Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
The nonlinear Schrodinger equation on star
graphs
Schrodinger and other dynamics on thin networks
Claudio Cacciapuoti
Hausdorff Center for MathematicsBonn Universitat
Workshop“Mathematical Aspects of Quantum Mechanics and Quantum Transport Theory”
Bielefeld, April 23 - 28, 2012
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
1. Metric graphs: introduction & linear operators
2. Nonlinear Schrodinger equation on graphs(joint work with R. Adami, D. Finco, D. Noja)
- Scattering of fast solitons
- Stationary states
3. Approximation of networks of thin tubes(joint work with S. Albeverio, D. Finco)
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Metric Graphs
A metric graph is realized by a set ofedges ejn
j=1 and vertices, with a met-ric structure on any edge.Self-adjoint, (one-dimensional) differ-ential operators can be defined on thegraph.
Graphs as one-dimensional approx-imations for constrained dynamicsin which transverse dimensions aresmall with respect to longitudinalones.
Graphs are somewhat simplified objects which neglect much of the relevant(i.e. geometric) structure, yet they exhibit non trivial features.
- G. Berkolaiko et al., Quantum graphs and their applications, 2006.
- P. Exner et al., Analysis on graphs and its applications, 2008.
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Metric Graphs
A metric graph is realized by a set ofedges ejn
j=1 and vertices, with a met-ric structure on any edge.Self-adjoint, (one-dimensional) differ-ential operators can be defined on thegraph.
Graphs as one-dimensional approx-imations for constrained dynamicsin which transverse dimensions aresmall with respect to longitudinalones.
Graphs are somewhat simplified objects which neglect much of the relevant(i.e. geometric) structure, yet they exhibit non trivial features.
- G. Berkolaiko et al., Quantum graphs and their applications, 2006.
- P. Exner et al., Analysis on graphs and its applications, 2008.
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Sobolev Spaces on graphs
Every edge of the graph is isomorphicto a (bounded or unbounded) orientedsegment, ej ∼ Ij
A function on a graph is a vector
Ψ = (ψ1, ..., ψN ) with ψj ≡ ψj (xj ) ; xj ∈ Ij
The spaces Lp(G) are defined by
Lp(G) =N⊕
j=1
Lp(Ij ) ; ‖Ψ‖p =
(N∑
j=1
‖ψj‖pLp (Ij )
) 1p
In a similar way one can define Hp(G) =⊕N
j=1 Hp(Ij )
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Laplacian on a star graph
For a Star Graph: ej ∼ (0,+∞), v ≡ 0
Hilbert Space
H =N⊕
j=1
L2((0,∞))
Ψ = (ψ1, . . . , ψN ) ∈ H
‖Ψ‖ =
( N∑j=1
‖ψj‖2L2(R+)
)1/2
The operator −∆G
−∆GΨ =
(−d2ψ1
dx21
, . . . ,−d2ψN
dx2N
)
D(−∆G) =N⊕
j=1
H2((0,∞)) + self-adjoint conditions in the vertex
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Laplacian on a star graph
For a Star Graph: ej ∼ (0,+∞), v ≡ 0
Hilbert Space
H =N⊕
j=1
L2((0,∞))
Ψ = (ψ1, . . . , ψN ) ∈ H
‖Ψ‖ =
( N∑j=1
‖ψj‖2L2(R+)
)1/2
The operator −∆G
−∆GΨ =
(−d2ψ1
dx21
, . . . ,−d2ψN
dx2N
)
D(−∆G) =N⊕
j=1
H2((0,∞)) + self-adjoint conditions in the vertex
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Laplacian on a star graph
For a Star Graph: ej ∼ (0,+∞), v ≡ 0
Hilbert Space
H =N⊕
j=1
L2((0,∞))
Ψ = (ψ1, . . . , ψN ) ∈ H
‖Ψ‖ =
( N∑j=1
‖ψj‖2L2(R+)
)1/2
The operator −∆G
−∆GΨ =
(−d2ψ1
dx21
, . . . ,−d2ψN
dx2N
)
D(−∆G) =N⊕
j=1
H2((0,∞)) + self-adjoint conditions in the vertex
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Laplacian on a star graph
For a Star Graph: ej ∼ (0,+∞), v ≡ 0
Hilbert Space
H =N⊕
j=1
L2((0,∞))
Ψ = (ψ1, . . . , ψN ) ∈ H
‖Ψ‖ =
( N∑j=1
‖ψj‖2L2(R+)
)1/2
The operator −∆G
−∆GΨ =
(−d2ψ1
dx21
, . . . ,−d2ψN
dx2N
)
D(−∆G) =N⊕
j=1
H2((0,∞)) + self-adjoint conditions in the vertex
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Laplacian on a Star Graph: Vertex Conditions
Any unitary N × N matrix, U, defines a selfadjoint Laplacian on G, −∆UG , by
the boundary conditions
(U − 1)
ψ1(v)...
ψN (v)
+ i(U + 1)
ψ′1(v)...
ψ′N (v)
= 0
- Dirichlet condition [decoupling condition]:
ψ1(v) = ψ2(v) = · · · = ψN (v) = 0
- Kirchhoff condition [standard condition]:
ψ1(v) = ψ2(v) = · · · = ψN (v) ,N∑
j=1
ψ′j (v) = 0
- delta-like condition: [Ujk = 2(N + iα)−1 − δjk ]
ψ(v) ≡ ψ1(v) = ψ2(v) = · · · = ψN (v) ,N∑
j=1
ψ′j (v) = αψ(v) α ∈ R
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Laplacian on a Star Graph: Vertex Conditions
Any unitary N × N matrix, U, defines a selfadjoint Laplacian on G, −∆UG , by
the boundary conditions
(U − 1)
ψ1(v)...
ψN (v)
+ i(U + 1)
ψ′1(v)...
ψ′N (v)
= 0
- Dirichlet condition [decoupling condition]:
ψ1(v) = ψ2(v) = · · · = ψN (v) = 0
- Kirchhoff condition [standard condition]:
ψ1(v) = ψ2(v) = · · · = ψN (v) ,N∑
j=1
ψ′j (v) = 0
- delta-like condition: [Ujk = 2(N + iα)−1 − δjk ]
ψ(v) ≡ ψ1(v) = ψ2(v) = · · · = ψN (v) ,N∑
j=1
ψ′j (v) = αψ(v) α ∈ R
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Laplacian on a Star Graph: Vertex Conditions
Any unitary N × N matrix, U, defines a selfadjoint Laplacian on G, −∆UG , by
the boundary conditions
(U − 1)
ψ1(v)...
ψN (v)
+ i(U + 1)
ψ′1(v)...
ψ′N (v)
= 0
- Dirichlet condition [decoupling condition]:
ψ1(v) = ψ2(v) = · · · = ψN (v) = 0
- Kirchhoff condition [standard condition]:
ψ1(v) = ψ2(v) = · · · = ψN (v) ,N∑
j=1
ψ′j (v) = 0
- delta-like condition: [Ujk = 2(N + iα)−1 − δjk ]
ψ(v) ≡ ψ1(v) = ψ2(v) = · · · = ψN (v) ,N∑
j=1
ψ′j (v) = αψ(v) α ∈ R
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Laplacian on a Star Graph: Vertex Conditions
Any unitary N × N matrix, U, defines a selfadjoint Laplacian on G, −∆UG , by
the boundary conditions
(U − 1)
ψ1(v)...
ψN (v)
+ i(U + 1)
ψ′1(v)...
