PhD thesis presentation of Nguyen Bich Van

NLS Normal form Matrices Non degeneracy

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

equation

Nguyen Bich Van

PhD thesis defenseSapienza università di Roma, 17–12–2012


In my thesis I have studied characteristic polynomials, associatedby some rules to a class of marked graphs.

Examplegraph G =

a 1,2

4,1

b2,3

d 4,3 cMatrix

CG =

0 −2

√ξ1ξ2 0 −2

√ξ4ξ1

−2√ξ1ξ2 ξ2 − ξ1 −2

√ξ2ξ3 0

0 −2√ξ2ξ3 −ξ1 + ξ3 −2

√ξ4ξ3

−2√ξ4ξ1 0 −2

√ξ4ξ3 ξ4 − ξ1


With the characteristic polynomial:

χG = det(tI − CG) =

= −4ξ31ξ2 + 4ξ2

1ξ2ξ3 − 4ξ31ξ

4 + 8ξ21ξ2ξ4 + 4ξ2

1ξ3ξ4 − 8ξ1ξ2ξ3ξ4+

+(ξ31−9ξ2

1ξ2−ξ21ξ3 +ξ1ξ2ξ3−9ξ2

1ξ4 +9ξ1ξ2ξ4 +ξ1ξ3ξ4 +7ξ2ξ3ξ4)t+

+ (3ξ21 − 6ξ1ξ2 − 2ξ1ξ3 − 3ξ2ξ3 − 6ξ1ξ4 + ξ2ξ4 − 3ξ3ξ4)t2+

+ (3ξ1 − ξ2 − ξ3 − ξ4)t3 + t4. (1)

The problem isto prove that a rather complicated infinite list of such polynomialsin a variable t, of degree increasing with the graph dimension, andwith coefficients polynomials in the parameters ξi have distinctroots for generic values of the parameters.


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


The plan of the talk:

1 Normal forms of NLS2 Construction of colored marked graphs and matrices3 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) onthe torus Tn.

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

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


The cubic NLS in dimension 1 is completely integrable and severalexplicit solutions are known. In higher dimensions we loose thecomplete integrability and all techniques associated to it, but wecan still use the following well-known fact

The NLS (2) can be written as an infinite dimensional Hamiltoniandynamical system u = {H, u},where the symplectic variables are Fourier coefficients of thefunctions

u(t, ϕ) =∑

k∈Znuk(t)ei(k,ϕ). (3)

the symplectic form is i∑

k∈Zn duk ∧ duk and the Hamiltonian is

H :=∑

k∈Zn|k|2uk uk ±

∑k∈Zn:

∑2q+2i=1 (−1)i ki =0

uk1 uk2uk3 uk4 ...u2q+1u2q+2

(4)up to rescaling of u.


In order to study the long-time behavior of the solutions ofHamiltonian PDEs close to an equilibriumit is necessary start from a suitably non degenerate normal formand 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 ellipticfixed point at zero, i.e. there exists a canonical system ofcoordinates (p, q) such that the Hamiltonian takes the form

H(p, q) =∑j∈I

λj(p2j + q2

j ) + H>2(p, q) , λj ∈ R

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


Normal form reduction

The normal form reduction at order Dis a symplectic change of variables ΨD which reduces H to itsresonant terms:

H(p, q) ◦ΨD =∑

jλj(p2

j + q2j ) + H>2

Res(p, q) + HD(p, q)

where H>2Res Poisson commutes with

∑j λj(p2

j + q2j ) while HD(p, q)

is a formal power series of minimal degree > D + 1.

There are two classes of problems in this scheme:1 Even though HD 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.


RemarkIf the λj are rationally independent then the normal formHBirk =

∑j λj(p2

j + q2j ) + H>2

Res(p, q) is integrable, a feature whichis used in proving for instance long time stability results.Otherwise HBirk may not be integrable but it is possible that itsdynamics is simpler than the one of the original Hamiltonian.

In the case NLS HBirk =∑

j λj(p2j + q2

j ) + H>2Res(p, q) has invariant

tori of the form

p2i + q2

i = 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) adiffeomorphism.S is called the tangential sites, Sc -the normal sites.


