PhD thesis presentation of Nguyen Bich Van

NLS Normal form Matrices Non degeneracy

Characteristic polynomials, associated to the
energy graph of the non–linear Schrödinger
equation

Nguyen Bich Van

PhD thesis defense
Sapienza università di Roma, 17–12–2012


In my thesis I have studied characteristic polynomials, associated
by some rules to a class of marked graphs.
Example
graph G =
1,2
a b
4,1 2,3
4,3
d c
Matrix
 √ √ 
0 −2 ξ1 ξ2 0 −2 ξ4 ξ1
−2√ξ ξ −
√
ξ2√ ξ1 −2 ξ2 ξ3 0 
1 2 √
CG = 
 
0 −2 ξ2 ξ3 −ξ1 + ξ3 −2 ξ4 ξ3 

√ √

−2 ξ4 ξ1 0 −2 ξ4 ξ3 ξ4 − ξ1


With the characteristic polynomial:

χG = det(tI − CG ) =
= −4ξ1 ξ2 + 4ξ1 ξ2 ξ3 − 4ξ1 ξ 4 + 8ξ1 ξ2 ξ4 + 4ξ1 ξ3 ξ4 − 8ξ1 ξ2 ξ3 ξ4 +
3 2 3 2 2

3 2 2 2
+(ξ1 −9ξ1 ξ2 −ξ1 ξ3 +ξ1 ξ2 ξ3 −9ξ1 ξ4 +9ξ1 ξ2 ξ4 +ξ1 ξ3 ξ4 +7ξ2 ξ3 ξ4 )t+
+ (3ξ1 − 6ξ1 ξ2 − 2ξ1 ξ3 − 3ξ2 ξ3 − 6ξ1 ξ4 + ξ2 ξ4 − 3ξ3 ξ4 )t 2 +
2

+ (3ξ1 − ξ2 − ξ3 − ξ4 )t 3 + t 4 . (1)

The problem is
to prove that a rather complicated inﬁnite list of such polynomials
in a variable t, of degree increasing with the graph dimension, and
with coeﬃcients polynomials in the parameters ξi have distinct
roots for generic values of the parameters.


This is a combinatorial algebraic problem which arises from the
study of a normal form for the nonlinear Schrödinger equation on a
torus.
In my thesis I have solved completely this problem
by showing a stronger property(separation an irreducibility) of
these polynomials.


The plan of the talk:

1 Normal forms of NLS
2 Construction of colored marked graphs and matrices
3 Separation and irreducibility of characteristic polynomials


The NLS

The Nonlinear Schrödinger equation

Normal forms


Nonlinear Schrödinger equation

Consider the Nonlinear Schrödinger equation (NLS for short) on
the torus Tn .

iut − ∆u = κ|u|2q u, q = 1, 2, . . . (2)

where u := u(t, ϕ), ϕ ∈ Tn .
-The NLS describes how the wavefunction of a physical system
evolves over time.
-The case q = 1 is associated to the cubic NLS.
-When κ = 0, this is the linear Schrödinger equation. It has many
PERIODIC solutions.


The cubic NLS in dimension 1 is completely integrable and several
explicit solutions are known. In higher dimensions we loose the
complete integrability and all techniques associated to it, but we
can still use the following well-known fact
The NLS (2) can be written as an inﬁnite dimensional Hamiltonian
dynamical system u = {H, u},
˙
where the symplectic variables are Fourier coeﬃcients of the
functions
u(t, ϕ) = uk (t)e i(k,ϕ) . (3)
k∈Zn

the symplectic form is i k∈Zn duk ∧ d uk and the Hamiltonian is
¯

H := |k|2 uk uk ±
¯ uk1 uk2 uk3 uk4 ...u2q+1 u2q+2
¯ ¯ ¯
k∈Zn k∈Zn :
2q+2
(−1)i ki =0
i=1
(4)
up to rescaling of u.


