Stabilization of high-order solutions of the cubic Nonlinear Schrodinger Equation

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by  Adrian Alexandrescu

Stabilization of high-order solutions of the cubic Nonlinear Schr¨odinger Equation
Adrian Alexandrescu,∗ Gaspar D. Montesinos, and V´ıctor M. P´erez-Garc´ıa†
Departamento de Matem´aticas, E. T. S. Ingenieros Industriales and
Instituto de Matem´atica Aplicada a la Ciencia y la Ingenier´ıa (IMACI)
Avda. Camilo Jos´e Cela, 3, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain
(Dated: February 5, 2008)
Inthispaperweconsiderthestabilizationofnon-fundamentalunstablestationarysolutionsofthe
cubic nonlinear Schr¨odinger equation. Specifically we study the stabilization of radially symmetric
solutionswithnodesandasymmetriccomplexstationarysolutions. Forthefirstoneswefindpartial
stabilization similar tothatrecentlyfoundforvortexsolutions whileforthelateronesstabilization
does not seem possible.
7 PACSnumbers: 42.65.Tg,05.45.Yv,03.75.Lm
0
0
2 I. INTRODUCTION a pulsating Townes soliton after some rearrangement of

n initial data [15, 16] that is able to propagate without es-
a NonlinearSchr¨odingerEquations(NLS) areoneofthe sentialdistortionsforverylargedistances. Somerigorous
J most important models of mathematical physics arising results concerning early time collapse (i.e. the situation
4 in a greatarrayofcontexts [1, 2] as for example in semi- inwhichcollapsecannotbeavoided)wherefoundin[17].

conductor electronics [3], optics in nonlinear media [4], It has been only recently that the theoretical concept
]
S photonics [5], plasmas [6], fundamentation of quantum of stabilized solitons has been demonstrated in the labo-
P mechanics [7], dynamics of accelerators [8], mean-field ratory in Optical experiments [18, 19].
. theoryofBose-Einsteincondensates[9]orinbiomolecule Stabilized solitons have also been studied in three-
n
i dynamics [10], to cite a few examples. dimensional scenarios [15, 20, 21]. Also, they have been
nl It is well known that in multidimensional scenarios considered in vector media leading to the so-called sta-
[ theremayappearconcentrationphenomenaandcollapse bilized vector solitons [22, 23]. Finally, the possibility of
depending on the initial configuration [1]. Let us write existence of stabilized solitons with different nonlineari-
1
the model equation in the form ties and dimensionalities has been studied in Ref. [24].
v
The idea of stabilized solitons and nonlinearity man-
9
0 iu =−1∆u+g(z)|u|2u, (1) agement has also inspired some related mathematical
z
0 2 research, with a focus on the averaged and collapse-
1 preventing properties [25, 26, 27]. However, a rigorous
0 on R2 with ∆ = ∂2/∂x2 + ∂2/∂y2 and initial data theoretical mathematical description of the stabilization
7 u0(x,y). It is well-known that, when g(z) = g0 < 0, process is still missing [28].
0 the solutions of Eq. (1) with L2 R2 norm
The reviews [29, 30] offer a panoramic vision of the
/
n (cid:0) (cid:1) field of stabilized solitons.
li N(u)≡kuk22 = |u|2, (2) In this paper we want to complement the present
n ZR2 knowledge on stabilized solitons with a numerical study
:
v such that g0kuk2 is larger than a critical value N0 may of the stabilization of more complex stationary solu-
i tions of the nonlinear Schr¨odinger equation. Although
X undergo collapse (i.e. catastrophic formation of very
the Townes soliton can be stabilized, vortices have been
sharp gradients and concentration phenomena for u).
r
a However, when g0kuk2 < N0 there cannot be collapse found more difficult to stabilize [31] since the periodic
modulation of the non-linear coefficient alone is not
[1].
enough to achieve stabilization.