ψ′N (v)
= 0
- Dirichlet condition [decoupling condition]:
ψ1(v) = ψ2(v) = · · · = ψN (v) = 0
- Kirchhoff condition [standard condition]:
ψ1(v) = ψ2(v) = · · · = ψN (v) ,N∑
j=1
ψ′j (v) = 0
- delta-like condition: [Ujk = 2(N + iα)−1 − δjk ]
ψ(v) ≡ ψ1(v) = ψ2(v) = · · · = ψN (v) ,N∑
j=1
ψ′j (v) = αψ(v) α ∈ R
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Nonlinear Schrodinger Equation on a Star Graph (focusing, withpower nonlinearity)
Let µ > 0
id
dtΨt = HΨt − |Ψt |2µΨt |Ψ|2µ ≡ diag [|ψj |2µ]
Or in integral form
Ψt = e−iHtΨ0 + i
∫ t
0
e−iH(t−s)|Ψs |2µΨs ds
Or in components
i∂
∂tψj (xj , t) = − ∂2
∂x2j
ψj (xj , t)− |ψj (xj , t)|2µψj (xj , t) xj > 0
but again do not forget the boundary conditions.
Nature of the problem: a system of N PDEs coupled through the vertexconditions.
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Nonlinear Schrodinger Equation on a Star Graph (focusing, withpower nonlinearity)
Let µ > 0
id
dtΨt = HΨt − |Ψt |2µΨt |Ψ|2µ ≡ diag [|ψj |2µ]
Or in integral form
Ψt = e−iHtΨ0 + i
∫ t
0
e−iH(t−s)|Ψs |2µΨs ds
Or in components
i∂
∂tψj (xj , t) = − ∂2
∂x2j
ψj (xj , t)− |ψj (xj , t)|2µψj (xj , t) xj > 0
but again do not forget the boundary conditions.
Nature of the problem: a system of N PDEs coupled through the vertexconditions.
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Nonlinear Schrodinger Equation on a Star Graph (focusing, withpower nonlinearity)
Let µ > 0
id
dtΨt = HΨt − |Ψt |2µΨt |Ψ|2µ ≡ diag [|ψj |2µ]
Or in integral form
Ψt = e−iHtΨ0 + i
∫ t
0
e−iH(t−s)|Ψs |2µΨs ds
Or in components
i∂
∂tψj (xj , t) = − ∂2
∂x2j
ψj (xj , t)− |ψj (xj , t)|2µψj (xj , t) xj > 0
but again do not forget the boundary conditions.
Nature of the problem: a system of N PDEs coupled through the vertexconditions.
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Laplacian with delta-like condition in the vertex
From now on we shall consider only delta-like conditions in the vertex.
ψ(v) ≡ ψ1(v) = ψ2(v) = · · · = ψN (v) ,N∑
j=1
ψ′j (v) = αψ(v) α ∈ R
We denote by Hα the corresponding Laplacian
When α = 0 (Kirchhoff condition) we write H ≡ Hα=0
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Soliton on the line
Consider the equation
i∂
∂tψ(x , t) = − ∂2
∂x2ψ(x , t)− |ψ(x , t)|2µψ(x , t) x ∈ R , t > 0
Given the functionφ(x) = [(µ+ 1)]
12µ sech
1µ (µx)
One has the family of solitary rotating/traveling waves:
φx0,v,ω(x , t) := e i v2
xe−it v2
4 e iωtω1
2µ φ(√ω(x − x0 − vt))
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Are there solitons on Start Graphs?
Yes, for star graphs with even number of edges and Kirchhoff conditions in thevertex.Think at the star graph as N
2lines which cross at the vertex
On each line put a copy of the soliton φx0,v,ω(x , t) with the same x0, v and ωThe function Φx0,v,ω constructed in this way satisfies the Kirchhoff conditionin the vertexΦx0,v,ω describes N
2identical solitons moving simultaneously on the graph
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Are there solitons on Start Graphs?
Yes, for star graphs with even number of edges and Kirchhoff conditions in thevertex.Think at the star graph as N
2lines which cross at the vertex
On each line put a copy of the soliton φx0,v,ω(x , t) with the same x0, v and ωThe function Φx0,v,ω constructed in this way satisfies the Kirchhoff conditionin the vertexΦx0,v,ω describes N
2identical solitons moving simultaneously on the graph
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Propagation of pulses in networks
Is this picture (approximately) realistic?
t = 0 t large
Inspired by the work: J. Holmer, J. Marzuola, and M. Zworski, Fast solitonscattering by delta impurities, Commun. Math. Phys. 274, 2007.
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Propagation of pulses in networks
Is this picture (approximately) realistic?
t = 0 t large
Setting:
Cubic NLS
Kirchhoff vertex (but more general conditions are allowed)
Initial stateΨ0(x) = (
√2χ(x)e−i v
2x sech(x − x0), 0, 0)
χ is a cutoff function
High velocity regime v 1
x0 > v1−δ, with 0 < δ < 1.
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Propagation of pulses in networks
Find an approximate solution of the equation:
Ψt = e−iHtΨ0 + i
∫ t
0
e−iH(t−s)|Ψs |2Ψs ds
Three step analysis
Phase 1: approaching the vertex
Phase 2: scattering through the vertex
Phase 3: persistence of the outgoing state
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Propagation of pulses in networks: Phase 1, approaching the vertex
t ∈ [0, t1 = x0/v − v−δ]
In this phase the incoming pulse moves from x0 to x0 − vt1 = v 1−δ,remaining at a distance of order v 1−δ from the vertex [0 < δ < 1]
During this phase only a small tail of the pulse intersects the vertex, thesolution Ψt behaves as the solitary solution of the NLS in R and remainssubstantially supported only on the edge e1
The approximating function is:
Φt(x) = (φx0,−v (x , t), 0, 0)
Lemma
For any t ∈ [0, t1]
‖Ψt − Φt‖ 6 Ce−v1−δ
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Propagation of pulses in networks: Phase 1, approaching the vertex
t ∈ [0, t1 = x0/v − v−δ]
In this phase the incoming pulse moves from x0 to x0 − vt1 = v 1−δ,remaining at a distance of order v 1−δ from the vertex [0 < δ < 1]
During this phase only a small tail of the pulse intersects the vertex, thesolution Ψt behaves as the solitary solution of the NLS in R and remainssubstantially supported only on the edge e1
The approximating function is:
Φt(x) = (φx0,−v (x , t), 0, 0)
Lemma
For any t ∈ [0, t1]
‖Ψt − Φt‖ 6 Ce−v1−δ
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Propagation of pulses in networks: Phase 1, approaching the vertex
t ∈ [0, t1 = x0/v − v−δ]
In this phase the incoming pulse moves from x0 to x0 − vt1 = v 1−δ,remaining at a distance of order v 1−δ from the vertex [0 < δ < 1]
During this phase only a small tail of the pulse intersects the vertex, thesolution Ψt behaves as the solitary solution of the NLS in R and remainssubstantially supported only on the edge e1
The approximating function is:
Φt(x) = (φx0,−v (x , t), 0, 0)
Lemma
For any t ∈ [0, t1]
‖Ψt − Φt‖ 6 Ce−v1−δ
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Propagation of pulses in networks: Phase 2, scattering through thevertex
t ∈ [t1 = x0/v − v−δ, t2 = x0/v + v−δ]
During this phase the “body” of the soliton crosses the vertex.
The time interval t2 − t1 is small (of order v−δ, 0 < δ < 1). Then theeffect of the nonlinear term is negligible
The pulse travels for a large distance (of order v(t2 − t1) = v 1−δ,0 < δ < 1). Then the linear dynamics can be described by using ascattering approximation.
