
X ∞
(24b) ρ (x,y)=ζµ+1 dzK (z+y)W(x,z),
µ µ+1
Zx
2Tobeholdhowboth(15)and(16)arecertainlyconsistentwith(7).
6 J.FUENTES,P.GALAVIZ,ANDT.MATOS
and the µ–th iterated kernel
∞
′ ′ ′
(25) Kµ(z+y)= dz Kµ−1(z+z )K(z +y).
Zz
As the latter prescribes, provided that the function K suffices the Picard it-
µ
eration, then it is assured the existence and uniqueness of the solution to (18) by
Lindelo¨f’s apophthegm, thusly, the convergence of the von Neumann series (24)
remains to be tested.
3.1. Analysis of convergence. Theexpansion(23)comprisesthatif K(x+y) is
bounded by a number ǫ R in any closed interval in x of lenght l, hen|ce, inequ|ity
∈
K(x + y) ǫ implies that K (x + y) is also bounded for all µ 2, that is,
µ
|K (x+y)| ≤ ǫ(ǫl)µ−1. Therefore, each term of (24a) satisfies ≥
µ
| |≤
∞
(26) ζµ−1 dzK (z+y)K(x+z) ǫ( ζ ǫl)µ−1 K(x+z) .
µ
≤ | | k k
(cid:12) Zx (cid:12)
(cid:12) (cid:12)
In partic(cid:12)ular, when ζ <1, the sequence(cid:12) σ (x,y) of partial sums is a Cauchy
(cid:12) | | (cid:12){ µ }
sequence for any positive real number δ iff
ν
σ (x,y) σ (x,y) ǫ ( ζ ǫl)κ−1 K(x+y)
ν µ
| − |≤ " | | #k k
κ=µ+1
X
( ζ ǫl)µ
ǫ | | K(x+y)
≤ 1 ζ ǫl k k
−| |
δ,
≤
is held in the very limit µ . Quite in the same way for
→∞
(27) ρ (x,y) ǫ( ζ ǫl)µ W(x,z) ,
µ
| |≤ | | k k
which means that ρ (x,y) will vanish uniformly along the interval l while µ .
µ
→∞
Thereupon, the sequence σ (x,y) of continuous functions converges absolute and
µ
uniformly to the function
∞ ∞
σ(x,y)= ζµ−1 dzK (z+y)K(x+z)
µ
µ=1 Zx
X
∞ ∞
(28) = dz ζµ−1K (z+y) K(x+z)
µ
Zx "µ=1 #
X
∞
= dzΞ(z,y;ζ)K(x+z),
Zx
given Ξ(z,y;ζ) as the dissolvent kernel, analogous to the higher order deformation
constraint (10) in the ambiance of homotopy analysis studied in section 2.
Thus, the solution to (18) has the form
(29) W(x,y)=K(x+y) ζσ(x,y), y >x,
− −
and, on account of that, from (21) we obtain
∞
1 1
(30) W(x,y)=qΓ , p(x+y)2 qζ dzΞ(z,y;ζ)Γ , p(x+z)2 ,
− 2 − 2
(cid:20) (cid:21) Zx (cid:20) (cid:21)
PSEUDO SPECTRAL TRANSFORM FOR SCHRO¨DINGER–POISSON EQUATIONS 7
whilst the solution to equations (3) is computed through the formulae:
w(x)=2W(x,x),
∞
(31) w(x)= dyu(y),
Zx
u(x)=w (x).
x
In our particular case, we claim for a real valued, regular function u(x) that
vanishes asymptotically, exponentially,
(32) lim [u(x)exp( 2δ(±)x)]=0, 0<δ(±).
x→±∞ ±
Inasmuch as this scenario holds, the reflection coefficient R(k) is meromorphic
in the Bargmann strip
min(δ(−),δ(+))<Imk <δ(+),
−
where there is a bijection between the poles of R(k) and the discrete eigenvalues
k = ip [37, pp. 68–79]. For all these poles and all the discrete eigenvalues, the
α α
correspondence
(33) lim [(k ip )R(k)]=iq ,
α α
k→ipα −
is prevailed.
On the other hand, as we have settled in (20), the reflection coefficient R(k) is
demanded to be zero for all k, thus, the last relationship is trivial and, whether or
not our solutions are normalizable, it constitutes a relaxed constraint.