In the context of the analysis of the propagation of
Theaimofthisworkistoinvestigatethe possibilityof
Kerrbeams inlayeredopticalmediaitwasfirstexplored
stabilizing different types of stationary solutions of the
the possibility that making the nonlinearcoefficient g(z)
NLS equation. Specifically, we will study the stabiliza-
tooscillatewiththeindependentvariablebetweenvalues
tionofhigherorderradiallysymmetricsolutionsinscalar
corresponding to the collapsing and expansion regimes
andvectormediatocheckifthesimplerstructureofthese
could lead to collapse supresion [11].
solutions with respect to vortices allows them to be sta-
This idea was later explored [12] in the same context
bilized. Wewillalsostudythestabilizationofmorecom-
and exported to the field of Bose-Einstein condensation
plex asymmetricsolutionsofthe NLSE describedin Ref.
[13, 14, 15]. The stabilized structure was identified as
[32].
The paper is organized as follows: in Sec. II we study
the stabilizationofhigher-orderradiallysymmetric solu-
∗Electronicaddress: [email protected] tionsinscalarlayeredmediaandfindthatitisnotposible
†Electronicaddress: [email protected] to achieve stabilization of these structures in this simple
2
way. In Sec. III we show the stabilizationof these struc-
tures by the addition of a second stabilizing component,
4
i.e. in the vector case. In Sec. IV we consider the pos-
sibility of stabilizing higher order asymmetric solutions.
3
Finally, in Sec. V we summarize our conclusions.
2
II. STABILIZATION OF HIGHER-ORDER
RADIALLY SYMMETRIC SOLUTIONS IN 1
SCALAR LAYERED MEDIA
0
First we start by analyzing the simple case of scalar
fieldswhosepropagationisgovernedbytheNLSequation −1
(1), with g(z) = g0 constant. In this case, the equation
for stationary solutions u(z,x,y)=φ(x,y)eiλz, is −2
0 2 4 6 8 10 12
r
1
2
−λφ=− ∆φ+g0|φ| φ. (3)
2 FIG. 1: Plot of R (solid line) and R (dashed line), nor-
1 2
malized to unity, for g = −0.5,λ = 0.5. For these solutions
0
It is well known [1], that Eq. (3) has an infinite num-
N ≃77.17 and N ≃195.84, respectively.
1 2
ber of solutions with radial symmetry each having a fi-
nite number of nodes. We will denote these solutions by
R0,R1,R2,... labelling them by their number of nodes.
Their norms satisfy N0 < N1 < N2.... R0 is the ground
state, also called the Townes soliton, which plays an im-
) 1
portantroleinthetheorysinceN0setsacriticalvaluefor (z
the norm below which collapse cannot exist. Due to the N0.5
scaling symmetry of the nonlinear Schr¨odingerequation,
these solutions exist for any value of λ. 0
0 5 10 15
Thefirsttwohigherorderstationarysolutionswithra-
dialsymmetry,i.e. R1,andR2calculatedbyusingastan- 30
dard shooting method, are depicted in Fig. 1. All of our
radially symmetric solutions to be presented in this pa- )20
z
per have been computed by this method and then inter- (
A
polated onto a two-dimensional rectangular grid. After 10
this injection,wehaveusedaNewtonrelaxationmethod
0
in order to increase the accuracy of the computed sta- 0 5 10 15
z
tionary solutions.
We havecheckedthe “stationarity”ofournumerically
found solutions by propagating them through an homo- FIG. 2: Propagation of the norm given by Eq. (2) and max-
geneous nonlinear medium subject to the perturbation imum amplitude A(z) = max(x,y)|u(x,y,z)| for the solution
of Eq. (1) with initial data u(x,y,0) = R (r) through an
coming from the numerical errors (both on the initial 1
homogeneous nonlinear media, i.e. g(z)=−0.5.
dataandduetotheroundofferrorduringtheevolution).
All our numericalsimulations to be presentedinthis pa-
perhavebeendoneusingasecondorderintimeandspec-
tral in space split-step Fourier algorithm with absorbing to the first radially excited solution by setting g(z) =
boundaryconditionstogetridofthe outgoingradiation. g0+g1cos(Ωz), with g0 =−0.5, g1 =−1.5 and Ω=100
In Fig. 2 we present the propagation of the norm and (theparametersweresetaccordingtocriteriaestablished
amplitude taking u(x,y,0) = R1(r). As the stationary in Ref. [15, 16]) and taking u(x,y,0) = R1(r). This
solutionsofEq. (3)areallunstableinthe contextofEq. mechanism allows a stable propagation of the Townes
(1), one expects that sooner or later the instability will solitonoverlongdistancesofmorethan400units inadi-
set in. In the case of the Townes and vortex solitons the mensional units [15] but the same mechanism enhances
instability sets in by z ≃15 and z ≃3, respectively [31]. only slightly the stability of singly-charged vortex soli-
From the propagationof the norm and amplitude shown tons [31]. In principle our stationary solution has a sim-
in Fig. 2, we can guess that, in our case, a collapse de- pler structure thanthe vortexand one could expect bet-
stroyingthestructureofthisstationarystateoccursnear ter results, but we will see that this is not true.
