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UDC 517. 958
R. Adami1, D. Noja1
1Dipartimento di Matematica e Applicazioni, Universita di Milano Bicocca, Italy riccardo. adami@unimib. it, diego. noj a@unimib. it
PACS 05. 45. -a, 03. 65. -W, 05. 45. Yv
We illustrate some new results and comment on perspectives of a recent research line, focused on the stability of stationary states of nonlinear NLS with point interactions. We describe in detail the case of a & quot-S'-"- interaction, that provides a rich model endowed with a pitchfork bifurcation with symmetry breaking in the family of ground states. Finally, we give a direct proof of the stability of the ground states in the cases of a subcritical and critical (in the sense of the blow-up) nonlinearity power.
Keywords: nonlinear dynamics, quantum mechanics, solitons, symmetry breaking, pitchfork bifurcation.
1. A new range of application for point interactions
One of the most celebrated features of the point interactions (PI) lies in their capability of supplying exactly solvable models. For this reason, PI have been widely employed to construct toy models and for pedagogical purposes. Nonetheless, they prove useful also when used to model real physical systems: the more a physically relevant quantity (e.g. energy spectrum or time evolution) is explicitly known, the more information can be extracted. General theory and reference to physical applications with extended bibliography are in (see [10,11]). In particular, PI fit well the needs of modeling the so-called defects, namely, small inhomogeneities in a medium where a wave propagates, under the hypothesis that the details of the internal structure of the inhomogeneity are not relevant, so that its action can be modeled as concentrated at a point. More precisely, the smallness of the inhomogeneity is to be evaluated with respect to the typical wavelength of the incoming waves, or equivalently, in the case of a quantum system, to the width of the wave function. In this paper we address the analysis of effects of the interaction between nonlinearity and point defects in the behaviour of solutions of nonlinear Schrodinger (NLS) equation. We prefer not to enter in a description of the vast field of application of the NLS equation, from the theory of integrable systems and inverse scattering to the propagation of amplitude envelope of waves. We cite just two relevant applications of the NLS as an effective model for real physical systems: dynamics of Bose-Einstein condensates (BEC) and laser beam propagation in nonlinear (Kerr) media. In both cases it is physically meaningful to consider the propagation of NLS waves in the presence of defects. In particular, the recent spectacular development of both theoretical research and experimental technology involving BEC (see [45] and references therein, and [13,14,16]) provides point interactions with a wide range of applications.
As widely known, in current experiments the formation of a BEC is induced in bounded region of spaces, usually delimited by magnetic and/or optical traps. In such situations, the condensate lies in a one-particle quantum state, whose corresponding wavefunction is characterized
as the minimizer of the Gross-Pitaevskii functional. When the trap is removed, the wavefunction of the BEC spreads out according to the evolution prescribed by the cubic Schrodinger equation
idt^(t, x) = -dfy (t, x) + a^(t, x)2^(t, x), (1. 1)
where we denoted by ^ the wave function of the condensate, and the space variable x belongs to R, R2 or R3 according to the fact that we are modeling a cigar-shaped or a disc-shaped or a genuinely three-dimensional BEC. We recall that the nonlinearity carries the information that, even though its state is an actual one-particle state, the condensate consists of a large number of interacting particles (in the experimentally realized condensates, at least thousands) — the fact that the nonlinearity is cubic means that the dynamical effects of the two-particle interactions overwhelm the effects of many-body collisions. The strength of the nonlinear term, given by the constant a, is proportional to the scattering length of the two-body interaction between the particles. Here we do not summarize the progress in the derivation of (1. 1) as an effective equation for a many-body quantum system. The interested reader is referred to [20−22] for the three-dimensional problem, to [34] for the two-dimensional case, and to [1,2,12] for the case of cigar-shaped condensates. In the following we focus on this last case, in which, on one hand, the nonlinearity is milder, while, on the other hand, the family of point interactions is richer.