Let T and R be the transmission and reflection coefficients of the linearoperator
T =2
NR = −N − 2
N
Recall that the function Ψ(k)
Ψ(k, x1, x2, x3) = (e−ikx1 + Re ikx1 ,T e ikx2 ,T e ikx3 )
satisfies the Kirchhoff boundary conditions, and is a solution ofHΨ = k2Ψ, in distributional sense.
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Propagation of pulses in networks: Phase 2, scattering through thevertex
t ∈ [t1 = x0/v − v−δ, t2 = x0/v + v−δ]
During this phase the “body” of the soliton crosses the vertex.
The time interval t2 − t1 is small (of order v−δ, 0 < δ < 1). Then theeffect of the nonlinear term is negligible
The pulse travels for a large distance (of order v(t2 − t1) = v 1−δ,0 < δ < 1). Then the linear dynamics can be described by using ascattering approximation.
Let T and R be the transmission and reflection coefficients of the linearoperator
T =2
NR = −N − 2
N
Recall that the function Ψ(k)
Ψ(k, x1, x2, x3) = (e−ikx1 + Re ikx1 ,T e ikx2 ,T e ikx3 )
satisfies the Kirchhoff boundary conditions, and is a solution ofHΨ = k2Ψ, in distributional sense.
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Propagation of pulses in networks: Phase 2, scattering through thevertex
t ∈ [t1 = x0/v − v−δ, t2 = x0/v + v−δ]
Approximating function:
ΦSt =
(φx0,−v (t) + R φ−x0,v (t),T φ−x0,v (t),T φ−x0,v (t)
)
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Propagation of pulses in networks: Phase 2, scattering through thevertex
t ∈ [t1 = x0/v − v−δ, t2 = x0/v + v−δ]
Approximating function:
ΦSt =
(φx0,−v (t) + R φ−x0,v (t),T φ−x0,v (t),T φ−x0,v (t)
)
On the left side the state ΦS at time t1. On the right side the state ΦS at time t2.
Each edge of the graph is extended to a line by ideally adding the half line (−∞, 0]
represented by the dashed lines. Dashed lines are not part of the real graph.
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Propagation of pulses in networks: Phase 2, scattering through thevertex
t ∈ [t1 = x0/v − v−δ, t2 = x0/v + v−δ]
Approximating function:
ΦSt =
(φx0,−v (t) + R φ−x0,v (t),T φ−x0,v (t),T φ−x0,v (t)
)
Lemma
For any t ∈ [t1, t2]
‖Ψt − ΦSt ‖ 6 Cv−δ/2
for v large enough and some constant δ ∈ (0, 1).
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Propagation of pulses in networks: Phase 3, persistence of theoutgoing state
t ∈ [t2 = x0/v + v−δ, t3 = t2 + T ln v ]
Phase 3 begins with three low solitons which move away from the vertex
For t > t2 and x ∈ R we define the functions φtr and φref byi∂
∂tφtr = − ∂2
∂x2φtr − |φtr |2φtr
φtr (x , t2) = T φ−x0,v (x , t2)
i∂
∂tφref = − ∂2
∂x2φref − |φref |2φref
φref (x , t2) = R φ−x0,v (x , t2)
The outgoing state is approximated by
Φoutt :=
(φref (t), φtr (t), φtr (t)
)
Lemma
Fix T > 0, then for any time t ∈ [t2, t2 + T ln v ], there exists 0 < η < 1/2such that
‖Ψt − Φoutt ‖ 6 Cv−η
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Propagation of pulses in networks: Phase 3, persistence of theoutgoing state
t ∈ [t2 = x0/v + v−δ, t3 = t2 + T ln v ]
Phase 3 begins with three low solitons which move away from the vertexFor t > t2 and x ∈ R we define the functions φtr and φref by
i∂
∂tφtr = − ∂2
∂x2φtr − |φtr |2φtr
φtr (x , t2) = T φ−x0,v (x , t2)
i∂
∂tφref = − ∂2
∂x2φref − |φref |2φref
φref (x , t2) = R φ−x0,v (x , t2)
The outgoing state is approximated by
Φoutt :=
(φref (t), φtr (t), φtr (t)
)
Lemma
Fix T > 0, then for any time t ∈ [t2, t2 + T ln v ], there exists 0 < η < 1/2such that
‖Ψt − Φoutt ‖ 6 Cv−η
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Propagation of pulses in networks: Phase 3, persistence of theoutgoing state
t ∈ [t2 = x0/v + v−δ, t3 = t2 + T ln v ]
Phase 3 begins with three low solitons which move away from the vertexFor t > t2 and x ∈ R we define the functions φtr and φref by
i∂
∂tφtr = − ∂2
∂x2φtr − |φtr |2φtr
φtr (x , t2) = T φ−x0,v (x , t2)
i∂
∂tφref = − ∂2
∂x2φref − |φref |2φref
φref (x , t2) = R φ−x0,v (x , t2)
The outgoing state is approximated by
Φoutt :=
(φref (t), φtr (t), φtr (t)
)
Lemma
Fix T > 0, then for any time t ∈ [t2, t2 + T ln v ], there exists 0 < η < 1/2such that
‖Ψt − Φoutt ‖ 6 Cv−η
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Propagation of pulses in networks: Phase 3, persistence of theoutgoing state
t ∈ [t2 = x0/v + v−δ, t3 = t2 + T ln v ]
Phase 3 begins with three low solitons which move away from the vertexFor t > t2 and x ∈ R we define the functions φtr and φref by
i∂
∂tφtr = − ∂2
∂x2φtr − |φtr |2φtr
φtr (x , t2) = T φ−x0,v (x , t2)
i∂
∂tφref = − ∂2
∂x2φref − |φref |2φref
φref (x , t2) = R φ−x0,v (x , t2)
The outgoing state is approximated by
Φoutt :=
(φref (t), φtr (t), φtr (t)
)
Lemma
Fix T > 0, then for any time t ∈ [t2, t2 + T ln v ], there exists 0 < η < 1/2such that
‖Ψt − Φoutt ‖ 6 Cv−η
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Propagation of pulses in networks: Phase 3, persistence of theoutgoing state
Proposition (HMZ)
For 0 < |γ| < 1, let φγt be defined byi∂
∂tφγ = − ∂2
∂x2φγ − |φγ |2φγ
φγ(x , 0) = γ√
2 cosh−1(x)
x ∈ R
then the function φγt has the following asymptotic behavior for t →∞
φγ(x , t) =
a√
2 cosh−1(ax) e iϕ(γ) +OL∞(t−1/2) 1/2 < |γ| < 1
OL∞(t−1/2) 0 < |γ| < 1/2
with a = 2|γ| − 1.
For N = 3, T = 2/3 and |R| = 1/2. The transmitted low solitons survive,the reflected low soliton disappears.For N ≥ 4, T 6 1/2 and |R| > 1/2. The transmitted low solitons disappear,only the reflected low soliton survives.
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Propagation of pulses in networks: Phase 3, persistence of theoutgoing state
Final outcome of the scattering
N = 3 N ≥ 4
Warning:
Our result holds up to times of order ln v , with an error OL2 (v−η). Theresult in Proposition [HMZ] approximates φtr (t) and φref (t) for t →∞with an error OL∞(t−1/2).