3.2. Sketching the solutions. In general, the dissolvent kernel Ξ(z,y;ζ) can be
built as the quotient of two functions P(z,y;µ) and Q(µ) prescribed by the series
expansion:
∞
( ζ)µ
P(z,y;ζ)= − Λ (z,y),
µ
µ!
µ=0
(34) X
∞
( ζ)µ
Q(ζ)= − λ ,
µ
µ!
µ=0
X
with the proviso of initial data Λ (z,y)=K(z+y) and λ =1. As for the coming
0 0
terms µ 1, the function Λ (z,y) is computed by means of the recursive relation:
µ
≥
∞
(35) Λµ(x,z)=λµK(z+y) ζ dsKµ−1(z+s)Λµ−1(s,y),
−
Zx
with
(36) λµ = dzΛµ−1(z,z),
Zl
recalling that l ReX.
∈
8 J.FUENTES,P.GALAVIZ,ANDT.MATOS
After the algorithm (35)–(36) has been iterated one time, fixing ζ = 1 without
loss of generality, the integral in (30) is explicitly calculated as
∞
1
dzΞ(z,y;1)Γ , p(x+z)2 =
2
Zx (cid:20) (cid:21)
1 q 1
q(x+y)Γ2 , p(x+y)2 + Γ , p(x+y)2 exp p2(x+y)2
− 2 p√π 2 −
(cid:20) (cid:21) (cid:20) (cid:21)
∞
2qp 1 q
+ dz exp p2(x+z)2 (z+y)Γ , p(z+y)2 exp p2(z+y)2 ,
√π − 2 − p√π −
Zx (cid:26) (cid:20) (cid:21) (cid:27)
this integral must converge to zero in the limit µ of iterations since σ(x,y)
→ ∞
satisfiesthe Ho¨lder conditionforsome positiverealconstantδ whenbecomes char-
acterized as from (19). Then, as it decays monotonically with each iteration, the
samereasonsstatedin3.1areconcedable. Asaresultwedropoutthesecondterm
from (30) and, therefore, the solution to (21) is simply
1
(37) W(x,y)=qΓ , p(x+y)2 .
− 2
(cid:20) (cid:21)
Now,itisadmitedlysuggestedthatbythe directapplicationofformulae(31)to
(37), the solution to the system (3) has the general form
(38) u(x)=2qE(x,x ) exp[ pc (x x )2]+c ,
0 1 0 2
− ±
where E(x,x ) is a real valued, continuous function to be determined (as we will
0
see in section 4), x is a point in the neighbourhood of x and c ,c are integration
0 1 2
constants.
Furthermore, the function E(x,x ) must suffice (by itself) equation (14) and,
0
whenever it vanishes faster than (32), namely
lim [E(x,x )exp(δx)]=0, δ =min(δ(−),δ(+)),
0
x→∞
therelationship(33)holds;consequentlyu(x)tobenormalizableisnotanecessary
condition and the factor q is not relevant in our quest for exact solutions.
4. Detachable solutions
Let us consider the solution (38) and let us assume that E = E (x,x ) and
1 1 0
E = E (x′,x′) satisfy by themselves the nonlinear system (3). We can look for
2 2 0
separablesolutionsbystartingwiththe stationarySchr¨odingerequation(14),from
which the product E E has to be solvable, i.e.
1 2
(39) −∂x2−∂x2′ +zφ12(|E1E2|2) E1E2 =ωE1E2,
such that both φ(cid:2)= φ = φ and φ = φ (cid:3)+ φ = φ + φ must be fulfilled.
12 21 1 2 2 1
The general solutions to the latter functional equation are φ = aln E 2 and
− | |
φ = aln E 2, for some real constant a and ν = 1,2. More than that, it is
ν ν
− | |
requiredfrom(38)thatE satisfiessimultaneouslythePoissonequation,thatmeans
(40) ( aln E 2) = E 2.
xx
− | | | |
In spiritofsection2, assumethat E is a continuousrealfunction which behaves
according to the law (7), hence (40) has a family of parametric solutions E
→
PSEUDO SPECTRAL TRANSFORM FOR SCHRO¨DINGER–POISSON EQUATIONS 9
E(;ξ ,ξ ), namely,
1 2
·
(41)
exp(pξ )
1 , ξ R,
2ap2+exp[2p(x x +ξ )] 1 ∈
E(x,x ;ξ ,ξ )=4ap2exp[p(x x )]  ± 0 1
0 1 2 0
± ×1+2ap2exepx[p2(pp(ξx2) x +ξ )], ξ2 ∈R,
0 2
±
withξ1,ξ2 therealparameterswhichcharacterizethesolutions. Noticethat,inthe
veryparticularcaseξ =(2p)−1ln(2ap2)andξ = (2p)−1ln(2ap2),theexpression
1 2
−
in (41) is reduced to
p (2a)sech[p(x x )], for ξ ,
0 1
(42) E(x,x ;ξ ,ξ )= ±
0 1 2
(pp(2a)sech[p(x x )], for ξ .