In order to study the long-time behavior of the solutions of
Hamiltonian PDEs close to an equilibrium
it is necessary start from a suitably non degenerate normal form
and the existence of a such normal form is not obvious for (2).


Theory of Poincare-Birkhoff normal form

Consider a non-linear Hamitonian dynamical system with an elliptic
fixed point at zero, i.e. there exists a canonical system of
coordinates (p, q) such that the Hamiltonian takes the form

H(p, q) = λj (pj2 + qj2 ) + H >2 (p, q) , λj ∈ R
j∈I

here the index set I is finite or possibly denumberable while
H >2 (p, q) is some polynomial with minimal degree > 2.


Normal form reduction

The normal form reduction at order D
is a symplectic change of variables ΨD which reduces H to its
resonant terms:

H(p, q) ◦ ΨD = λj (pj2 + qj2 ) + HRes (p, q) + H D (p, q)
>2

j

>2
where HRes Poisson commutes with j λj (pj2 + qj2 ) while H D (p, q)
is a formal power series of minimal degree > D + 1.

There are two classes of problems in this scheme:
1 Even though H D is of minimal order D + 1 its norm diverges
as D → ∞, due to the presence of small divisors.
2 If I is an inﬁnite set it is not trivial, even when D = 1, to
show that ΨD is an analytic change of variables.


Remark
If the λj are rationally independent then the normal form
>2
HBirk = j λj (pj2 + qj2 ) + HRes (p, q) is integrable, a feature which
is used in proving for instance long time stability results.
Otherwise HBirk may not be integrable but it is possible that its
dynamics is simpler than the one of the original Hamiltonian.
>2
In the case NLS HBirk = j λj (pj2 + qj2 ) + HRes (p, q) has invariant
tori of the form

pi2 + qi2 = ξi , i ∈ S ⊂ I; pj = qj = 0 , j ∈ Sc = I S (5)

on which the dynamics is of the form ψ → ψ + ω(ξ)t with ω(ξ) a
diﬀeomorphism.
S is called the tangential sites, S c -the normal sites.


In order to obtain information on the solutions of the complete
Hamiltonian close to these tori one needs to study the Hamilton
equations of H linearized at a family of invariant tori. In terms of
equations this is described by a quadratic Hamiltonian with
coeﬃcients depending on the parameters ξ and on the angle
variables of the tori.
The matrix obtained by linearizing HBirk at the solutions (5)
is referred to as the normal form matrix (or normal form).


Stability for the NLS

In a recent work [1] M. Procesi and C. Procesi constructed a
normal form for the NLS.
This normal form of the NLS is described by an infinite dimensional
Hamiltonian which determines a linear operator ad(N) = {N, ∗}
(Poisson bracket), depending on a finite number of parameters ξi
(the actions of certain excited frequencies), and acting on a certain
infinite dimensional vector space F (0,1) of functions.
Stability for this infinite dimensional operator
will be interpreted in the same way as it appears for finite
dimensional linear systems, that is the property that the linear
operator is semisimple with distinct eigenvalues.


The normal form matrix is infinite dimensional. But the condition
of its semisimplicity makes at all sense because it decomposes into
an infinite direct sum of finite dimensional blocks.

Figure : The normal form matrix

We need to show that these finite dimensional matrix blocks have
distinct eigenvalues.


In my thesis I have proved:
Theorem
For generic choices of tangential sites S and parameters ξ the
normal form N constructed in [1] in the case of cubic NLS in all
dimensions is non-degenerate in the sense that it is semisimple
with non-zero and distinct eigenvalues. The same result for all
higher degree NLS in dimension 1 and 2.

The problem arises from the study of NLS, but one could
formulate it as a purely algebraic question. And in fact the proof is
essentially combinatorial and algebraic in nature.