Our estimates are in L2-norm. The result in Proposition [HMZ] is inL∞-norm. Our result is rigorous for what concerns “mass transmission”not for the “profile” of the outcoming pulses
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Propagation of pulses in networks: Phase 3, persistence of theoutgoing state
Final outcome of the scattering
N = 3 N ≥ 4Warning:
Our result holds up to times of order ln v , with an error OL2 (v−η). Theresult in Proposition [HMZ] approximates φtr (t) and φref (t) for t →∞with an error OL∞(t−1/2).
Our estimates are in L2-norm. The result in Proposition [HMZ] is inL∞-norm. Our result is rigorous for what concerns “mass transmission”not for the “profile” of the outcoming pulses
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
References
W. K. Abu Salem, J. Frohlich, and I. M. Sigal, Colliding solitons for the
nonlinear Schrodinger equation, Comm. Math. Phys. 291 (2009), 151–176.
R. Adami, C. C., D. Finco, and D. Noja, Fast solitons on star graphs, Rev.
Math. Phys 23 (2011), 409–451.
K. Datchev and J. Holmer, Fast soliton scattering by attractive delta impurities,
Comm. Part. Diff. Eq. 34 (2009), 1074–1113.
J. Holmer, J. Marzuola, and M. Zworski, Fast soliton scattering by delta
impurities, Commun. Math. Phys. 274 (2007), 187–216.
P. G. Kevrekidis, D. J. Frantzeskakis, G. Theocharis, and I. G. Kevrekidis,
Guidance of matter waves through Y-junctions, Phys. Lett. A 317 (2003),513–522.
G. Perelman, A remark on soliton-potential interaction for nonlinear Schrodinger
equations, Math. Res. Lett. 16 (2009), no. 3, 477–486.
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Standing waves
Consider the focusing NLS equation with a δ vertex of strength α
id
dtΨt = HαΨt − |Ψt |2µΨt
where Hα denotes the Laplacian on the graph with a delta-like condition inthe vertex
We want to study possible existence and properties of stationary solutions(standing waves):
Ψt = e iωt Φω
The amplitude Φω satisfies the “elliptic” equation
HαΦω − |Φω|2µΦω = −ωΦω
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Standing waves
Now on every edge
−φ′′ − |φ|2µφ = −ωφ φ ∈ L2(R+)
and the most general solution is
φ(σ, a; x) = σ [(µ+ 1)ω]1
2µ sech1µ (µ√ω(x − a)) , |σ| = 1 a ∈ R
so that(Φω)i = φ(σi , ai ) ,
where σi , ai have to be chosen to satisfy Φω ∈ D(Hα)
The continuity at the vertex implies σj = 1
aj = εja, εi = ±1 , a ≥ 0
For a > 0 there are “bumps” and “tails”- εj = 1: there is a “bump” on the edge j- εj = −1: there is a “tail” on the edge j
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Standing waves
The condition on the derivative
n∑j=1
(Φω)′j (0) = α (Φω)1 (0)
gives
tanh(µ√ωa)
N∑i=1
εi =α√ω
(1)
Consequence:∑Ni=1 εi must have the same sign of α
- α > 0 strictly more bumps than tails- α < 0 strictly more tails than bumps- α = 0 same number of tails and bumps or a = 0
For every configuration of ε1, ..., εN (up to permutations of the edges) thecondition (1) fixes uniquely a
We index the stationary states with the number j of bumps
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Standing waves: α 6= 0 (delta-like vertex)
Stationary solutions exist and (their amplitudes) are given by
(Φjω
)i
=
φ(aj ) i = 1, . . . , j
φ(−aj ) i = j + 1, . . . ,N
aj =1
µ√ω
arctanh
(α
(2j − N)√ω
)
Due to the constraint between number of bumps and the sign of α
α > 0 : Φjω with j = [N/2 + 1], . . . ,N
α < 0 : Φjω with j = 0, . . . , [(N − 1)/2]
For any value of α 6= 0 there are
[N + 1
2
]states
Finally, there is a lower bound on the allowed frequencies:
α2
N2< ω
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Standing waves: α 6= 0 (delta-like vertex)
Nonlinear stationary states:α < 0 , N = 3 , j = 0, 1
Nonlinear stationary states:
α > 0 , N = 3 , j = 2, 3
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Standing waves: α = 0 (Kirchhoff vertex)
Let us consider the Kirchhoff case, α = 0.From the boundary condition
tanh(µ√ωa)
N∑i=1
εi = 0 (2)
- First case (N odd): (2) =⇒ a = 0The stationary state is unique
(Φω)i (x) = φ(0, x) i = 1, . . . ,N
N half solitons continuously joint at the vertex
- Second case (N even): (2) =⇒ a ∈ R,∑N
i=1 εi = 0There is a one-parameter family of stationary states
(Φaω)i (x) =
φ(−a, x) i = 1, . . .N/2
φ(+a, x) i = N/2 + 1, . . .Na ∈ R
N/2 identical solitons
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Standing waves: α = 0 (Kirchhoff vertex)
N odd
N even
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Variational properties
Minimize the energy at constant mass
infE [Ψ] s.t. Ψ ∈ E ,M[Ψ] ≡ ||Ψ||2 = m (3)
Recall that
D(E) = Ψ ∈ H1(G) s.t. ψ1(0) = ... = ψN (0)
and
E(Ψ) =1
2‖Ψ′‖2 + α|ψ1(0)|2 − 1
2µ+ 2‖Ψt‖2µ+2
2µ+2
Theorem (Work in progress)
Let µ > 0, α < α∗ < 0, ω > α2
N2 . Then the minimum problem (3) attains a
solution coinciding with the N tail state Φ0ω
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Variational properties
A technical difficulty: use of symmetric decreasing rearrangements Ψ∗ of Ψ toreduce the minimization problem to the set of symmetric states and prove theexistence of the minimum.
Proposition (Modified Polya-Szego inequality)
Assume that Ψ ∈ D(E). Then ‖Ψ∗‖Lp (G) = ‖Ψ‖Lp (G), for any 1 6 p 6∞, and
‖Ψ∗′‖2 6 N2
4‖Ψ′‖2
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Variational properties
A simple example.
Same L∞ and L1-norms, slopes grow to have ‖Ψ∗′‖2 = N2
4‖Ψ′‖2.
‖Ψ′‖2 = 2 ; ‖Ψ∗′‖2 =9
2
so that
‖Ψ∗′‖2 =N2
4‖Ψ′‖2 (N = 3)
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Variational properties: energy minima (N=3, Cubic NLS, Kirchhoffvertex)
It turns out that for cubic nonlinearity, α = 0 and N = 3 the energy does notattain a minimum value on the set of states with fixed mass
The stationary state is indeed a saddle point
Take Ψ ∈ D(E) such that ‖Ψ‖2 = m
Assume ‖ψ1‖2L2(R+) 6 ‖ψ2‖2
L2(R+) 6 ‖ψ3‖2L2(R+) and set
m1 = ‖ψ1‖2L2(R+) ; m2 = ‖ψ2‖2
L2(R+) + ‖ψ3‖2L2(R+)
Define
Φxm1,m2
(x1, x2, x3) :=
m1√
2sech
(m12x1
)m2
2√
2sech
(m24
(x2 − x))
m2
2√
2sech
(m24
(x3 + x))
where 0 < 2m1 6 m2, x ≥ 0, and the following condition of “continuity at thevertex” holds:
m1 =m2
2sech
(m2
4x)
moreoverm1 + m2 = m
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Variational properties: energy minima (N=3, Cubic NLS, Kirchhoffvertex)
ThenE(Ψ) ≥ E(Φx
m1,m2)
and E is minimized on states of the form Φxm1,m2
Moreover E(Φxm1,m2
) is increasing in m1 and limm1→0 E(Φxm1,m2
) = −m3
96then
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Variational properties: energy minima (N=3, Cubic NLS, Kirchhoffvertex)
Theorem
For any Ψ ∈ D(E) such that ‖Ψ‖2 = m, the following inequality holds
E(Ψ) > −m3
96.