In order to obtain information on the solutions of the completeHamiltonian close to these tori one needs to study the Hamiltonequations of H linearized at a family of invariant tori. In terms ofequations this is described by a quadratic Hamiltonian withcoefficients depending on the parameters ξ and on the anglevariables 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 anormal form for the NLS.This normal form of the NLS is described by an infinite dimensionalHamiltonian 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 certaininfinite dimensional vector space F (0,1) of functions.

Stability for this infinite dimensional operatorwill be interpreted in the same way as it appears for finitedimensional linear systems, that is the property that the linearoperator is semisimple with distinct eigenvalues.


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

Figure : The normal form matrix

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


In my thesis I have proved:

TheoremFor generic choices of tangential sites S and parameters ξ thenormal form N constructed in [1] in the case of cubic NLS in alldimensions is non-degenerate in the sense that it is semisimplewith non-zero and distinct eigenvalues. The same result for allhigher degree NLS in dimension 1 and 2.

The problem arises from the study of NLS, but one couldformulate it as a purely algebraic question. And in fact the proof isessentially 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, Sc = Zn \ S be thenormal sites.We start from the space V 0,1 of functions with basis the elements{ei

∑j νj xj zk , e−i

∑j νj xj zk}, k ∈ Sc .

In this space the conditions of commuting with momentum, resp.with mass select the elements, called frequency basis

FB = {ei∑

j νj xj zk , e−i∑

j νj xj zk , k ∈ Sc}; k ∈ Sc∑jνjvj + k = π(ν) + k = 0 resp.

∑jνj + 1 = 0. (6)

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


Cayley graph

We recall how we describe the operator ad(N) = {N, ∗} into thelanguage 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 thematrix, and an edge between i , j if and only if ci ,j 6= 0.

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


Cayley graph

Let G be a group and X = X−1 ⊂ G . Consider an actionG × 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, givena, b ∈ A we join them by an oriented edge a x // b , marked x , ifb = xa, x ∈ X .


Set Zm = {∑m

i=1 aiei , ai ∈ Z}-the lattice with basis elements ei . Inour setting the relevant group is the group G := Zm o Z/(2) thesemidirect product, denote by τ := (0,−1) so G = Zm ∪ Zmτ .

An element a = ei∑

j νj xj zk is associated to the group elementa =

∑j νjej ∈ Zm. Then a = e−i

∑j νj xj zk is associated to the

group element aτ = (∑

j νjej)τ ∈ Zmτ .

Thus the frequency basis is indexed by elements ofG1 \

⋃mi=1{−ei ,−eiτ}, where

G1 := {a, aτ, a ∈ Zm | η(a) = −1}.


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

X 0 = {ei−ej , i 6= j ∈ {1, ...,m}},X−2 = {(−ei−ej)τ, i 6= 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 werepresent red edges with a double line:g = (−ei − ej)τ, a g ga .


Given a =∑

i aiei , σ = ±1 set for u = (a, σ)

K ((a, σ)) :=σ

2 (|∑

iaivi |2 +

∑iai |vi |2). (7)

Sometimes we call K (u) the quadratic energy of u.

Definition

Given an edge u x // v , u = (a, σ), v = (b, ρ) = xu, x ∈ Xq, wesay that the edge is compatible with S if K (u) = K (v).


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

We have for a, b ∈ Zm

Ma,a = K (a)−∑

jajξj , Maτ,aτ = K (aτ) +

∑jajξj (8)

Maτ,bτ = −2√ξiξj , Ma,b = 2

√ξiξj ,

if a, b are connected by a compatible edge ei − ej (9)

Ma,bτ = −2√ξiξj , Maτ,b = 2

√ξiξj ,

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 offinite dimensional blocks. With respect to the frequency basis theblocks are described as the connected components of a graph ΛSwhich we now describe. Let π : Zm → Zn, ei 7→ vi . SetΘ = Ker(π).

Definition

The graph ΛS is the subgraph of G1 \⋃

i{−ei + Θ, (−ei + Θ)τ} inwhich we only keep the compatible edges.