A natural question in this context is the following: what happens when a wave (i.e. a condensate) is sent against a defect? One would guess (and it has been shown for some models, see e.g. [19,29,32,44]) that the incoming wave splits into a reflected wave, a transmitted wave and a captured component. Similar results have been proven for propagation on graphs also (see [4]), in the case of a repulsive vertex, where no capture occurs. Indeed, it seems reasonable to conjecture that a capture can occur only if a nonlinear stationary state exists. Since equation (1. 1) is dispersive, the presence of a nonlinear stationary state (or more than one) must be related to the defect. This is the reason why such possible stationary states are called defect modes. Even though at this stage it is an unproven fact, it is plausible to link the persistence of a captured wave with some sort of stability (in a sense to be made precise) of the defect mode. For this reason the interest in determining the stability of the defect modes lies not only in the problem itself, but extends to models of reality too.
As a short review on results on stability and instability of defect modes in the presence of a power nonlinearity we recall results proved in [26,27,40], where the effects of a
i-like defect are analysed. The first cited work deals with an attractive defect, and shows that, for any frequency u above the proper frequency of the unique bound state of the delta potential, there is a unique defect mode that oscillates at frequency u. It turns out that the wavefunction of such a defect mode is nothing but the nonlinear deformation of the linear bound state. The stability (more exactly, the orbital stability, see Definition 2. 2) of such a mode depends on fi and u: if ^ ^ 2, then the defect mode is stable for any u- if? i& gt- 2, then it becomes unstable at high frequencies. References [26,40] extend the analysis to a repulsive delta-like defect. The situation becomes more involved in the case of a more singular defect, for instance, the so-called 5'- defect. The following sections are devoted to this case. For more details see also the comprehensive review [7] and the forthcoming paper [6].
The established theoretical framework for the study of stability is provided by Weinstein and Grillakis-Shatah-Strauss theory (see [30,31,49,50]) or, alternatively, by Lions concentration-compactness method (see [41,42] and [17] for a review). The occurrence of bifurcation in the ground state has been investigated in [33] and more recently in [28,35,36,43,46].
2. Results
2.1. The 6'- & quot-potential"-
The so-called 5'--defect, with strength, located (just to be definite) at zero, is defined imposing the boundary condition
¦0(0+) — ^(0-) = -^ (0+) = -^ (0-) (2. 1)
to the solutions to (1. 1) (see [9,23]). The parameter 7 is real- when positive, the defect is called attractive, otherwise repulsive. More formally, one defines a linear Hamiltonian operator H1 as the operator that acts as -d2 on the domain D (H1) made of functions in H2(R~) © H2(R+) satisfying (2. 1). Note that the only continuous elements of the domain of H1 have a vanishing derivative at the origin. The operator H1 is a self-adjoint operator with the following spectral features: singular continuous spectrum is empty, absolutely continuous spectrum is given by the positive halfline and point spectrum is empty in the repulsive case, and coincides with {-4/ry2} in the attractive case. In the last case the corresponding (non-normalized) eigenfunction is
& lt-f7(:x) = e (x)e t'-^'-,
where we denoted the sign function by e. Notice that is odd. The quadratic form F1 associated
to is ^^^^^^ ^^ ^^^nrM mM /^l • - Ul /TCP + /T Ul /TCP ^
and reads
to H1 is defined on the domain Q := Hl (R+) © Hl (E) (we stress that Q is independent of 7)
F, WO = WII2 — 7& quot-V (0+) — ^(0-)|2,
where we made the following abuse of notation
Iip'-(x)l2dx + lim / l^'-(x)l2dx,
that will be extensively repeated.
At variance with the delta potential, the Schrodinger operator with a 5'- interaction cannot be derived from a form sum, because the 5'- is not small with respect to the laplacian. Nevertheless it can be obtained as the norm-resolvent limit of the sum of three S potentials (see [18,24]) with a fine tuned rescaling, defined as follows
[H,+u]-1 = lim ?^0

7? e j e? e2 j 7
for any -v in the resolvent set of H, (see Figure 1).