0 2
±
Finally, with the aim of (38) anpd (41), and recalling the form of our initial
ansatz ψ(x,t) ψ(x,t;ξ ,ξ ) = u(x;ξ ,ξ )exp( iωt), we write down the general
1 2 1 2
→ ±
stationary family of solutions to the nonlinear system (3) as
(43) ψ(x,t;ξ ,ξ )=2qE(x,x ;ξ ,ξ ) exp[ pc (x x )2]exp( iωt)+c ,
1 2 0 1 2 1 0 2
− ± ±
whichareSchr¨odinger–Poisson,stationary,single,parametricsolitonssubjectto
the conservative dispersion relation:
p2+12ap4 4a2p6+2(1+2ap2)2 p abln 8ap2
− − − 1+2ap2
(44) ω = ,
(1+2ap2)2 h (cid:16) (cid:17)i
where we have neglected ξ ,ξ and x for a tight writing.
1 2 0
The wave function in (43) corresponds to a quiescent soliton centred at x , os-
0
cillating with speed ω and decaying according to the eigenvalue p associated to a
single bounded state.
IncaseA=1in(20),wewouldobtaintheequivalentmultiple–stationarysoliton
6
solutionsonceappliedtheproceduredescribedabove. Thisisnotaneasytask: the
more bounded states, the more demanding to solve a system of A GLM equations
in a closed form. For this purpose, a numerical approach might be an alternative
to achieve the results.
Asafinalremark,notethatthenumberofdiscreteeigenvaluesAoftheSchr¨odinger
operator H in (14) is computed through a function J = J(x) defined by the non-
linear equation
(45) cucos2J c−1 sin2J +J =0,
x
−
subject to the boundary condition
(46) J( )=0,
−∞
c standing for any positive constant, then
(47) A= J( )/π .
⌊ ∞ ⌋
Certainly,ωisdiscretizedsinceaandbareconstantsandpisadiscreteeigenvalue
of H, that is, because we have considered a single soliton (A=1) there is a single
value for ω. In consequence, for these particular case, (38) must satisfy (45) such
that1= J( )/π ,just asitis. Properly,eachmultiple solitonsolutionto(1)has
⌊ ∞ ⌋
10 J.FUENTES,P.GALAVIZ,ANDT.MATOS
to fulfill (45) so the number of nonlinear superpositions (or Bianchi permutations)
coincides with (47).
5. Discussion
Wepresentedexact,stationary,parametricsolutionstotheSchr¨odinger–Poisson
nonlinear system of partial differential equations. In the first part of our study, we
showed how the homotophy analysis method suffices to integrate the system (1).
However, we found that to compute the integrals in (13) is not straightforward,
but instead, a limitation carried with this programme. As obvious, this approach
might provide first numerical computations.
In the second part of our study, we found a family of exact solutions to the
Schr¨odinger–Poissonsystem through the inverse scattering method. The inherent
relation between these two techniques has been tangentially depicted.
In both cases, we addressed our discussion to the theory of integral equations,
eithernonlinearandlinear. Tosolvetheseequationsisnotaneasytask,nonetheless,
the techniques come up with a deep understanding about the very nature of the
problem in itself. At this point, we have not been able to seek for exact solutions
by any algebraic method. Also, without counting the inverse scattering method,
the mainstream techniques described in soliton theory are insufficient to solve our
problem in a closed form.
Oursolutionsjustcoverthe stationaryconfigurationofthe Schr¨odinger–Poisson
system and are related to a single eigenvalue of the Schr¨odinger operator. To seek
forclosedsolutionsoutofequilibriumisnotagoalwehaveachievedyet. Quitethe
same for multiple Bianchi permutations, i.e. more than one eigenvalue (bounded
state) of the Schr¨odinger operator. In that case, we should find something at the
samelevelofaB¨acklund(orDarboux)transformationinordertoinduceanonlinear
superposition of bounded states.
Acknowledgments
The authors are glad to thank the valuable remarks and discussions by their
colleagues at the Department of Physics at CINVESTAV, Mexico City.
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