The infimum cannot be reached, as for m1 = 0 the conditions
m1 =m2
2sech
(m2
4x)
m1 + m2 = m
do not correspond to an admissible state.
The state Φxm1,m2
with x = 0, m1 = m3
and m2 = 2m3
is a minimum of theenergy on the manifold of states with Ψ1 = Ψ2 = Ψ3, and it is a maximum onthe manifold of states of the form Φx
m1,m2. Then it is a saddle point.
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Summary on NLS on graphs
- Summary
Analysis of the scattering of fast solitons
Existence of stationary states
Proof of the nonexistence of the energy minimum (Kirchhoff graph)
- Work in progress
Minimization of the energy functional for fixed mass(concentration-compactess)
Existence of the ground state and characterization as minimizer of theaction functional
Stability of the ground state (Grillakis-Shatah-Strauss)
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
References
R. Adami, C. C., D. Finco, and D. Noja, arXiv:1202.2890 [math-ph] (to appear
on J.Phys. A).
R. Adami, C. C., D. Finco, and D. Noja, arXiv:1104.3839 [math-ph].
T. Cazenave, Semilinear Schrodinger equations, Courant Lecture Notes inMathematics, AMS, vol 10, Providence, 2003.
Fukuizumi R., Ohta M., Ozawa T.: Ann. I.H.Poincare, AN, 25, 837-845 (2008)
Grillakis M., Shatah J., Strauss W., J.Funct.Anal., 74, 160-197 (1987)
Vakhitov M.G., Kolokolov A.A., Radiophys. Quantum Electron 16, 783 (1973)
Weinstein M., Comm.Pure Appl.Math. 39, 51-68 (1986)
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Vertex coupling approximation
Any unitary N × N matrix, U, defines a selfadjoint Laplacian on G, −∆UG , by
the boundary conditions
(U − 1)
ψ1(v)...
ψN (v)
+ i(U + 1)
ψ′1(v)...
ψ′N (v)
= 0
Problem: Which unitary matrices U define “good” conditions in the vertex?
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Vertex coupling approximation
Consider a manifold Ω surrounding the graph G
Define −∆Ω (boundary conditions on ∂Ω)
Analyze the convergence of −∆Ω to −∆UG as Ω collapses onto G
(convergence of the spectrum, resolvent convergence, convergence of thesolution of a Cauchy problem, ...)
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Vertex coupling approximation
Consider a manifold Ω surrounding the graph GDefine −∆Ω (boundary conditions on ∂Ω)
Analyze the convergence of −∆Ω to −∆UG as Ω collapses onto G
(convergence of the spectrum, resolvent convergence, convergence of thesolution of a Cauchy problem, ...)
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Vertex coupling approximation
Consider a manifold Ω surrounding the graph GDefine −∆Ω (boundary conditions on ∂Ω)
Analyze the convergence of −∆Ω to −∆UG as Ω collapses onto G
(convergence of the spectrum, resolvent convergence, convergence of thesolution of a Cauchy problem, ...)
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Vertex coupling approximation
Neumann boundary conditions on ∂Ω:M. I. Freidlin and A. D. Wentzel ’93; G. Raugel ’95; S. Kosugi ’00;P. Kuchment and H. Zeng ’01; J. Rubinstein and M. Schatzman ’01; Y. Saito’01; P. Exner and O. Post ’05; O. Post ’06; A. I. Bonciocat ’08.
The essential spectrum of −∆Ω is [0,∞]
Let ψ be a function in D(−∆Ω) with finite energy, ‖ψ‖H1(Ω) 6 C then:
in the edges, far away from the vertex region, ψ is essentially constant inthe transverse direction
in vertex region ψ is essentially constant
In the limit one gets Kirchhoff conditions in the vertex
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Vertex coupling approximation
Dirichlet boundary conditions on ∂Ω:O. Post ’05; S. Molchanov and B. Vainberg ’06; D. Grieser ’07; S. Albeverio,C. C., and D. Finco ’07; C. C. and P. Exner ’07; C. C. and D. Finco ’08;G. Dell’Antonio and E. Costa ’10.
The essential spectrum of −∆Ω is[π2
d2 ,+∞)
where d is the diameter of
the tubes, d → 0
A rescaling is needed −∆Ω → −∆Ω − π2
d2
The operator −∆Ω− π2
d2 can have eigenvalues which go to −∞ as d → 0
No intuition for the shape of the function in the vertex region
The conditions in the vertex depend on the properties of theapproximating manifold
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Waveguide Collapsing onto a Graph
=⇒
Dirichlet Laplacian on Ω: D(−∆DΩ ) :=
ψ ∈ H2(Ω) s.t. ψ
∣∣∂Ω
= 0
The Waveguide Ω
C : R→ R2 C(s) := (γ1(s), γ2(s))| s ∈ R ; γ′1(s)2 + γ′2(s)2 = 1
C has no self-intersectionsSigned curvature: γ(s) := γ′2(s)γ′′1 (s)− γ′1(s)γ′′2 (s) ; γ ∈ C∞0 (R)
Ω :=
(x , y) ∈ R2∣∣∣ (x , y) = (γ1(s), γ2(s)) + un(s), ∀s ∈ R, u ∈ (−1, 1)
n(s) = (−γ′2, γ′1) is the vector (of unit norm) orthogonal to CΩ is a waveguide of constant width (Supp |γ| < 1)
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Waveguide Collapsing onto a Graph
Scaling of Ω: ε is a “small” dimensionless parameter
Global System of Coordinates: s ∈ R, u ∈ (0, 1)
Scaling of the Width: u → εau a > 3Scaling of the Curvature:
γ(s)→ γε(s) :=1
εγ( s
ε
); θ =
∫Rγ(s)ds =
∫Rγε(s)ds = θε
Scaling of the Waveguide: Ω→ Ωε
Ωε is net of waveguides which collapses onto a “prototypical” graph asε→ 0Two length scales
A small one εa (a > 3), Transversal “Fast Dynamics”A large one ε, Longitudinal “Slow Dynamics”
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Waveguide Collapsing onto a Graph
Scaling of Ω: ε is a “small” dimensionless parameter
Global System of Coordinates: s ∈ R, u ∈ (0, 1)Scaling of the Width: u → εau a > 3
Scaling of the Curvature:
γ(s)→ γε(s) :=1
εγ( s
ε
); θ =
∫Rγ(s)ds =
∫Rγε(s)ds = θε
Scaling of the Waveguide: Ω→ Ωε
Ωε is net of waveguides which collapses onto a “prototypical” graph asε→ 0Two length scales
A small one εa (a > 3), Transversal “Fast Dynamics”A large one ε, Longitudinal “Slow Dynamics”
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Waveguide Collapsing onto a Graph
Scaling of Ω: ε is a “small” dimensionless parameter
Global System of Coordinates: s ∈ R, u ∈ (0, 1)Scaling of the Width: u → εau a > 3Scaling of the Curvature:
γ(s)→ γε(s) :=1
εγ( s
ε
); θ =
∫Rγ(s)ds =
∫Rγε(s)ds = θε