Moreover, since any delta potential, in its turn, can be approximated by a strong limit of rescaled regular potentials, then it is possible to interpret a i-prime potential as the suitable limit of rescaled, well-behaved potentials. Let us remark that if ^ belongs to the operator domain of H, then the form associated to H, has the expression
= № 112 — ^'- (0+)I2,
which explains the questionable name of.
2.2. Combining nonlinearity and defect
Once constructed the operator H, the evolution in the presence of both a generic power nonlinearity and a defect is defined by
idt^(t, x) = H^(t, x) + al^(t, x) l2^(t, x). (2. 2)
Fig. 1. A regular approximation for an attractive i-prime potential centred at zero. To obtain an approximation for a repulsive i-prime, one must reverse the central well.
For such equation it is possible to prove global well-posedness if? i & lt- 2 (see [3]), local well-posedness if ^ ^ 2 (see [6]), and to provide examples of blow-up for this last case (see [8]). However, until the solution exists, L2-norm and energy
m = wn2 — - v (o-)i2 — ^imZXl
are conserved by time evolution.
Thanks to the existence of a conserved energy it is possible to introduce a notion of nonlinear ground state: intuitively, one would define it as a minimizer of the energy among the wavefunctions endowed with the same L2-norm, as this is the definition that naturally extends the more familiar notion of linear ground state.
As in the linear case, it is meaningful to search for stationary states of (2. 2), i.e. solutions of the form
^(x, t) = e1^ (x). (2. 3)
The amplitudes ^ are solutions of the stationary equation
H1 ^ + ^ - = 0. (2. 4)
This leads to the introduction of the so-called action functional
suvo = ?(VO + llHI2, (2. 5)
defined on the energy domain Q. It is immediate indeed that Euler-Lagrange equations for Su are given just by (2. 4). Note that the action (and the energy as well) is not bounded from below on Q. To overcome this problem, a ground state is usually defined as a minimizer of Su constrained on the Nehari manifold
1"№) = SL (W = ^ - + u^) = 0.
The above set is a codimension one manifold that obviously contains all stationary points of Su, and it tuns out that on it the action is bounded from below.
The relation between the constrained variational problem for? and Su is a byproduct of the Grillakis-Shatah-Strauss theory on stability of stationary states (see [30], [31]) applied to mini-mizers of Su: a minimizer of the action on the Nehari manifold is a local minimizer of the energy among the function with the same L2-norm || if and only if it is stable (in the sense of Definition (2. 2)).
The following preliminary result is obtained through variational techniques (we remove the subscript u from when not needed to avoid ambiguity):
Theorem 2.1. For any oj & gt- there exists at least one minimizer of Su among all functions on the Nehari manifold. Furthermore, the minimizer solves the stationary Schrodinger equation with defect:
H1 ^ + - = 0. (2. 6)
For oj ^ equation (2. 6) admits no solutions in D (ii7).
The line of the proof is standard, except that: first, the functional space of reference Q is larger than H1(M) — second, the problem is one-dimensional, so that one must cope with a lack of compactness when passing from weak convergence in Q to strong convergence in Lp- third, the boundary condition to be reconstructed is non standard. A complete proof is in [6].
An important point about Theorem 2.1 is that, in order to find the ground states, it suffices to determine which one among the solutions of (2. 6) has least action. This can be made directly, as the solutions to equation (2. 6) can be explicitly found. It has been said, however, that the variational analysis provides information beyond the one obtainable through the direct ODE approach- for example, the minimum is constrained to a finite codimension (one in this case) manifold, an information which is important for stability issues.
2.3. Symmetry breaking
Equation (2. 6) can be rephrased as follows:
-& lt-92 $ + - m2^ = 0, (2. 7)
with ^ e H2(R+) © H2(R-) satisfying the boundary condition (2. 1).
