Stationary States of NLS on Star Graphs

Abstract

We consider a generalized nonlinear Schr\"odinger equation (NLS) with a power
nonlinearity |\psi|^2\mu\psi, of focusing type, describing propagation on the
ramified structure given by N edges connected at a vertex (a star graph). To
model the interaction at the junction, it is there imposed a boundary condition
analogous to the \delta potential of strength \alpha on the line, including as
a special case (\alpha=0) the free propagation. We show that nonlinear
stationary states describing solitons sitting at the vertex exist both for
attractive (\alpha<0, representing a potential well) and repulsive (\alpha>0, a
potential barrier) interaction. In the case of sufficiently strong attractive
interaction at the vertex and power nonlinearity \mu<2, including the standard
cubic case, we characterize the ground state as minimizer of a constrained
action and we discuss its orbital stability. Finally we show that in the free
case, for even N only, the stationary states can be used to construct traveling
waves on the graph.

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Stationary States of NLS on Star Graphs
Riccardo Adami,1, ∗Claudio Cacciapuoti,2, †Domenico Finco,3, ‡and Diego Noja1, §
1Dipartimento di Matematica e Applicazioni, Universit` a di Milano Bicocca, via R. Cozzi, 53, 20125 Milano, Italy
2Hausdorff Center for Mathematics, Institut f¨ ur Angewandte Mathematik, Endenicher Allee, 60, 53115 Bonn, Germany
3Facolt` a di Ingegneria, UTIU UniNettuno, Corso V. Emanuele II, 39 00186, Roma, Italy
We consider a nonlinear Schr¨ odinger equation (NLS) with a power focusing nonlinearity on a star
graph with N edges and a vertex with boundary conditions of δ type, including the special case of
a Kirchhoff vertex. We show that nonlinear stationary states exist both for attractive and repulsive
δ interaction and we give explicitly their expression. In the case of attractive interaction at the
vertex and subcritical nonlinearity we characterize the ground state as suitably constrained action
minimum and we rigorously discuss its orbital stability. Finally we show that in the Kirchhoff case,
for even N only, the stationary states can be used to construct traveling waves on the graph.
Keywords: quantum graphs, non-linear Schr¨ odinger equation, solitary waves.
1.Introduction.
metric graphs is a well developed subject [4], but despite
the potential applications in nonlinear optics and Bose
Einstein condensates, the nonlinear Schr¨ odinger dynam-
ics on ramified structures is essentially unexplored up to
now (but see [6]). In the previous paper [1] the scattering
behavior of fast solitary waves from a free or interact-
ing vertex of a star graph was analyzed. Here we prove
that a star graph which sustains a nonlinear focusing
Schr¨ odinger dynamics, admits bound states (sometimes
called pinned normal modes or impurity solitons when
considered on the line or in optical lattices) in the pres-
ence of a δ interaction at the vertex. Moreover, bound
states exist for both attractive and repulsive interaction
at the vertex and in contrast with the δ impurity on the
line, there could exist excited states beside the ground
state. To fix notation let us consider a star graph G
made of N half lines with common origin. The associ-
ated Hilbert space is L2(G) =?N
tions in L2(R+), namely
The study of Schr¨ odinger equation on
j=1L2(R+). Functions
in L2(G) will be represented as column vectors of func-
Ψ =



ψ1
...
ψN



(1)
Moreover, for i = 1,2 we introduce the spaces Hi(G) =
?N
j=1Hi(R+) equipped with the norm
?Ψ?2
Hi(G)=
N
?
k=1
?ψk?2
Hi(R+), (2)
where Hi(R+) is the usual Sobolev space of the square in-
tegrable functions with (distributional) derivative square
integrable up to the order i. To define the dynamics on
the graph let us consider the Hamiltonian Hαon the do-
main
D(Hα) :=
N
?
with action given by
?
Ψ ∈ H2(G) s.t. ψ1(0) = ... = ψN(0),
?
k=1
ψ?
k= αψ1(0) ,α ∈ R
(3)
(HαΨ)i= −ψ??
i
(4)
It is well known that the operator Hαis a self-adjoint op-
erator on L2(G), see e.g. [8]. Moreover, Hαgeneralizes
the ordinary Dirac’s delta interaction with strength pa-
rameter α on the line (formally given by −d2
e.g. [3]). The case α = 0 in (3) plays a distinguished role
and defines what is usually given the name of Kirchhoff
boundary condition in the literature. To pose the NLS
on the graph we define the product of functions compo-
nentwise, i.e.