z ≃11. Our results are shown in Fig. 3. We can see how the
We have tried to apply the stabilization technique inclusion of this periodic modulation leads to a shorter
based on the modulation of the nonlinear coefficient stablepropagationdistanceofthestructureandthatcol-
3
lapse occurs by z ≃6.5 describedby the followingset ofcoupledNLS equations:
∂u1 1 2 2
i =− ∆u1+g(z) a11|u1| +a12|u2| u1, (4a)
∂z 2
(cid:0) (cid:1)
∂u2 1 2 2
i =− ∆u2+g(z) a21|u1| +a22|u2| u2. (4b)
∂z 2
) 1 (cid:0) (cid:1)
z
(
N0.5 Defining α = N(u2)/N(u1) and u˜i = ui/Ni, i = 1,2
Eqs. (4) become
0
0 2 4 6 8 i∂∂u˜z1 =−21∆u˜1+g(z)N1 a11|u˜1|2+αa12|u˜2|2 u˜1(,5a)
10 (cid:0) (cid:1)
i∂∂u˜z2 =−21∆u˜2+g(z)N1 a21|u˜1|2+αa22|u˜2|2 u˜2(,5b)
(z) 5 (cid:0) (cid:1)
A For simplicity, we will discard the tilde in what follows
and take both components to be normalized.
0 The parameter α is a measure of the strength of the
0 2 4 6 8
z interaction between both components, which could be
accomplished experimentally by launching beams of dif-
ferent energies.
FIG.3: Propagation of thenormgiven byEq. (2)andmaxi-
mum amplitude A(z) = max |u(x,y,z)| for the solution
(x,y)
of Eq. (1) with initial data u(x,y,0) = R (r) under the
1
effect of a periodic modulation of the nonlinear coefficient B. Case α=0
g(z) = g + g cos(Ωz) where g = −0.5, g = −1.5 and
0 1 0 1
Ω=100.
Inthelimitα→0,correspondingtothecasewhenthe
normof one ofthe components is much smaller than the
We have tried to stabilize this configurationby choos- other, Eqs. (5) become
ing different parameters for the modulation with similar
results. Wethinkthatthereductionofthelifetimeofthe i∂u1 = −1∆u1+g(z)N1a11|u1|2u1, (6a)
unstable stationary structure can be understood by con- ∂z 2
sideringtheexchangeofenergy,i.e. energyflow,between ∂u2 1 2
i = − ∆u2+g(z)N1a21|u1| u2. (6b)
the substructures of the first excited radially symmetric ∂z (cid:20) 2 (cid:21)
stationary solution: the central peak and its surround-
ing ring. During the defocusing (g(z) > 0) stages both Eq. (6a) is a scalar NLS equation with a modulated
substructuresspreadandoverlapintheregionwherethe nonlinear coefficient and thus admits solutions in the
fieldwaszeropreviously. Then,inthefocusing(g(z)<0) form of stabilized Townes solitons. Eq. (6b) is a lin-
stagetheenergyfromtheoverlappingregionflowsmainly ear Schr¨odinger equation for u2 in a trapping potential
to the central peak. Therefore, with each focusing step generated by u1. First we look for stationary solutions
the central peak is supplied with more and more energy, of Eqs. (6) when g(z)=g0 of the form
and this process leads finally to appearance of collapse
(see the sharp amplitude peaks for z ≃7 in Fig. 3). u1(r,z) = φλ(r)exp(−iλz), (7)
u2(r,z) = ϕµ(r)exp(−iµz), (8)
i.e. solutions of the nonlinear eigenvalue problem
1
III. STABILIZATION OF HIGHER-ORDER 2
λφ = − ∆φ+g0N1a11|φ| φ, (9a)
RADIALLY SYMMETRIC SOLUTIONS IN 2
VECTOR LAYERED MEDIA 1
2
µϕ = − ∆ϕ+g0N1a21|φ| ϕ. (9b)
(cid:20) 2 (cid:21)
A. Motivation and model
Obviously, Eq. (6a) is equivalent to Eq. (3) and thus
we get all of its stationary solutions, for instance, the
Our next idea is to try to stabilize the excited radi- Townes soliton. We will look for stationary solutions of
allysymmetricstationarysolutionssbyusingastabilized Eq. (9b) with radial symmetry beyond the nodeless one
Townes soliton as a guide in which the higher order so- ϕ0. This implies that the effective potential