The only solutions to (2. 7) that vanish at infinity are constructed by gluing together two pieces of a solitary wave for the NLS, namely
, T1T"., ±A & gt- (a + 1) Cosh vnJZj{x — x i) l, x & lt- 0 A & gt- (fi + 1) 2CW2C cosh M [pLy/uj (x — ?2)], X & gt- 0
where the parameters x1 and x2 are to be adjusted so that (2. 1) is satisfied. Now, it is immediately seen by (2. 5) that due to contribution of the point interaction energy, one has
su (C-2) & lt- Su (Cf2)
so we can restrict the search for minimizers to the functions fa1−2, i.e. solutions of (2. 6) that change sign at the origin (and only there).
For any such function, the boundary condition (2. 1) translates into the system
, 2^, 2^+2 _, 2^, 2^+2
?r1 + C1 = iVU
0^ = | tanh (n^fuxi) ^ 1, (2. 8)
whose solutions can be depicted as the intersection of the full and the dashed lines in Figure 2. One immediately finds that for & lt- oj ^ t'-1c unique solution is given by l
o li
Fig. 2. The full lines represent the solutions to the first equation in (2. 8): they consist of the line 0 ^ t1 = t2 ^ 1, and of a curve, that is concave if f is not too small. The dashed lines represent the solutions to the second equation of (2. 8): they consist of a family of hyperbola parametrized by u
that corresponds to an antisymmetric stationary state y, where
1. 7v^+2
y = X ! = -X2 = ---j= log--=--.
Zf^/u jy/u — 2
At uj = tV1^- two new solutions arise, giving birth to two new branches of stationary states that persist for u & gt- they correspond to the couple of asymmetric stationary states
^?i1'--y2,i2 ,-yi, with both y1 and y2 positive but, except in the cubic case / = 1, not in explicit form. A direct computation yields, for these values of u,
Su (№-y2) = (rj'--yi) & lt- suOft-).
We conclude that with the growth of the frequency u there exist two branches of asymmetric ground states which bifurcate from the branch of (anti)symmetric ones. We are then in the presence of a spontaneous symmetry breaking of the set of ground states.
2.4. Stability: a pitchfork bifurcation
The study of the stability for such a system can be made by applying the Grillakis-Shatah-Strauss theory (see [30,31]). This theory provides sufficient conditions for the orbital stability of stationary states, which is stability & quot-up to the symmetries& quot-. Roughly speaking, the notion of orbital stability coincides with the ordinary Ljapunov stability for orbits instead of states, where orbits are to be understood with respect to a symmetry group. In our case the symmetry group is U (1), corresponding to the well known phase invariance of the NLS, which persists in the presence of point perturbation too. So, a stationary state is said to be orbitally stable if at any time a solution to (2. 2) remains arbitrarily close to the orbit {e, 9 e [0, ?k)}, provided that it started sufficiently close to it. More rigorously,
Definition 2.2. A stationary state is called orbitally stable if for any e & gt- 0 there exists a a & gt- 0 s.t.
inf 110 — \qa ^ sup inf \ipt — \Q ^ e,
where is the solution corresponding to the initial condition0.
A stationary state is called orbitally unstable if it is not orbitally stable.
The Grillakis-Shatah-Strauss theory (see [30,31]) carries out a deep investigation of the orbital stability of stationary states of (infinite dimensional) hamiltonian systems with symmetries, generalizing previous work by the same authors and independently by Michael Weinstein (see [48−50]).
They succeeded in giving sufficient conditions for stability and instability by studying second-order approximation of the action (linearization) around a stationary state, and carefully controlling the nonlinear remainders exploiting symmetries and conservation laws. In the present situation, as it is well known, one gets a hamiltonian system from NLS equation passing to real variables (], p) = (Re0, Im-0). We confine ourself to a brief operative summary of the method, and so we omit the (however important) connection with hamiltonian systems referring to the original literature for details.