The matrices

Matrix blocks

Graphs


Spaces V 0,1 , F 0,1 on which the normal form acts

Let S = {v1 , ..., vm } be the tangential sites, S c = Zn S be the
normal sites.
We start from the space V 0,1 of functions with basis the elements
i νx −i j νj xj
{e j j j zk , e zk }, k ∈ S c .
¯
In this space the conditions of commuting with momentum, resp.
with mass select the elements, called frequency basis

i νj xj −i νj xj
FB = {e j zk , e j zk ,
¯ k ∈ S c }; k ∈ S c
νj vj + k = π(ν) + k = 0 resp. νj + 1 = 0. (6)
j j

Denote by F 0,1 the subspace of V 0,1 commuting with momentum
and mass.


Cayley graph

We recall how we describe the operator ad(N) = {N, ∗} into the
language of group theory and in particular of the Cayley graph.
In fact to a matrix C = (ci,j )
we can always associate a graph, with vertices the indices of the
matrix, and an edge between i, j if and only if ci,j = 0.

Thus the indecomposable blocks of the matrix will be associated to
connected components of a graph.
For the matrix of ad(N) in the frequency basis the relevant graph
comes from a special Cayley graph.
From now for simplicity of notations we will write formulas for the
cubic NLS. For higher degree NLS the formulas and combinatorics
are similar but more complicated.


Cayley graph

Let G be a group and X = X −1 ⊂ G. Consider an action
G × A → A of a group G on a set A, we then deﬁne.
Deﬁnition (Cayley graph)
The graph AX has as vertices the elements of A and, given
x /
a, b ∈ A we join them by an oriented edge a b , marked x , if
b = xa, x ∈ X .


Set Zm = { m ai ei , ai ∈ Z}-the lattice with basis elements ei . In
i=1
our setting the relevant group is the group G := Zm Z/(2) the
semidirect product, denote by τ := (0, −1) so G = Zm ∪ Zm τ .

i νx
An element a = e j j j zk is associated to the group element
−i j νj xj
a = j νj ej ∈ Zm . Then a = e
¯ zk is associated to the
¯
group element aτ = ( j νj ej )τ ∈ Zmτ .

Thus the frequency basis is indexed by elements of
G 1 m {−ei , −ei τ }, where
i=1

G 1 := {a, aτ, a ∈ Zm | η(a) = −1}.


The matrix structure of ad(N) := 2iM is encoded in part by the
Cayley graph GX of G with respect to the elements

X 0 = {ei −ej , i = j ∈ {1, ..., m}}, X −2 = {(−ei −ej )τ, i = j ∈ {1, ..., m}}

We distinguish the edges by color, as X 0 to be black and X −2 red,
hence the Cayley graph is accordingly colored; by convention we
represent red edges with a double line:
g
g = (−ei − ej )τ, a ga .


Given a = i ai ei , σ = ±1 set for u = (a, σ)
σ
K ((a, σ)) := (| ai vi |2 + ai |vi |2 ). (7)
2 i i

Sometimes we call K (u) the quadratic energy of u.
Deﬁnition
Given an edge u / v , u = (a, σ), v = (b, ρ) = xu, x ∈ Xq , we
x

say that the edge is compatible with S if K (u) = K (v ).


The matrix structure of ad(N) := 2iM:the matrix of the
action of N by Poisson bracket in the frequency basis

We have for a, b ∈ Zm

Ma,a = K (a) − aj ξ j , Maτ,aτ = K (aτ ) + aj ξj (8)
j j

Maτ,bτ = −2 ξi ξj , Ma,b = 2 ξi ξj ,
if a, b are connected by a compatible edge ei − ej (9)

Ma,bτ = −2 ξi ξj , Maτ,b = 2 ξi ξj ,
if a, bτ are connected by a compatible edge (−ei − ej )τ (10)

All other entries are zero.


It was shown in [1] that M decomposes as infinite direct sum of
finite dimensional blocks. With respect to the frequency basis the
blocks are described as the connected components of a graph ΛS
which we now describe. Let π : Zm → Zn , ei → vi . Set
Θ = Ker (π).
Definition
The graph ΛS is the subgraph of G 1 i {−ei + Θ, (−ei + Θ)τ } in
which we only keep the compatible edges.