lution could be stabilized as it was done in Ref. [31] for
2
vortices. Hence, we shift our attention to vector systems V(r)=a21g0N1|φλ| , (10)
4
must support at least two bound states. The number cally varying nonlinear coefficient. During the propaga-
of radially symmetric bound states supported by a two- tion the potential a21g(z)|R0|2 oscillates with frequency
dimensional potential is bounded by the inequality [34]: Ω due to the periodic modulation of coefficient g(z),
hence, the potentialwill change its behaviorperiodically
1 dr dr′ rr′V(r)V(r′) ln r from attractive to repulsive. In general, for the linear
N2D,l=0 <1+2RR2 drrV(r) (cid:12) (cid:0)r′(cid:1)(cid:12). (11) situation described by Eq. (6b), one may achieve a non-
(cid:12) (cid:12)
dispersive propagation for small oscillations of g(z), i.e.
R
This inequality sets a necessary condition to have the without being necessary to change its sign. However, as
required bound state (i.e. that with one node) for fixed weareinterestedtoextendtheseresultstothefullnonlin-
g0 = −2π by fixing a lower bound for the value of a21 earequations,wemustconstructg(z)withanalternating
required to obtain N2D,l=0 ≥ 2. In Table I we show the signbecauseanalyticalresults[15,17]andcomputersim-
lowerboundsandthenumericalresultsforthenumberof ulations [15] have revealed that this is a requirement to
boundstatesobtainedforseveralvaluesofthecoefficient get stabilized solitons.
a21. We can see that for a12 =4 we can have two modes Using results from the quantum mechanical theory of
intheeffectivepotentialgeneratedbyR0,thesecondone fast perturbations [35] we have chosen a set of param-
being a radially symmetric function with one node. eters, g0 = −2π, g1 = 8π and Ω = 100 for which the
oscillatingbehaviorofg(z)shouldmaintaintheprofileof
a21 N2D,l=0 Bound states eigenvalues the excited solution in u2. Moreover, this set of param-
1 <1.39 λ0≃0.575 eters leads to an stabilized Townes soliton in component
2 <1.79 λ0≃1.925 u1 [16]. InFig. 5wecanseethatalthoughthemaximum
3 <2.19 λ0≃3.558 valueoftheamplitudeexhibitsoscillationsrelatedtothe
4 <2.58 λ ≃5.335, λ ≃0.195
0 1 periodic modulation of g(z), the norm remains constant
which indicates that no radiation is emitted while the
TABLE I: Upper bound set by Eq. (11) on the number potential a21g(z)|u1|2 is switched from attractive to re-
of bound states in two dimensions with radial symmetry
pulsive.
and without angular momentum of the potential V(r) =
a g N |R |2. λ and λ are the approximate eigenvalues Next we take u1 =R0,u2 =U1 as initial data for Eqs.
21 0 1 0 0 1
of the ground and first excited states of V(r), respectively, (6). This situation is described by the case of α ≃ 0,
found numerically. whichmeansthattheenergyinjectedintothemediumby
the second component u2 is much smaller in comparison
Wehavecomputedtheprofileoftheradiallysymmetric withthatofthefirstcomponentu1[31]. Thedynamicsof
the most important parameters of the system is plotted
solutionofEq. (6b)withonenode inϕ1 whichexistsfor
in Figure 6.
a21 = 4 by using a standard shooting method, which is
shown in Fig. 4 Wewanttoremarkthatthetimeevolutionof|u1|2 ex-
hibits high and low frequency oscillations corresponding
to the periodic modulation of g(z) and to the internal
dynamicsof Eq. 6a, respectively[15]. Therefore,the po-
0.6 tentialexperiencedbyu2includesthisoscillationpattern.