Neglecting higher order terms, one has for the action expanded around the stationary state fa (we omit other superscripts for simplicity)
1 i (] f ]
Su,{lpu, + V + ip) = Su (fa) + - S& quot-(fa) ,
The Hessian operator S& quot- (fa) can be represented in matrix form as (we implicitly introduced the representation of a function ] + ip as the real vector function (], p))
SI (fa) := '- Ll 0
0 L2
0 L2
where Li and L2 are two selfadjoint operators with D (Li) = D (L2) = D (H1) given by
Li = H1 + u — A (2J + 1) fa2^
L2 = H1 + u — Afa2^.
Now, were S& quot- (fa) a positive operator, the (linear) stability of fa would be immediately established, as the situation would be analogous to what happens for a classical particle in a potential well. Unfortunately, this cannot be the case. First of all, the operator S& quot-(fa) is endowed with a non trivial kernel that consists of the linear span of (0,^), due to the symmetry. Second, recall that every ground state is a minimizer only on the constraint provided by the Nehari manifold, which has codimension one. On the space orthogonal to the Nehari manifold, S& quot- (fa) is surely negative, as
(^) & lt- 0.
It follows that there exists a cone on which S& quot- (^) is actually negative.
Nevertheless, it is possible that the dynamical constraints given by the conservation laws prevent the wave function from further evolving far inside that cone, finally forcing the solution to remain close to the orbit of the ground state. The Grillakis-Shatah-Strauss theory establishes that this is the case if a certain number of conditions are satisfied. In its easiest version, such a set of conditions can be collected as follows) Spectral conditions:
(1) KerL2 = Span{^w},
(2) L2 & gt- 0,
Fig. 3. Bifurcation diagram for ^ ^ 2. The full line denotes stable stationary states, while the dashed line represents unstable stationary states. Notice that ground states are always stable
(3) KerLi = {0},
(4) L1 has exactly one negative eigenvalue. ii) Vakhitov-Kolokolov'-s criterion (see [47]):
that, since dSu^u) = & amp- equivalent to
& gt- 0,
d2SUJ (ijjuj) n (2g)
duj* 1 j
In the case of interest, conditions i) and ii) are verified except for the the stationary states in the branch ipvj'-~y with uj & gt- t/z1^-, where a more sophisticated version of conditions /'-) and /'-/'-) is needed, again provided by the Grillakis-Shatah-Strauss theory (see [31]). The results on stability can be summed up as follows.
Theorem 2.3. For any /i & gt- 0
(1) If uj & lt- then the unique (up to a phase) ground state iljy~y is orbitally stable.
(2) If uj & gt- -^z1^, then the stationary state rt[)y~y is orbitally unstable.
For 0 ^ /j, ^ 2, uj & gt- -^z1^, the two ground states ip1~y2, ri) yJ~yi are orbitally stable.
For 2 & lt- /j, & lt- ?1* & lt- 2.5 there exist uji & gt- and uj2 & gt- uji, such that, if ^fz^-y- & lt- uj & lt- uj,
then ipyi~V2 and ipy2~Vl are orbitally stable- if u & gt- u2, then ipyi~V2 and -yi are orbitally
For /j, & gt- ?j*, there exist uji & gt-uj2 & gt- uj, such that, if & lt- uj & lt- uj or uj & gt- uj2, then
ipyi~V2 and ipy2~yi are orbitally unstable.
The bifurcation diagrams for the system are portrayed in Figures 3 and 4.