We then have
Theorem
The indecomposable blocks of the matrix M in the frequency basis
correspond to the connected components of the graph ΛS .
The entries of M are given by (8), (9), (10).

The fact that in the graph ΛS we keep only compatible edges
implies in particular that the scalar part K ((a, σ)) (which is an
integer) is constant on each block. On the other hand, in general,
there are inﬁnitely many blocks with the same scalar part. It will
be convenient to ignore the scalar term diag(K ((a, σ))), given a
compatible connected component A we hence deﬁne the matrix
CA = MA − diag(K ((a, σ))).


The ﬁnal goal

Characteristic polynomials

Irreducibility and separation


One of the main ingredients of our work is to understand the
possible connected components of the graph ΛS , we do this by
analyzing such a component as a translation Γ = Au where A is
some complete subgraph of the Cayley graph containing the
element (0, +) = 0. If u ∈ Zm the matrix CAu is obtained from CA
by adding the scalar matrix −u(ξ) = −(u, ξ).
Example: Consider the following complete subgraph containing
(0, +).

(−e1 −e2 )τ e1 −e2
A = (−e1 − e2 , −) (0, +) / (e1 − e2 , +) .

A translation by an element (u, +) is hence

(−e1 −e2 )τ e1 −e2
A(u, +) = (−e1 − e2 − u, −) (u, +) / (e1 − e2 + u, +)


Example

The matrices associated to these graphs are:
 √ 
−ξ1 − ξ2 2 ξ1 ξ2 0
 
 √ √ 
CA = −2 ξ1 ξ2 0 2 ξ1 ξ2 
 
 
√
 
0 2 ξ1 ξ2 ξ2 − ξ1
 √ 
−ξ1 − ξ2 − u(ξ) 2 ξ1 ξ2 0
 
 √ √ 
CAu =  −2 ξ1 ξ2 −u(ξ) 2 ξ1 ξ2
 

 
√
 
0 2 ξ1 ξ2 ξ2 − ξ1 − u(ξ)


In particular we have shown (cf. [1], §9) that

A can be chosen among a finite number of graphs which we call
combinatorial.
For cubic NLS we have the following Theorem from [2]
Theorem
For generic choices of S the connected components of graph ΛS ,
different from the special component −ei , −ei τ , are formed by
affinely independent points.


We also have (see [2])
Lemma
The characteristic polynomial of each matrix CA is in
Z[ξ1 , . . . , ξm , t] (the roots disappear).

We wish to prove
Outside a countable union of real algebraic hypersurfaces in the
space of parameters
eigenvalues of the matrix CA for connected components A that we
described above are all distinct .
This fact will be useful in [3] in order to prove, by a KAM
algorithm, the existence and stability of quasi–periodic solutions
for the NLS (not just the normal form).


A direct method

In fact eigenvalues of a matrix CG are roots of characteristic
polynomials χG = det(tI − CG ). One should compute
discriminants and resultants of them, which are polynomials in
variables ξi and show that they are not identically zero. This can
be done by direct computations only for small cases. In general
case, even in dimension n = 3, the total number of these
polynomials is quite high (in the order of the hundreds or
thousands) so that the algorithm becomes quickly non practical!
Hence
we will prove that roots of characteristic polynomials are all
distinct by showing a stronger algebraic property of them!


Irreducibility and Separation

Theorem (Separation and Irreducibility)

The characteristic polynomials of blocks of the normal form matrix
are all distinct and irreducible as polynomials with integer
coeﬃcients, that is in Z[ξ1 , . . . , ξm , t] ⊂ Q(ξ1 , . . . , ξm )[t].

Following the fact that an irreducible polynomial f (t) over a ﬁeld
F of characteristic 0 is uniquely determined as the minimal
¯
polynomial of each of its roots (in the algebraic closure F ) and its
derivative f (t) is non-zero, g.c.d(f , f ) = 1 we have
Implication
Outside the countable union of algebraic hypersurfaces in the
space of parameters ξ all eigenvalues are non-zero and distinct.