Asthelowfrequencyoscillationsdonotfulfilltherequire-
ments [31] derivedfromthe theory offast perturbations,
0.4 they leadtoenergyemissionfromthe secondcomponent
u2. The radiation emission is linked to the decreasing
value of the u2 norm: as the outgoing radiation hits the
0.2
boundary of the computational domain it is removed by
u1 an absorbing potential. Nevertheless, the first compo-
0 nent evolves unperturbed as a stabilized Townes soliton
[15].
The spatial intensity profile of the propagating beam
−0.2 u2 u2, see Fig. 7, has similar shape to the initial one (see
Fig. 1). However, the maximum of the intensity distri-
0 2 4 6 8 10
r butioninFig. 6(b) isaboutthreetimes smallerthanthe
corresponding value of the initial one due to the energy
loss during the propagation.
FIG. 4: [Color online] Spatial radial profile (normalized to
one and for λ=0.5) of the stationary solution of Eqs. (9) of
the form φ = R .(solid line) and the first excited state with
0
radial symmetry ϕ=ϕ (dashed line). C. Weak nonlinear coupling
1
We have propagated numerically the initial data u2 = Finally we turn on the nonlinear coupling in system
ϕ1 according to Eq. (6b) alone but now with a periodi- (5) by setting α = 0.1. The dynamics of the system pa-
5
) 1
z
(
N
0.5
0
0 50 100 150 200 250 300 350 400
1.5
) 1
z
(
A
0.5
0
0 50 100 150 200 250 300 350 400
z
FIG.6: [Color online] Propagation ofinitial dataoftheform
u = R ,u = U under Eq. (6) with α = 0, with g =
1 0 2 1 0
−2π, g = 8π and Ω = 100. Shown are the norm N(z) and
1
maximum amplitude A(z) for u (blue) and u (red).
1 2
FIG.5: [Coloronline]Evolutionofthe(a)normN(z)and(b)
maximum value of the amplitude A(z) described by the Eq.
(6b) for parameter values g =2π, g =8π and Ω=100. (c)
0 1
Detailedviewoftheamplitudeoscillationsinthepropagation
range z ∈[70,72].
FIG. 7: [Color online] Pseudocolor plot of the intensity
|u (x,y,z)|2forthesamesimulationasinFig. 6onthespatial
2
region [−10,10]×[−10,10] for (a) z =0 and (b) z=400.
rameters is presented in Figure 8. One can see that the
propagationofthe excitedstate is drasticallychangedin
IV. SOLUTIONS WITHOUT RADIAL
comparisonwith the previous analyzedcases. Under the
SYMMETRY
effectofthenonlinearcoupling,bothcomponentsu1 and
u2 reshape their transverse spatial profile by emitting
energy. After a short propagation distance, z = 25, the Equation (3) has more solutions beyond those hav-
excited solution losses its spatial shape, acquiring a pro- ing radial symmetry. For instance, Alfimov and cowork-
filewhichresemblestotheTownessolitonandwhichwill ers [32], using branching-off techniques from the theory
bepropagatealongwiththeTownessoliton. Wemaysay of dynamical systems constructed solutions having non-
that both initial profiles decay or readjust (with energy trivial discrete rotational symmetries. Two examples of
loss) their shape, eventually leading to the formation of thosesolutionsareshowninFig. 10. Inallofthemweob-
nodeless stabilized vector solitons [22]. Nevertheless, for serve that a large central peak is surrounded by smaller
smaller values of α, e.g. α = 0.05, we recover the be- ones having the prescribed discrete symmetry plus an
haviourdescribedinthe previoussubsection, with shape outer ring.
preservation accompanied by energy emission. Sincethesesolutionsarealsounstable,theirfreeprop-
agation leads to collapse as shown in Figs. 11 and 12.
We think that the mechanism described at the end Since their norms are much above the critical one both
of Section II is responsable for the decay of component solutionssufferafastinstabilitytocollapse,theconstant
u2 when the nonlinear coupling is large enough. The behavior of the norm indicating the fact that outgoing
changes taking place in the component u2 are then cou- radiation waves have no time to escape from this sys-
pled back to component u1, which leads to energy emis- tem before the instability develops. The sensitivity of
sion, as seen by the decreasing norm of u1 in Figure 8. those configurations to collapse manifests itself in the
6
) 1
z
(
N
0.5
0
0 50 100 150 200 250 300 350 400
2
)
z
(1 FIG. 10: [Color online] Pseudocolor plots of asymmetric so-
A
lutions of Eq. (3) (a) C invariant solution with norm N ≃
3 3
627.68 and(b)C invariantsolution with normN ≃723.91.