ψ3φ3
dx2+αδ0; see,
ΨΦ ≡

ψ1φ1
ψ2φ2

, so that
|Ψ|2µΨ ≡


|ψ1|2µψ1
|ψ2|2µψ2
|ψ3|2µψ3

.
(5)
It is well defined the NLS equation on the graph,
id
dtΨt = HαΨt− |Ψt|2µΨt
(6)
where µ > 0. This amounts to a system of scalar NLS
equations on the halfline, coupled through the boundary
condition at the origin included in the domain (3) which
physically represents the interaction at the vertex of the
graph. In [1] for the cubic NLS it has been shown the
well posedness of the dynamics, i.e. local existence and
uniqueness for initial data in the energy domain.
analogous result holds true for every µ > 0 and strong
solutions in the operator domain, and the well posedness
is global (i.e. solution exists for all times) for power non-
linearities less than critical, i.e. µ < 2. Moreover, as in
the standard NLS on the line, mass M(Ψ) = ||Ψ||2
An
2and
arXiv:1104.3839v1 [math-ph] 19 Apr 2011

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energy, given here by
E[Ψ] =1
2
N
?
i=1
??+∞
0
|ψ?
i|2−|ψi|2µ+2
µ + 1
dx
?
+α
2|ψ1(0)|2
(7)
are conserved by the flow.
2.Stationary states.
ary solutions of the previous equation, by which we mean
solutions of (6) of the form
We are interested in the station-
Ψt= eiωtΨω. (8)
The parameter ω is interpreted as propagation constant
in nonlinear optics and as chemical potential in BEC. It
is immediate that the amplitude Ψωhas to satisfy
HαΨω− |Ψω|2µΨω= −ωΨω
Locally Hαacts as the Laplacian and so on every edge
we must seek L2(R+) solutions of the equation
ω > 0.(9)
− φ??− |φ|2µφ = −ωφ ω > 0. (10)
The most general one is given by the function
φ(σ,a;x) = σ [(µ + 1)ω]
1
2µsech
1
µ(µ√ω(x − a)) (11)
where |σ| = 1 and a ∈ R. Therefore the components
(Ψω)iof a stationary state Ψωmust satisfy
(Ψω)i(x) = φ(σi,ai;x).(12)
In order to have a solution of (9) we have to determine σi
and aisuch that Ψω∈ D(Hα). The continuity condition
in (3) immediately implies σ1= ... = σN and ai= εia
with εi= ±1 and a > 0. Due to the bell shape of the
function φ, we shall say that in the i-th edge: there is
a bump if ai > 0, that is, if εi = +1; there is a tail if
ai< 0, that is, if εi= −1. Now we determine εiand a.
The second boundary condition in (3) implies
tanh(µ√ωa)
N
?
i=1
εi=
α
√ω.
(13)
Equation (13) gives as a first constraint that
must have the same sign of α. That is for α > 0 the sta-
tionary state must have strictly more bumps than tails
while for α < 0 we must have more tails than bumps. For
every such a configuration condition (13) fixes uniquely a.
We choose to index the stationary states by the number
j of bumps. Let Ψj
ωbe given by
?
φ(−aj;x)
?N
i=1εi
?Ψj
ω
?
i(x) =
φ(aj;x)i = 1,...j
i = j + 1,...N
(14)
aj=
1
µ√ωarctanh
?
α
(2j − N)√ω
?
. (15)
FIG. 1. Nonlinear stationary states: α < 0,N = 3,j = 0,1
FIG. 2. Nonlinear stationary states: α > 0,N = 3,j = 2,3
To summarize, solutions of (9) for α > 0 are given by
Ψj
with j = 0,...,[(N − 1)/2].
number of bumps j there are
satisfying (14) and (15) differing for a permutation of
edges, a degeneracy reflecting the discrete symmetry of
the boundary conditions at the vertex. As a last remark,
note moreover that from (15) follows the lower bound on
the nonlinear stationary states frequencies
ωwith j = [N/2 + 1],...,N and for α < 0 by Ψj
ω
Notice that for a given
?N
j
?
stationary states Ψj
ω
α2
N2< ω . (16)
3.Ground state and variational properties.
want to identify the ground state of the system
among the domain E of finite energy states, E
?Ψ ∈ H1(G) s.t. ψ1(0) = ··· = ψN(0)?. The stationary
Now we
=