Proof of separation and irreducibility theorem

For a given polynomial with integer coefficients there exist
reasonable computer algebra algorithms to test irreducibility but
this is not a practical method in our case where the polynomials
are infinite and their degrees also tend to infinity. So we shall use
combinatorics. The fact that the polynomials are distinct is based
by induction on the irreducibility theorem and it is relatively easy
to prove. Meanwhile
The proof of irreducibility is very complicated.
One needs to classify graphs by the appearance of indices and apply
induction on the size of matrix and on the number of variables ξi .


Induction tool

We shall prove irreducibility of a characteristic polynomial by the
following algorithm
Remark
If we set one variable ξi = 0 in the matrix associated to a graph G
we get the matrix associated to the graph obtained from G by
deleting all edges which have index i in the markings. Hence the
characteristic polynomial of G specializes to the product of
characteristic polynomials of the connected components of the
obtained graph. By induction these factors are irreducible, so we
obtain a factorization of the specialized polynomial

If we repeat the argument with a different variable obtaining a
different specialization and a different factorization. If these two
factorizations are not compatible then we are sure that the
polynomial we started with is irreducible!


Example

In
(1,2) (i,j) (h,k)
G := a b c d
setting ξ1 = 0 we get

(i,j) (h,k)
a b c d

χG |ξ1 =0 = χa χb∪c∪d |ξ1 =0 |
from this one deduces that if χG is not irreducible, then it must
factor into a linear factor and an irreducible cubic factor.


On the other hand, setting ξi = 0 we get

(1,2) (h,k)
a b c d

and
χG |ξi =0 = χa∪b χc∪d |ξi =0 |
it is the product of two quadratic irreducible factors!
So
χG is an irreducible polynomial.


This argument does not work for:

Example: graph
1,2
G := a b
4,1 2,3
4,3
d c
whichever variable we set equal to zero we get a linear and a cubic
term!
To treat all cases, we need further many lemmas:


Lemma "Super test": Suppose we have a connected marked graph
G in which we ﬁnd a vertex a and an index, say 1, so that

c
1,i
1,h
... d a b... ...
1,k
1,j

e
we have:
1 appears in all and only the edges having a as vertex.
When we remove a (and the edges meeting a) we have a
connected graph with at least 2 vertices.
When we remove the edges associated to any index, the
characteristic polynomials of connected components of the
obtained graph are irreducible.
Then the polynomial χG (t) is irreducible.


Some key lemmas

Lemma
If in the maximal tree T of G there are two blocks A, B and two
indices i, j such that:
1 i, j do not appear in the edges of the blocks A, B.
2

χA |ξi =ξj =0 = χB |ξi =ξj =0
¯ ¯ (11)
Then A, B are reduced to points:|B| = |A| = 1, A = {a}, B = {b}
and b ± a = ni ei + nj ej . The sign and the numbers ni , nj are
determined by the path in T from a to b.

Starting from two factorizations of χG |ξi =0 , χG |ξj =0 , i = j we get
possible equalities between specialized characteristic polynomials of
blocks in T and by this lemma we can simplify the graph.


Lemma
If there exists a pair of indices, say (1, i), such that 1 appears only
once in the maximal tree T and T has the form:

1,h
A_ _ _B

Figure :

where i = h, and i appears only in the block B. Then χG is
irreducible.


Due to the linear independent of edges in the maximal tree T we
see that we have to treat 3 cases by the appearance of indices in
T:
1 There are two indices which appear only once.
2 There is only one index that appears once.
3 Every index appears twice.
And every case contains a great number of subcases. So the
analysis is very deep and complicated!