We then haveTheorem

The indecomposable blocks of the matrix M in the frequency basiscorrespond 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 edgesimplies in particular that the scalar part K ((a, σ)) (which is aninteger) is constant on each block. On the other hand, in general,there are infinitely many blocks with the same scalar part. It willbe convenient to ignore the scalar term diag(K ((a, σ))), given acompatible connected component A we hence define the matrixCA = MA − diag(K ((a, σ))).


The final goal

Characteristic polynomials

Irreducibility and separation


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

A = (−e1 − e2,−)(−e1−e2)τ

(0,+)e1−e2// (e1 − e2,+) .

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

A(u,+) = (−e1 − e2 − u,−)(−e1−e2)τ

(u,+)e1−e2// (e1 − e2 + u,+)


Example

The matrices associated to these graphs are:

CA =

−ξ1 − ξ2 2

√ξ1ξ2 0

−2√ξ1ξ2 0 2

√ξ1ξ2

0 2√ξ1ξ2 ξ2 − ξ1

CAu =

−ξ1 − ξ2 − u(ξ) 2

√ξ1ξ2 0

−2√ξ1ξ2 −u(ξ) 2

√ξ1ξ2

0 2√ξ1ξ2 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 callcombinatorial.

For cubic NLS we have the following Theorem from [2]

TheoremFor generic choices of S the connected components of graph ΛS ,different from the special component −ei ,−eiτ , are formed byaffinely independent points.


We also have (see [2])

Lemma

The characteristic polynomial of each matrix CA is inZ[ξ1, . . . , ξm, t] (the roots disappear).

We wish to prove

Outside a countable union of real algebraic hypersurfaces in thespace of parameterseigenvalues of the matrix CA for connected components A that wedescribed above are all distinct .

This fact will be useful in [3] in order to prove, by a KAMalgorithm, the existence and stability of quasi–periodic solutionsfor the NLS (not just the normal form).


A direct method

In fact eigenvalues of a matrix CG are roots of characteristicpolynomials χG = det(tI − CG). One should computediscriminants and resultants of them, which are polynomials invariables ξi and show that they are not identically zero. This canbe done by direct computations only for small cases. In generalcase, even in dimension n = 3, the total number of thesepolynomials is quite high (in the order of the hundreds orthousands) so that the algorithm becomes quickly non practical!

Hencewe will prove that roots of characteristic polynomials are alldistinct by showing a stronger algebraic property of them!


Irreducibility and Separation

Theorem (Separation and Irreducibility)

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

Following the fact that an irreducible polynomial f (t) over a fieldF of characteristic 0 is uniquely determined as the minimalpolynomial of each of its roots (in the algebraic closure F ) and itsderivative f ′(t) is non-zero, g .c.d(f , f ′) = 1 we have

ImplicationOutside the countable union of algebraic hypersurfaces in thespace of parameters ξ all eigenvalues are non-zero and distinct.


Proof of separation and irreducibility theorem

For a given polynomial with integer coefficients there existreasonable computer algebra algorithms to test irreducibility butthis is not a practical method in our case where the polynomialsare infinite and their degrees also tend to infinity. So we shall usecombinatorics. The fact that the polynomials are distinct is basedby induction on the irreducibility theorem and it is relatively easyto prove. Meanwhile

The proof of irreducibility is very complicated.One needs to classify graphs by the appearance of indices and applyinduction on the size of matrix and on the number of variables ξi .


Induction tool

We shall prove irreducibility of a characteristic polynomial by thefollowing algorithm

RemarkIf we set one variable ξi = 0 in the matrix associated to a graph Gwe get the matrix associated to the graph obtained from G bydeleting all edges which have index i in the markings. Hence thecharacteristic polynomial of G specializes to the product ofcharacteristic polynomials of the connected components of theobtained graph. By induction these factors are irreducible, so weobtain a factorization of the specialized polynomial

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


Example

InG := a

(1,2)b

(i ,j)c

(h,k)d

setting ξ1 = 0 we get

a b(i ,j)

c(h,k)

d

χG |ξ1=0 = χaχb∪c∪d |ξ1=0|

from this one deduces that if χG is not irreducible, then it mustfactor into a linear factor and an irreducible cubic factor.