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states of the NLS on graphs just determined turn out to
be critical points of the so called action functional Sω
Sω[Ψ] = E[Ψ] +ω
2||Ψ||2
2
(17)
and so they can be interpreted as critical points of the
energy E[Ψ] constrained to have fixed mass. To deter-
mine possible existence and properties of minima of the
action in the space E, it proves useful to study a related
constrained problem. Let us introduce the so called Ne-
hari manifold (sometimes called the natural constraint,
because it obviously contains all the stationary states)
{Ψ ∈ E s.t. Iω[Ψ] ≡ ?S?
Ground states, if they exist, are solutions of the following
constrained minimization problem
ω[Ψ],Ψ? = 0} .
inf{Sω[Ψ] s.t. Ψ ∈ E,Iω[Ψ] = 0} ≡ d(ω) .
The analysis, well established in the case of regular ex-
ternal potential and on open sets in Rn, has been recently
carried out in the case of single δ interaction on the line,
see [2, 5]. For the present case, we have the following
Theorem 1 Let µ > 0, α < 0, ω >
variational problem (18) attains a solution coinciding (up
to phase) with the N tail state Ψ0
A sketch of the proof (details will be given elsewhere)
is as follows. The constrained functional Sωis bounded
from below. Let us consider a minimizing sequence Ψn;
this is bounded in the energy space and admits a subse-
quence, again denoted as Ψn, weakly convergent to some
Ψ∞. This solves the minimum problem if it is shown that
Ψ∞is non vanishing and Iω(Ψ∞) = 0. The first prop-
erty follows preliminarily considering the α = 0 prob-
lem; this can be reduced to the analogous problem on
the halfline adapting to the case of graphs a symmet-
ric rearrangement argument, and it turns out that the
solutionˆΨω is given by half a soliton φ on each edge.
Then the thesis Ψ∞ ?= 0 follows by contradiction ex-
ploiting weak lower semicontinuity of Sω and the easy
inequality d(ω) < Sω(ˆΨ0) . The second property follows
from Brezis-Lieb inequalities. Finally, the minimizer Ψ∞
turns out to be a solution in D(Hα) of the stationary
state equation S?
metric rearrangement one concludes that Ψ∞≡ Ψ0
this ends the proof.
A standard reasoning allows to conclude that the ground
state Ψ0
ωalso solves the alternative variational problem
(18)
α2
N2. Then the
ω.
ω(Ψ) = 0, i.e. (9), and again by sym-
ω, and
inf{E[Ψ] s.t. Ψ ∈ E,M[Ψ] = M[Ψ0
Concerning the energy ordering above the ground state,
let us consider the special but physically relevant case
of the cubic NLS. An explicit calculation shows that the
mass of the bound states Ψj
every α):
ω]} . (19)
ωis independent on j (for
||Ψj
ω||2
2= 2Nω
1
2+ 2α (20)
The energy spectrum at fixed mass is then
E[Ψj
ω] = −N
3ω
3
2−1
3
α3
(2j − N)2
(21)
Notice that the N bound states for both sign of α have
an energy increasing with the number of bumps. It is a
conjecture that an analogous ordering for the energy at
fixed mass holds true for every µ < 2.
4.Stability of ground state.
bility of the ground state. Due to the U(1) invariance of
(6) the stability has to be considered as orbital stability
[7, 10, 13]. As in the scalar case, the NLS on a graph
turns out to be a hamiltonian system on the real Hilbert
space of the couples of real and imaginary part of the
wavefunction. We decompose Ψ = u + iv. Note that the
real vectors u and v satisfy the same boundary conditions
as Ψ and the Schr¨ odinger equation is equivalent to the
canonical system
?u
where the Hamiltonian E is
??+∞
N
?
Now we come to the sta-
d
dt
v
?
= JdE[u,v] ,
J =
?0 I
−I 0
?
. (22)
1
2
N
?
k=1
0
(|u?
??∞
k|2+ |v?
k|2) dx
?
+α
2(|u1(0)|2+ |v1(0)|2)
?
−
1
2µ + 2
k=1
0
(|uk|2+ |vk|2)µ+1dx
≡ E[u,v] .
(23)
Linearization of the hamiltonian system (22) around the
stationary state is achieved by substituting
(Ψt)k= (Ψ0
ω,k+ ηk+ iρk)eiωt
(24)
and neglecting higher order terms than linear in (22).
The real vector functions η and ρ satisfy
?η
where L = diag(L−,L+) and
?
?
L− and L+ are matrix self adjoint operators acting on
the real vector functions η and ρ belonging to D(Hα).
Note that L+Ψ0
bital stability (and instability) for general hamiltonian
systems and in particular for NLS equations are provided
by the classical papers of Weinstein [12, 13] and Grillakis-
Shatah-Strauss [7]. They are based on the properties of
linearization L and on the so called Vakhitov-Kolokolov
criterion (see [11]). These results imply in particular that
d
dt
ρ
?
= JL
?η
ρ
?
.(25)
(L+)i,k=
−d2
−d2
dx2+ ω − |Ψ0
ω,k|2µ
?
δi,k
(L−)i,k=
dx2+ ω − (2µ + 1)|Ψ0
ω,k|2µ
?
δi,k.
(26)
ω= 0. Precise conditions to have or-