Fig. 4. Bifurcation diagram for 2 & lt- ?i & lt- ?j*. We have no results for the interval (u1}u2), but we conjecture that it is always possible to choose = u2
3. Proof of stability
The content of this section is technical. Here we we give a proof of the stability of all ground states in the case ^ ^ 2. Under such a restriction, every ground state satisfies the Vakhitov-Kolokolov'-s criterion. The proof we present here differs from the one given in [6], as it does not use the Grillakis-Shatah-Strauss theory and so it does not refer to linearization. We decided to include in this report such a technical part in order to convey some information on the method of proofs and on the techniques employed. An analogous analysis is given for the case of a NLS with 5 interaction in [27], and both are inspired by [25]. In order to proceed we need some preliminary definitions and results.
First, the definition of orbital stability can be reformulated using the notion of orbital neighbourhood.
Definition 3.1. The set
Uv (0) := e Q, s. t inf U — eidt\Q ^ V]
is called the orbital neighbourhood with radius rq of the function 0.
It is convenient to introduce a function that associates to any frequency uj & gt- tV the value of the minimum attained by evaluated on functions in the Nehari manifold corresponding to that frequency. Namely,
d: (-±,+0o)-& gt-R
u i-& gt- d (u) := min[S'-we Q, (0) = 0}.
It is then important to stress other points that we did not mention explicitly so far.
Remark 3.2.
(1) In the energy space Q the following norm is defined:
^|2 dx + ?lim y WI2 dx.
(2) For any 9 e [0, 2n) and any (f) e Q, one has Su (e%d4& gt-) = Su (0). As a consequence, if fa is a ground state, then all the functions e%d fa in its orbit are ground states too. In the proof of Theorem (3. 4) we will make the phase explicit by denoting
faJ'-X2'-e : — ei& amp-fa1'-*2'-0.
The result we need to go through the proof are summarized in the following Proposition. Their proof is contained in [6].
Proposition 3.3.
(1) For any function (f) in the Nehari manifold one has (4& gt-) = S (4& gt-), where S is the functional defined by
Xfj, 21+2'-
(2) Any minimizer fa of the functional on the Nehari manifold minimizes also the functional S on the region ^ 0.
(3) Following Fibich and Wang (see [25]) we recall that the map
m: =^mttl
Q: Q -& gt- E, 0 ^
is well-defined. Notice that Q maps a function 0 into the frequency of a ground state having the same L2^+2-norm as 0. Such a ground state may not be unique, but the L2^+2-norm always is.
(4) If fa minimizes Su on the Nehari manifold Iu — 0, then minimizes Su on the set
W e Q, II0IU+2 = WfaW+2}-
(5) For any q & gt- 0, the function minimizes the functional among the functions in Q that satisfy
(0): — II2 + qI0I2 -mWt+t — 0.
(6) If ^ ^ 2, then any ground state satisfies the Vakhitov-Kolokolov'-s condition (2. 9). Now we can prove the
Theorem 3.4. If 1 ^ ^ ^ 2, then any ground state is stable.
Proof. We specialize to the case with u & gt- namely, beyond the frequency of bifurcation.
In fact, for u & lt- 77^-!-, this proof can be easily adapted and one recovers essentially the argument given in [27].
Fix u-0 & gt- and suppose that the stationary solution r^^'-tn^ is orbitally unstable.
This means that there exists e0 & gt- 0 and a sequence e Ui {fa1'-^2'-0) such that
sup inf II^fc (t) -faJfa^IIQ & gt- 60 where pk (t) is the solution to equation (2. 2) with initial data pfc.
Nonlinearity-defect interaction: symmetry breaking bifurcation in a NLS with a 5'- impurity 15 With no loss of generality, we assume
^ ^ f №-w& gt-° - c-^\q = m-2'-0 — rj (ryi'-0h. (3. 1)
Let tk be the smallest positive time for which
f. (3−2)
and let us use the notation = '--Pk (4). By conservation laws,
(60 = ?(60 + yll& amp-lll = + f Ml -+ yl№~w, 0lll = = ?M-
(3. 3)
where we used the fact that, by construction, the sequence ipk converges to ^yi1'-~y2'-0 strongly in Q, that implies the convergence of the energy and of the L2-norm.