Example: The proof of a subcase of the second case
In this case in the maximal tree there is one index, say 1, which
appears only once, there is another index, say 3, which appears
three times.Other indices appear twice. Consider the subcase when
1, 3 appear together in an edge and T has the form

2,k1 1,3 2,k2
A_ _ _B_ _ _C _ _ _D

Figure :

1) If A, D are not joined by an edge then:
χG |ξ1 =0 = χA∪B χC ∪D |ξ1 =0 , (12)
χG |ξ2 =0 = χA χB∪C |ξ2 =0 χD |ξ2 =0 .
¯ ¯ (13)


By symmetry we need to consider only case (15). By lemma 12 we
get get |A| = |C | = 1, A = {0}, C = {c}, c = τn1 e1 +n2 e2 (0). By
inspection of Figure (3) n1 , n2 ∈ {±1}.

η(c) ∈ {0, −2} =⇒ c = ±(e1 − e2 ), −e1 − e2 (17)

i. e. there exists an edge marked (1, 2) that connects 0 and c.
Moreover, all indices, diﬀerent from 1, 2 must appear an even
number of times in every path from 0 to c. Consider the index k1 .
i) If k1 = 3, then k1 must appear once more in the block B like:

2,k1 k1 ,s 1,3 2,k2
0 _ _ _ B1 _ _ _ B2 _ _ _ c _ _ _ D

Now we can apply Lemma 13 to the pair (1, k1 ) and get the
irreducibility of χG .
ii) If k1 = 3, consider the index k2 .


If k2 = 3, then either k2 appears in the block D as in ﬁgure (4), or
it appears in the block B as in ﬁgure (5).

Figure :

Figure :


In the case of ﬁgure (4), by lemma 13 for the pair (1, k2 ), χG is
irreducible.
Now consider the case of ﬁgure (5). By factorizations of χG |ξ1 =0 |
and χG |ξk2 =0 | one deduces χB2 |ξk2 =0 = χc |ξ1 =0 . Then by lemma
¯
12 we have B2 = {b2 }, c = τ±e1 ±ek2 (±b2 ). We have in the case
σb2 = σc =⇒ c = b2 ± (e1 − ek2 ), i. e. there exists a black edge
with the marking (1, k2 ) that connects c and b2 ; and in the case
σb2 = −σc =⇒ η(b2 + c) = −2 =⇒ c = −b2 − e1 − ek2 , i. e.
there exists a red edge with the marking (1, k2 ) that connects c
and b2 .
+) If s = 3 and B1 = {b1 }, then, by Lemma "Super test" for the
vertex b1 and the index 3, χG is irreducible.
+) If s = 3 and |B1 | > 1, let i be an index that appears in the
block B1 . If i appears twice in the block B1 then by Lemma 13 for
the pair (1, k2 ), χG is irreducible.


Hence, since i appears only twice, we need to consider the case,
when i appears once in the block B1 and once in the block D as in
ﬁgure (6).

Figure :


Higher degree NLS

For higher degree NLS formulas are more complicated and we do
not have aﬃnely independence of vertices in graphs.
So
we prove the separation and irreducibility directly by arithmetical
arguments!

In [4] I have proved for graphs of dimensions 1 and 2.
The main idea is
that we suppose that characteristic polynomials are not irreducible,
we can consider their possible factorizations, divisibility of
coeﬃcients and then we shall get a contradiction.


M.Procesi and C.Procesi.
A normal form for the schrödinger equation with analytic
non-linearities.
Communications in Mathematical Physics, 312(2):501–557,
2012.
arXiv: 1012.0446v6 [math. AP].
C.Procesi M.Procesi and Nguyen Bich Van.
The energy graph of the non linear schrödinger equation.
To appear in Rendiconti Lincei: Matematica e Applicazioni,
arXiv: 1205.1751 [math AP].
M. Procesi and C. Procesi.
A KAM algorithm for the resonant non-linear schrödinger
equation.
Preprint 2012, arXiv: 1211.4242v1[math AP].
Nguyen Bich Van.
Characteristic polynomials, related to the normal form of the
non linear schrödinger equation.


GRAZIE PER LA VOSTRA ATTENZIONE!

PhD thesis presentation of Nguyen Bich Van

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