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solitary solutions of (22) are orbitally stable if i) spectral
conditions hold: i1) kerL+= {Ψ0
spectrum is positive; i2) n(L−) = 1 where the left hand
side is the number of negative eigenvalues. ii) Vakhitov-
Kolokolov condition
Applying these conditions one has the following
Theorem 2 Let µ ∈ [0,2], α < 0, ω >
ground state Ψ0
Concerning Vakhitov-Kolokolov condition one has
?
?
ω} and the rest of the
d
dω||Ψ0
ω||2
2> 0 holds.
α2
N2. Then the
ω is orbitally stable in H1(G) .
d
dω||Ψ0
ω||2
2= C(1
µ−1
2)
?1
?1
µ−1?
|α|
N√ω
(1 − t2)
1
µ−1dt
+
|α|
2N√ω
1 −|α|2
N2ω
(27)
with C = C(N,µ,ω) = N(µ+1)
is positive thanks to the lower bound on ω given in (16).
Concerning i1), Ψ0
is the (simple) ground state of L+. The quadratic form
of L+ can be rewritten in ground state representation.
An integration by parts gives in fact
1
µ
µ
ω
1
µ−3
2; the r.h.s of (27)
ωis positive, and we prove now that it
?L+Ψ,Ψ? =
N
?
N
?
k=1
?+∞
?
0
(Ψ0ω,k)2|d
dx(ψk
Ψ0
ω,k
)|2dx
+
k=1
ψk(0)ψ?
k(0) − |ψk(0)|2Ψ0?
ω,k(0)
Ψ0ω,k(0)
?
(28)
and the last term is vanishing due to the δ boundary
condition. So ?L+Ψ,Ψ? > 0 for every Ψ ∈ D(Hα) not
coinciding with Ψ0
ω, which is the only eigenvector with
eigenvalue 0. Concerning i2), note that the ground state
is a strict local minimum of Sωconstrained on the codi-
mension one natural manifold I−1
the tangent space to I−1
is a codimension 1 subspace of the energy space. Corre-
spondingly, L could have at most one negative eigenvalue.
In fact it has one, being ?L−Ψ0
the Weinstein and Grillakis-Shatah-Strauss theory, the
ground state Ψ0
ωis orbitally stable.
Note that from formula (27) it follows that for µ > 2 there
exists ω∗such that Ψ0
and is orbitally unstable for ω > ω∗.
5. Kirchhoff vertex and traveling waves.
sider the Kirchhoff case, α = 0. Equation (13) reads
now
ω(0). So L is positive on
ω(0) at the ground state, which
ω,Ψ0
ω? < 0. According to
ωis orbitally stable for ω ∈ (α2
N2,ω∗)
Let us con-
tanh(µ√ωa)
N
?
i=1
εi= 0 .(29)
If N is odd the only solution is a = 0, the stationary
stateˆΨωis unique and it is given by
?ˆΨω
?
i(x) = φ(0,x)i = 1,...,N.(30)
Notice that all the stationary states Ψj
wise and in energy toˆΨωfor α → 0. If N is even a family
of solutions is given by
?
φ(+a,x)i = N/2 + 1,...N
ωconverge point-
?ˆΨa
ω
?
i(x) =
φ(−a,x)i = 1,...N/2
a ∈ R
(31)
For this family a is a not fixed by the boundary conditions
and it is a free parameter. Since α = 0, no interaction
is present in the vertex and translational symmetry is
restored by edge pairing. In this case the graph can be
considered just as a set of N/2 copies of the real line,
and on each copy of the real line a unperturbed NLS
is given. The solutionsˆΨa
ωcan be interpreted as N/2
identical NLS solitary waves on each real line translated
by a quantity a. Therefore it is clear that mass and
energy of the states Ψa
0do not depend on a. We remark
finally that from this last family of stationary states one
can construct, through Galilean invariance and on each
copy of the line, traveling waves of the form
Ψtr(t) = ei(v
2x−v2
4t+θ)ˆΨa(t)
ω
a(t) = a + vt.(32)
Acknowledgments
G.N.F.M. (Gruppo Nazionale per la Fisica Matematica)
for financial support.
The authors would like to thank
∗riccardo.adami@unimib.it
†cacciapuoti@him.uni-bonn.de
‡d.finco@uninettunouniversity.net
§diego.noja@unimib.it
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