Scaling of the Waveguide: Ω→ Ωε
Ωε is net of waveguides which collapses onto a “prototypical” graph asε→ 0Two length scales
A small one εa (a > 3), Transversal “Fast Dynamics”A large one ε, Longitudinal “Slow Dynamics”
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Waveguide Collapsing onto a Graph
Scaling of Ω: ε is a “small” dimensionless parameter
Global System of Coordinates: s ∈ R, u ∈ (0, 1)Scaling of the Width: u → εau a > 3Scaling of the Curvature:
γ(s)→ γε(s) :=1
εγ( s
ε
); θ =
∫Rγ(s)ds =
∫Rγε(s)ds = θε
Scaling of the Waveguide: Ω→ Ωε
Ωε is net of waveguides which collapses onto a “prototypical” graph asε→ 0Two length scales
A small one εa (a > 3), Transversal “Fast Dynamics”A large one ε, Longitudinal “Slow Dynamics”
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Waveguide Collapsing onto a Graph
Scaling of Ω: ε is a “small” dimensionless parameter
Global System of Coordinates: s ∈ R, u ∈ (0, 1)Scaling of the Width: u → εau a > 3Scaling of the Curvature:
γ(s)→ γε(s) :=1
εγ( s
ε
); θ =
∫Rγ(s)ds =
∫Rγε(s)ds = θε
Scaling of the Waveguide: Ω→ Ωε
Ωε is net of waveguides which collapses onto a “prototypical” graph asε→ 0
Two length scalesA small one εa (a > 3), Transversal “Fast Dynamics”A large one ε, Longitudinal “Slow Dynamics”
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Waveguide Collapsing onto a Graph
Scaling of Ω: ε is a “small” dimensionless parameter
Global System of Coordinates: s ∈ R, u ∈ (0, 1)Scaling of the Width: u → εau a > 3Scaling of the Curvature:
γ(s)→ γε(s) :=1
εγ( s
ε
); θ =
∫Rγ(s)ds =
∫Rγε(s)ds = θε
Scaling of the Waveguide: Ω→ Ωε
Ωε is net of waveguides which collapses onto a “prototypical” graph asε→ 0Two length scales
A small one εa (a > 3), Transversal “Fast Dynamics”A large one ε, Longitudinal “Slow Dynamics”
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Reduction to a One Dimensional Dynamics
The operator −∆DΩε is unitarily equivalent to Hε in L2(R× (0, 1))
Hε = − ∂
∂s
1
(1 + εa−1uγ(s/ε))2
∂
∂s− 1
ε2a∂2
∂u2+
1
ε2Wε(s, u) ,
Wε(s, u) = −γ(s/ε)2
4+O(εa−1)
ψ ∈ D(Hε) ⊂ L2(R× (0, 1)) =⇒ ψ∣∣
u=0,1= 0
Dimensional reduction(φ0,
(Hε −
π2
ε2a
)φ0
)L2((0,1))
'(− d2
ds2− 1
ε2γ2(s/ε)
4
)with
φ0(u) =
√2
εasin(πu/εa)
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Reduction to a One Dimensional Dynamics
The operator −∆DΩε is unitarily equivalent to Hε in L2(R× (0, 1))
Hε = − ∂
∂s
1
(1 + εa−1uγ(s/ε))2
∂
∂s− 1
ε2a∂2
∂u2+
1
ε2Wε(s, u) ,
Wε(s, u) = −γ(s/ε)2
4+O(εa−1)
ψ ∈ D(Hε) ⊂ L2(R× (0, 1)) =⇒ ψ∣∣
u=0,1= 0
Dimensional reduction(φ0,
(Hε −
π2
ε2a
)φ0
)L2((0,1))
'(− d2
ds2− 1
ε2γ2(s/ε)
4
)with
φ0(u) =
√2
εasin(πu/εa)
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Analysis of the One Dimensional Problem
Definition (Zero energy resonance)
Assume that ec|·|v ∈ L1(R) for some c > 0. We say that the Hamiltonian
h = − d2
ds2+ v(s)
has a zero energy resonance if there exists ψr ∈ L∞(R), ψr /∈ L2(R) such thathψr = 0 in distributional sense
If h has a zero energy resonance we define
ρ1 = limx→−∞
ψr (x) and ρ2 = limx→+∞
ψr (x)
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Main Theorem
Theorem (Albeverio, C.C., Finco ’07)
Assume that γ ∈ C∞0 (R) and take a > 3, then two cases can occur:
1. H = − d2
ds2 − γ2/4 has no zero energy resonances(φ0,
(Hε −
π2
ε2a
)φ0
)L2((0,1))
ε→0−−−→ −∆DG
−∆DG f := (−f ′′1 ,−f ′′2 ) ; f1(0) = f2(0) = 0
2. H = − d2
ds2 − γ2/4 has a zero energy resonance ψr(φ0,
(Hε −
π2
ε2a
)φ0
)L2((0,1))
ε→0−−−→ −∆ρ1ρ2G
−∆ρ1ρ2G f := (−f ′′1 ,−f ′′2 ) ;
ρ2f1(0) = ρ1f2(0)
ρ1f′
1 (0) + ρ2f′
2 (0) = 0
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Conclusions
Dirichlet case (Albeverio, C.C., Finco ’07)the result does not change if the projection is taken on excited transversestates
Generic case: no zero energy resonance, decoupling conditionsNon-generic case: existence of a zero energy resonance, coupling betweenthe edgesSmall perturbations of the geometry (C.C., Exner ’07)
γε(s) :=
√1 + λε
εγ( s
ε
)λ ∈ R
θε =
(1 +
λ
2ε
)θ +O(ε2)
In the non generic case the parameter λ enters into the definition of theboundary conditions of the operator on the graph
ρ2f1(0) = ρ1f2(0)
ρ1f ′1 (0) + ρ2f ′2 (0) =λ
2
(ρ1f1(0) + ρ2f2(0)
)λ := −λ
∫ ∞−∞
γ(s)2
4|ψr (s)|ds
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Conclusions
Dirichlet case (Albeverio, C.C., Finco ’07)the result does not change if the projection is taken on excited transversestatesGeneric case: no zero energy resonance, decoupling conditionsNon-generic case: existence of a zero energy resonance, coupling betweenthe edges
Small perturbations of the geometry (C.C., Exner ’07)
γε(s) :=
√1 + λε
εγ( s
ε
)λ ∈ R
θε =
(1 +
λ
2ε
)θ +O(ε2)
In the non generic case the parameter λ enters into the definition of theboundary conditions of the operator on the graph
ρ2f1(0) = ρ1f2(0)
ρ1f ′1 (0) + ρ2f ′2 (0) =λ
2
(ρ1f1(0) + ρ2f2(0)
)λ := −λ
∫ ∞−∞
γ(s)2
4|ψr (s)|ds
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Conclusions
Dirichlet case (Albeverio, C.C., Finco ’07)the result does not change if the projection is taken on excited transversestatesGeneric case: no zero energy resonance, decoupling conditionsNon-generic case: existence of a zero energy resonance, coupling betweenthe edgesSmall perturbations of the geometry (C.C., Exner ’07)
γε(s) :=
√1 + λε
εγ( s
ε
)λ ∈ R
θε =
(1 +
λ
2ε
)θ +O(ε2)
In the non generic case the parameter λ enters into the definition of theboundary conditions of the operator on the graph
ρ2f1(0) = ρ1f2(0)
ρ1f ′1 (0) + ρ2f ′2 (0) =λ
2
(ρ1f1(0) + ρ2f2(0)
)λ := −λ
∫ ∞−∞
γ(s)2
4|ψr (s)|ds
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Conclusions
Neumann case (C.C., Finco ’10):
ψ ∈ D(−∆NΩ) , ∂nψ|∂Ω = 0
Reduction of the dynamics with respect to the ground transverse state:standard condition in the vertex
(φN0 ,H
Nε φ
N0 )L2((−1,1)) ' −
d2
ds2
Reduction of the dynamics with respect to excited transverse states:decoupling conditions in the vertex (no possibility of zero energyresonances!)