Let us denote uk — uj (^k). We recall the following result from [25], used in [27] also:
Suk (?k) — Sujk (4& gt-o) ^ ^d& quot-(uQ)(uk — Uo)2 (3. 4)
where we denoted uk — w (?k). The fact that the Vakhitov-Kolokolov'-s condition is satisfied (see [6]), together with (3. 4) and (3. 3), implies uk — u0, and therefore, by the definition of the function uj, we have
k || 2^+2
1 _ __1
* A — M_
Uk ||M+2
2^ + 2 0
^ vPuo J
— im'--y2'-oh"+2. (3. 5)
, y 2, W, 1n
We define the sequence (k := «& quot-M+2?fc. By (3. 5),
Kk — -

M+2 _ 1
Mkh -- 0. (3. 6)
As a consequence, ((k) — Suo (?k) ^ 0, so ((k) ^ Suo (Vu). For this reason, and as Kk^+2 = ^i10'--y2'-°2^+2, point (4) in Proposition 3.3 implies that {(k} is a minimizing sequence for the problem
min^ty), ^ e Q{0}, \^2M+2 = № 2ryi'-°h"+2}.
By Banach-Alaoglu theorem there exists a subsequence, whose elements we denote by (k too, that converges weakly in Q and therefore in L2^+2. Let us call its weak limit.
First, notice that = 0. Indeed, were it zero, then weak convergence in Q would imply C~(0±) ^ 0, and therefore ((k) — S°0 (00 ^ 0, so
Then, employing the fact that 0° := X[0,+^)^u!0 (0) = 0, points (4) and (5) in Proposition 3.3 yield
= iimS2o (a) ^ s°o (x+4& gt-o) & gt- s^ix+to) ^
which is absurd. So it must be = 0.
We claim that = 0. We proceed by contradiction.
Suppose indeed that & lt- 0. Then, by point (2) in Remark 3.2 and points (1) and (2) in
Proposition 3.3 we know that the minimizers of the functional S on the region ^ 0 are given
by fa10'--y2'-e and fa20'--yi'-e, for all 9 e [0, 2^). Furthermore, all such functions lie on the set Iuo — 0. As a consequence, recalling the definition of the functional S, one obtains

& gt-

= 11 ^u!'--^2112^+2 •

But this is not possible, as ^ is the weak limit of functions having the same L2^+2-norm as
J, yi'--y2'-0
ru o
On the other hand, suppose that Iuo ((& amp-,) & gt- 0. By (3. 3) and (3. 5)
lim Iuo (?k) = 2 lim Suo (€k) — lim Uk\ZXl
kox kox U + 1 kiL ^
Therefore, by (3. 6),
lim Iuo ((k) — 0.
ki& lt-x
From the following inequality (see [15])
IunIl — IIun -u^IIl -IMIl 0, V1 & lt-p<- (3. 7)
one easily has
((k — -& gt- -U (& lt- 0
As a consequence, eventually in k we obtain IUo ((k — () & lt- 0 and then, using point (2) in Proposition 3. 3
iiCk — au+2 & gt- urur^I^. (3. 8)
But from (3. 7), and knowing that — 0, we have that the following inequality holds eventually in
II Ck — au+2 ^ wru-0'--y2'-0W2,+2,
that contradicts (3. 8).
We conclude that Iuo ((& amp->-) cannot be strictly positive and, as we already proved that it cannot be negative, it must vanish.
As a consequence, from point (2) in Proposition 3.3 again, we get || 112^+2 ^ W^U0'--y2'-0112^+2. But, since is a weak limit, it must be
\(^2^+2 — \fal'- V2'-°W2v+2.
This fact has the following relevant consequences:
• Owing to (3. 7), the sequence {(nj converges strongly to in the topology of L2^+2.