4 4
0 The spatial region shown is (x,y)∈[−10,10]×[−10,10].
0 50 100 150 200 250 300 350 400
z
FIG.8: [Color online]Propagation of initialdataoftheform
u = R ,u = U under Eq. (6) with α = 0.1, with g =
1 0 2 1 0
−2π, g = 8π and Ω = 100. Shownare the norm N(z) and ) 1
1 z
maximum amplitude A(z) for u (blue) and u (red). (
1 2 N
0.5
0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
60
)40
z
(
A
20
0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
z
FIG.11: [Coloronline]Propagationthroughanhomogeneous
nonlinearmediumofthenormN(z)andmaximumamplitude
FIG.9: Pseudocolor plotoftheintensity|u (x,y,z)|2 forthe
2 A(z) for thestationary solution depicted in Fig. 10(a).
same simulation as in Fig. 8 on the spatial region for (a)
z =0 on the spatial region [−4,4]×[−4,4] and (b) z =400.
on thespatial region [−2,2]×[−2,2].
V. CONCLUSIONS
In this paper we have complemented previous knowl-
edge on stabilized solitons of the Nonlinear Schr¨odinger
Equation by studying numerically the possibility of sta-
fact that the instability distances are two ordersof mag- bilizing excited stationary solutions with radial symme-
nitudesmallerthanthoseofsimplersolutionssuchasthe try in both (i) scalar and (ii) vectorial layered media.
Townes, vortex or first radially excited solitons. The scalar system is unable to stabilize radially excited
states due to the internal dynamics, i.e. energy flow, of
We have tried to use the modulation of the nonlinear this state which leads to shorter stable propagation dis-
coefficient to stabilize the stationary solutions presented tancesofthebeamandcollapsewhencomparedwiththe
in Fig. 10. However, the time evolution of the beam propagationofthesamebeamthroughhomogeneousme-
parameters does not change appreciably and in partic- dia. In the case of vector layered media we have shown
ular, the emergence of collapse cannot be delayed. We how weakly coupled beams when one of the components
have tried unsucessfully different sets of parameters to is chosen to be the Townes soliton and the other an un-
achieve stabilization. Intuitively, it seems difficult to be stable radially excited solution can be stabilized. In this
able to stabilize those complicated structures because of situation it is found that the radially excited state ra-
thecoexistenceofdifferenttypesofsubstructures(peaks, diates continuously energy while preserving an intensity
rings, etc) and the fact that their norm ar many times shape similar to the initial profile.
the critical one (see the caption of Fig. 10). In both scalar and vectorial layered media, the stabi-
7
cult to achieve. This fact is due to the ireversibility of
theinternalenergyflowsbetweenthewavesubstructures,
e.g. rings, peaks, during propagation. Additional stabi-
) 1
z lizing mechanisms, like spatially inhomogeneous nonlin-
(
N earitiesdependingonthetransversevariables,mighthelp
0.5
in stabilizing states with nontrivial structures as it hap-
0 pens inthe caseofnon-oscillatingstructures [36,37, 38].
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Finally, we have tried unsuccesfully to stabilize com-
60 plexasymmetricsolutionsofEq. (1). Thecomplexstruc-
ture of these solutions and the fact that their power is
)40 many times the critical power makes stabilization prob-
z
( ably impossible.
A
20
0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
z
Acknowledgments
FIG.12: [Coloronline]Propagationthroughanhomogeneous
nonlinearmediumofthenormN(z)andmaximumamplitude We want to thank G. Alfimov for discussions. This
A(z) for the stationary solution depicted in Fig. 10(b). work has been partially supported by grants FIS2006-
04190 (Ministerio de Educaci´on y Ciencia, Spain) and
PAI-05-001 (Consejer´ıa de Educaci´on y Ciencia de la
lization of solutions bearing nontrivial structure is diffi- Junta de Comunidades de Castilla-La Mancha, Spain),
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