(φNn ,H
Nε φ
Nn )L2((−1,1)) '
(−
d2
ds2+
3
4
1
ε2γ2(s/ε) +
E Nn
ε2a
)n > 1
Robin case (C.C., Finco ’10): ψ ∈ D(−∆RΩ), ∂nψ|∂Ω +αψ|∂Ω = 0, α ∈ R
Dependence on the transverse energyExistence of a generic and non-generic case
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Conclusions
Neumann case (C.C., Finco ’10):
ψ ∈ D(−∆NΩ) , ∂nψ|∂Ω = 0
Reduction of the dynamics with respect to the ground transverse state:standard condition in the vertex
(φN0 ,H
Nε φ
N0 )L2((−1,1)) ' −
d2
ds2
Reduction of the dynamics with respect to excited transverse states:decoupling conditions in the vertex (no possibility of zero energyresonances!)
(φNn ,H
Nε φ
Nn )L2((−1,1)) '
(−
d2
ds2+
3
4
1
ε2γ2(s/ε) +
E Nn
ε2a
)n > 1
Robin case (C.C., Finco ’10): ψ ∈ D(−∆RΩ), ∂nψ|∂Ω +αψ|∂Ω = 0, α ∈ R
Dependence on the transverse energyExistence of a generic and non-generic case
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Conclusions
Neumann case (C.C., Finco ’10):
ψ ∈ D(−∆NΩ) , ∂nψ|∂Ω = 0
Reduction of the dynamics with respect to the ground transverse state:standard condition in the vertex
(φN0 ,H
Nε φ
N0 )L2((−1,1)) ' −
d2
ds2
Reduction of the dynamics with respect to excited transverse states:decoupling conditions in the vertex (no possibility of zero energyresonances!)
(φNn ,H
Nε φ
Nn )L2((−1,1)) '
(−
d2
ds2+
3
4
1
ε2γ2(s/ε) +
E Nn
ε2a
)n > 1
Robin case (C.C., Finco ’10): ψ ∈ D(−∆RΩ), ∂nψ|∂Ω +αψ|∂Ω = 0, α ∈ R
Dependence on the transverse energyExistence of a generic and non-generic case
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
References
Dirichlet boundary:
O. Post, Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case, J. Phys.
A: Math. Gen. 38 (2005), no. 22, 4917–4931.
G. Dell’Antonio and L. Tenuta, Quantum graphs as holonomic constraints, J. Math. Phys. 47 (2006),
072102.
S. Albeverio, C. C., and D. Finco, Coupling in the singular limit of thin quantum waveguides, J. Math.
Phys. 48 (2007), 032103.
C. C. and P. Exner, Nontrivial edge coupling from a Dirichlet network squeezing: the case of a bent
waveguide, J. Phys. A: Math. Theor. 40 (2007), no. 26, F511–F523.
C. C. and D. Finco, Graph-like models for thin waveguides with Robin boundary conditions,
arxiv:0803.4314 [math-ph] (2008), 27pp, accepted for publication in Asymptotic Analysis.
S. Molchanov and B. Vainberg, Laplace operator in networks of thin fibers: spectrum near the threshold,
Stochastic analysis in mathematical physics, Proceedings of a Satellite Conference of ICM 2006 (2006),69–93, edited by G. Ben Arous, A.-B. Cruzeiro, Y. Le Jan, J.-C. Zambrini.
S. Molchanov and B. Vainberg, Transition from a network of thin fibers to the quantum graph: an
explicitly solvable model, Contemp. Math., AMS 415 (2006), 227–239.
S. Molchanov and B. Vainberg, Scattering solutions in a network of thin fibers: small diameter
asymptotics, Commun. Math. Phys. 273 (2007), 533–559.
D. Grieser, Spectra of graph neighborhoods and scattering, Proc. London Math. Soc. 97 (2008), no. 3,
718–752.
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
References
Dirichlet boundary (continues):
M. Harmer, B. Pavlov, and A. Yafyasov, Boundary conditions at the junction, J. Comput. Electron. 6
(2007), 153–157.
G. Dell’Antonio and E. Costa, Effective Schroedinger dynamics on ε-thin Dirichlet waveguides via
Quantum Graphs I: star-shaped graphs, arXiv:1004.4750 [math-ph] (2010), 23pp.
S. Albeverio and S. Kusuoka, Diffusion processes in thin tubes and their limits on graphs, (2010).
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
References
Neumann boundary:
M. I. Freidlin and A. D. Wentzel, Diffusion processes on graphs and averaging principle, Ann. Probab. 21
(1993), no. 4, 2215–2245.
S. Kosugi, A semilinear elliptic equation in a thin network-shaped domain, J. Math. Soc. Japan 52 (2000),
no. 3, 673–697.
Y. Saito, The limiting equation for Neumann Laplacians on shrinking domains, Electron. J. Diff. Equations
2000 (2000), no. 31, 1–25.
Y. Saito, Convergence of the Neumann Laplacian on shrinking domains, Analysis (Munich) 21 (2001),
no. 2, 171–204.
J. Rubinstein and M. Schatzman, Variational problems on multiply connected thin strips. I. Basic estimates
and convergence of the Laplacian spectrum, Arch. Ration. Mech. Anal. 160 (2001), no. 4, 271–308.
P. Kuchment and H. Zeng, Convergence of spectra of mesoscopic systems collapsing onto a graph, J.
Math. Anal. Appl. 258 (2001), no. 2, 671–700.
K. Kuwae and T. Shioya, Convergence of spectral structures: a functional analytic theory and its
applications to spectral geometry, Comm. Anal. Geom. 11 (2003), no. 4, 599–673.
G. Raugel, Dynamics of partial differential equations on thin domains, Dynamical systems, Lecture Notes
in Math. 1609 (1995), 208–315.
P. Exner and O. Post, Convergence of spectra of graph-like thin manifolds, J. Geom. Phys. 54 (2005),
77–115.
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
References
Neumann boundary (continues):
O. Post, Spectral convergence of quasi-one-dimensional spaces, Ann. Henri Poincare 7 (2006), 933–973.
P. Exner and O. Post, Convergence of resonances on thin branched quantum waveguides, J. Math. Phys.
48 (2007), 092104, 43pp.
A. I. Bonciocat, Curvature bounds and heat kernels: discrete versus continuous spaces, Ph.D. Thesis,
Universitat Bonn, http://hss.ulb.uni-bonn.de:90/2008/1497/1497.htm, 2008.
P. Exner and O. Post, Approximation of quantum graph vertex couplings by scaled Schrodinger operators
on thin branched manifolds, J. Phys. A: Math. Theo. 42 (2009), 415305, 22pp.
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Variational properties
We want to identify the ground state of the system among the finite energystates
E =
Ψ ∈ H1(G) s.t. ψ1(0) = · · · = ψN (0)
Recall that
E(Ψ) =1
2‖Ψ′‖2 + α|ψ1(0)|2 − 1
2µ+ 2‖Ψt‖2µ+2
2µ+2
The stationary states of the NLS on graphs turn out to be critical points(S ′ω[Ψ] = 0 coincides with the stationarity equation) of the so called actionfunctional Sω
Sω[Ψ] = E [Ψ] +ω
2||Ψ||22 .