On the other hand, setting ξi = 0 we get

a(1,2)

b c(h,k)

d

andχG |ξi =0 = χa∪bχc∪d |ξi =0|

it is the product of two quadratic irreducible factors!

SoχG is an irreducible polynomial.


This argument does not work for:

Example: graphG := a 1,2

4,1

b2,3

d 4,3 cwhichever variable we set equal to zero we get a linear and a cubicterm!To treat all cases, we need further many lemmas:


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

c

. . . d a 1,h1,h1,k

1,j

1,i

b . . . . . .

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 aconnected graph with at least 2 vertices.When we remove the edges associated to any index, thecharacteristic polynomials of connected components of theobtained 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 twoindices 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 = niei + njej . The sign and the numbers ni , nj aredetermined by the path in T from a to b.

Starting from two factorizations of χG |ξi =0, χG |ξj =0, i 6= j we getpossible equalities between specialized characteristic polynomials ofblocks 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 onlyonce in the maximal tree T and T has the form:

A 1,h___ B

Figure :

where i 6= h, and i appears only in the block B. Then χG isirreducible.


Due to the linear independent of edges in the maximal tree T wesee that we have to treat 3 cases by the appearance of indices inT :

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 theanalysis is very deep and complicated!


Example: The proof of a subcase of the second caseIn this case in the maximal tree there is one index, say 1, whichappears only once, there is another index, say 3, which appearsthree times.Other indices appear twice. Consider the subcase when1, 3 appear together in an edge and T has the form

A 2,k1___ B 1,3 ___ C 2,k2___ D

Figure :

1) If A,D are not joined by an edge then:

χG |ξ1=0 = χA∪BχC∪D|ξ1=0, (12)χG |ξ2=0 = χAχB∪C |ξ2=0χD|ξ2=0. (13)


By symmetry we need to consider only case (15). By lemma 12 weget get |A| = |C | = 1,A = {0},C = {c}, c = τn1e1+n2e2(0). Byinspection 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 evennumber of times in every path from 0 to c. Consider the index k1.i) If k1 6= 3, then k1 must appear once more in the block B like:

0 2,k1___ B1k1,s___ B2

1,3 ___ c 2,k2___ D

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


If k2 6= 3, then either k2 appears in the block D as in figure (4), orit 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 isirreducible.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 lemma12 we have B2 = {b2}, c = τ±e1±ek2

(±b2). We have in the caseσb2 = σc =⇒ c = b2 ± (e1 − ek2), i. e. there exists a black edgewith the marking (1, k2) that connects c and b2; and in the caseσb2 = −σc =⇒ η(b2 + c) = −2 =⇒ c = −b2 − e1 − ek2 , i. e.there exists a red edge with the marking (1, k2) that connects cand b2.+) If s = 3 and B1 = {b1}, then, by Lemma "Super test" for thevertex b1 and the index 3, χG is irreducible.+) If s = 3 and |B1| > 1, let i be an index that appears in theblock B1. If i appears twice in the block B1 then by Lemma 13 forthe 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 infigure (6).

Figure :


Higher degree NLS

For higher degree NLS formulas are more complicated and we donot have affinely independence of vertices in graphs.

Sowe prove the separation and irreducibility directly by arithmeticalarguments!

In [4] I have proved for graphs of dimensions 1 and 2.

The main idea isthat we suppose that characteristic polynomials are not irreducible,we can consider their possible factorizations, divisibility ofcoefficients and then we shall get a contradiction.


M.Procesi and C.Procesi.A normal form for the schrödinger equation with analyticnon-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 schrödingerequation.Preprint 2012, arXiv: 1211.4242v1[math AP].

Nguyen Bich Van.Characteristic polynomials, related to the normal form of thenon linear schrödinger equation.Preprint 2012, arXiv:1203.6015[math. CO].


GRAZIE PER LA VOSTRA ATTENZIONE!

PhD thesis presentation of Nguyen Bich Van

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Transcript of PhD thesis presentation of Nguyen Bich Van