• The sequence {(k} converges to in the strong topology of Q. Indeed, by the convergence of SUo ((k) to SUo (C~), the weak convergence in Q, and the strong convergence of
{(k} in L2^+2, we have
WCkII2 + Q0wCkW2 IIC, W2 + QoWUI2. (3. 9)
So the convergence is strong in the space Q endowed with the norm given by (3. 9), that is equivalent to the usual Q-norm.
• The sequence {?k} also converges to in the strong topology of Q. Indeed, applying (3. 6), we have
IIa — ClIIq & lt- IIa — CkIq + IICk — ClIIq 0. (3. 10)
— jyi '--y2'-0
• The function minimizes SUo with the constraint IUo — 0, so, either — ^ or — fa20'--yi'-e for some value of 9 in [0, 2^).

Let us suppose that — ^yi0'--y2'-e, for a certain value of 6. By (3. 10) we obtain ^yi0 ~y2'-& amp- strongly in Q, that contradicts inequality (3. 2), and thus the assumption of the orbital instability of the stationary state ^yi0'--y2'-0 proves false.
On the other hand, consider the case with — fa20'--yi'-e for some value of 6. By (3. 2) there exists a sequence 6k such that
Uk-rjrm'-dkh ^ 3? o- (3. ii)
Using elementary triangular identity, (3. 1) and (3. 11), we obtain, for any d e [0, 2^),
'- & quot-"-HQ — & quot-"-"- -SfcllQ ^
This contradicts (3. 10), so the proof is complete.
Uk-rj0'--yi'-eh & gt- IIC'-^-C'-^llo-IIC'-^-^llo ^ 3
4. Perspectives
The interplay between nonlinearity and defects is, in our opinion, a promising and worth developing field. In particular, already in simple models highly non trivial behaviour can emerge. An enlightening example has been supplied by means of the '- defect, in which the occurrence of a pitchfork bifurcation with symmetry breaking has been proved for the family of nonlinear ground states.
Such results have to be considered as the first achievements of our research project. Many non trivial variations on the theme could be given by studying the entire family of one-dimensional defects (a four parameters family, see [9]) and thus investigate the effect of various self-adjoint boundary conditions, in particular, of those that give rise to two bound states. We expect that, in the nonlinear problem, each of the two linear modes could be deformed into nonlinear modes for any frequency greater than the energy of the corresponding linear mode. Think, for instance, of a point interaction that, roughly speaking, is the sum of a and a '- defect at the same point. It exhibits two bound states, one of which is even (as the ground state for a Dirac'-s delta), while the other is odd (as the ground state for a delta prime). A number of question then arises: how do the corresponding nonlinear mode interact? Does it exist a third family of stationary (possibly ground) states that does not preserve any parity symmetry?
However, all these steps are only preliminary to the problem of studying the detailed evolution of a travelling soliton that meets an impurity.
It remains completely open the problem of defining analogous models in higher dimension. We recall that in dimension two and three, the only point interaction is the delta interaction, and in dimension higher than three there are no point perturbations of the laplacian. For instance, in the three dimensional case a bare power nonlinearity seems to be too strong to be added to a Dirac'-s 5 potential- so a different type of nonlinearity with a moderated behaviour at infinity should be considered. Conversely, in space dimension two the nai'-f power nonlinearity could be not necessarily in conflict with the domain of a delta interaction, but up to now no rigorous result exists on this problem.
Another related topic is given by quantum graphs (see [37−39] for the relevant definitions and analysis in the linear case). Also in the relatively simple case of a NLS on a star graph, the richer structure provides a larger number of nonlinear stationary states, for example two stationary states for a three edge star graph with a delta vertex, both attractive and repulsive, and the number increases with the number of edges (see [5]). In this respect, besides the determination of the ground state, it is an open interesting problem the analysis of stability of excited states, here
explicitly known. Nothing is known for the a star graph with more general vertex conditions, for
example the boundary condition of 5'- type.
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