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Variational properties
The action Sω[Ψ] is unbounded from below in E , but stationary states arecontained in the manifold
Ψ ∈ E s.t. Iω[Ψ] ≡ 〈S ′ω[Ψ],Ψ〉 = 0
Ground states, if they exist, are solutions of the following constrainedminimization problem
inf Sω[Ψ] s.t. Ψ ∈ E , Iω[Ψ] = 0 ≡ d(ω)
Theorem
Let µ > 0, α < α∗ < 0, ω > α2
N2 . Then the minimum problem for Sω[Ψ]constrained on the natural manifold Iω[Ψ] = 0 attains a solution coincidingwith the N tail state Φ0
ω
In this sense we consider the N tail state the ground state
Other examples of minimization for NLS on the line with point interactionsyeld similar problems (Fukuizumi-Ohta-Ozawa 2008, Fukuizumi-Jeanjean2008, Adami-Noja 2011)
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Variational properties
A tecnical difficulty: use of symmetric decreasing rearrangements Ψ∗ of Ψ toreduce to the halfline and calculate the exact value of the inf Sω[Ψ] in theKirchhoff case, needed in the proof of existence of ground state.
Theorem (Modified Polya-Szego inequality)
Assume that Ψ ∈ E . Then ‖Ψ∗‖Lp (G) = ‖Ψ‖Lp (G), for any 1 6 p 6∞, and
‖Ψ∗‖2H1(G) 6
N2
4‖Ψ‖2
H1(G)
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Variational properties
A simple example.
Same L∞ and L1-norms, slopes grow to have ‖Ψ∗‖2H1(G) 6
N2
4‖Ψ‖2
H1(G).
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Variational properties: energy minima
A slightly different variational problem: Minimize the energy at constant mass
infE [Ψ] s.t. Ψ ∈ E ,M[Ψ] ≡ ||Ψ||22 = m
Recall thatE = Ψ ∈ H1(G) s.t. ψ1(0) = ... = ψN (0)
and
E(Ψ) =1
2‖Ψ′‖2 + α|ψ1(0)|2 − 1
2µ+ 2‖Ψt‖2µ+2
2µ+2
This second variational problem can be studied using the concentrationcompactness method of P. L. Lions (around 1982, problems in Rn).In the case of a star graphs there is no translation invariance, and the c.c.method must be adapted.
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Variational properties: energy minima (N=3, Cubic NLS, Kirchhoffvertex)
It turns out that for cubic nonlinearity, α = 0 and N = 3 the energy does notattain a minimum value on the set of states with fixed mass
The stationary state is indeed a stationary point
Take Ψ ∈ E such that ‖Ψ‖2 = m
Assume ‖ψ1‖2L2(R+) 6 ‖ψ2‖2
L2(R+) 6 ‖ψ3‖2L2(R+) and set
m1 = ‖ψ1‖2L2(R+) ; m2 = ‖ψ2‖2
L2(R+) + ‖ψ3‖2L2(R+)
Define
Φxm1,m2
(x1, x2, x3) :=
m1√
2sech
(m12x1
)m2
2√
2sech
(m24
(x2 − x))
m2
2√
2sech
(m24
(x3 + x))
where 0 < m1 6 m2, x ≥ 0, and the following condition of “continuity at thevertex” holds:
m1 =m2
2sech
(m2
4x)
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Variational properties: energy minima (N=3, Cubic NLS, Kirchhoffvertex)
It turns out thatE(Ψ) ≥ E(Φx
m1,m2)
Then E attains its minimum on states of the form Φxm1,m2
Moreover E(Φxm1,m2
) is increasing in m1 and limm1→0 E(Φxm1,m2
) = −m3
96then
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Variational properties: energy minima (N=3, Cubic NLS, Kirchhoffvertex)
Theorem
For any Ψ ∈ H1(G) such that ‖Ψ‖2 = m, the following inequality holds
E(Ψ) > −m3
96.
The infimum cannot be achieved, as for m1 = 0 the condition
m1 =m2
2sech
(m2
4x)
does not correspond to an admissible state.
The state Φxm1,m2
with x = 0, m1 = m3
and m2 = 2m3
is a minimum of theenergy on the manifold of states with Ψ1 = Ψ2 = Ψ3, and it is a maximum onthe manifold of states of the form Φx
m1,m2. Then it is a saddle point.
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Orbital stability of the ground state
Due to the U(1) invariance of the dynamics, the stability has to be consideredas orbital stability.The orbit of Φω is defined as O(Φω) = e iθΦω(x), θ ∈ R.
The state Φω is orbitally stable if for every ε > 0 there exists δ > 0 such that
d(Ψ(0),O(Φω)) < δ ⇒ d(Ψ(t),O(Φω)) < ε ∀t > 0
whered(ψ,O(Φω)) = inf
u∈O(Φω)||ψ − u||E
and the norm || · ||E is the energy norm.
We shall use classical results on the analysis of orbital stability of solitarysolutions of nonlinear equations due to Weinstein, Grillakis-Shatah-Strauss(’80, KG, NLS, other equations).
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Orbital stability of the ground state
The NLS on a graph turns out to be a hamiltonian system on the real Hilbertspace of the couples of real and imaginary part of the wavefunction.Decomposing Ψ = u + iv , one obtains the canonical system
d
dt
(uv
)= JE ′[u, v ] , J =
(0 I−I 0
)where the Hamiltonian E coincides with the energy up to substitution of realvariables, E ≡ E [u, v ].Linearization of the hamiltonian system around the stationary state is achievedby posing
(Ψt)k = (Φ0ω,k + ηk + iρk )e iωt
and neglecting higher order terms than linear. The η and ρ satisfy
d
dt
(ηρ
)= JL
(ηρ
).
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Orbital stability of the ground state
OperatorL = diag(L−,L+)
and
(L+)i,k =
(− d2
dx2+ ω − |Φ0
ω,k |2µ)δi,k
(L−)i,k =
(− d2
dx2+ ω − (2µ+ 1)|Φ0
ω,k |2µ)δi,k .
L− and L+ are matrix self adjoint operators acting on the real vectorfunctions η and ρ belonging to D(Hα).
The Weinstein and GSS theory implies that solitary solutions are orbitallystable if
i) spectral conditions hold:i1) kerL+ = Φ0
ω and the rest of the spectrum is positive;i2) n(L−) = 1 where the left hand side is the number of negative eigenvalues.
ii) Vakhitov-Kolokolov condition ddω‖Φ0
ω‖22 > 0 holds.
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Orbital stability of the ground state
Theorem
Let µ ∈ [0, 2], α < α∗ < 0, ω > α2
N2 .
Then the ground state Φ0ω is orbitally stable in H1(G)
Note that from Vakhitov-Kolokolov condition, ddω‖Φ0
ω‖22 > 0, it follows that
for µ > 2 there exists ω∗ such that Ψ0ω is orbitally stable for ω ∈ (α
2
N2 , ω∗)
and is orbitally unstable for ω > ω∗ .
Introduction
MetricGraphs
Laplacian(s)on graphs
NLS ongraphs
Propagationof pulses innetworks
Standingwaves
Squeezingof fatmanifolds
Conclusions
There exist nonlinear standing waves for the NLS on a star graphs
The standing waves are explicit in some relevant case (δ star graph,including Kirchhoff, and others not treated here)
The ground state is variationally characterized
It is (orbitally) stable in the case of attractive δ star graph