Probabilistic methods for discrete nonlinearSchrodinger equations

Sourav Chatterjee(NYU)

Kay Kirkpatrick(Universite Paris IX)

Sourav Chatterjee Probability with discrete NLS

The cubic nonlinear Schrodinger equation

I Linear Schrodinger equation:

i∂tψ = −∆ψ,

where ψ : Rd × R→ C.

I Cubic nonlinear Schrodinger equation:

i∂tψ = −∆ψ + κ|ψ|2ψ.

I κ can be +1 (defocusing equation) or −1 (focusing equation).

I In this talk, we will concentrate on the focusing equation, thatis, κ = −1.

I Crucial to understanding a variety of physical phenomena:I Bose-Einstein condensationI Langmuir waves in plasmasI Nonlinear opticsI Envelopes of water wavesI etc...

Sourav Chatterjee Probability with discrete NLS

The Gibbs measure

I The cubic NLS is a Hamiltonian system, with Hamiltonian

H(ψ) = 2

∫|∇ψ|2dx + κ

∫|ψ|4dx .

I One method of studying the behavior of Hamiltonian systemsis through invariant measures for the dynamics.

I Gibbs measures, i.e. measures that have density of the form

e−βH(ψ)

are (formally) invariant by Liouville’s theorem.

I Here β is a parameter, sometimes called the ‘inversetemperature’.

Sourav Chatterjee Probability with discrete NLS

The mass cut-off

I The Gibbs measure for the focusing NLS has infinite mass(even in 1D) and hence cannot be normalized to give aprobability measure.

I This necessitates a mass cut-off: consider densities of the form

e−βH(ψ)1{N(ψ)≤B}

where B is another parameter and N(ψ) =∫|ψ|2dx .

I Even after mass cut-off, it is not obvious that everythingmakes sense.

I Following the pioneering work of Lebowitz-Rose-Speer, theapproach was given a rigorous foundation through the worksof McKean-Vaninsky, Bourgain and Rider.

I Most of the work till now in 1D. Bourgain and Tzvetkov didthe 2D defocusing case. Brydges-Slade found some problemsin defining the Gibbs measure in the 2D focusing case.

I Analysis of Gibbs measures can lead to conclusions aboutwell-posedness (Bourgain).

Sourav Chatterjee Probability with discrete NLS

3D and higher

I It is not clear (and probably not true) that Gibbs measures forthe focusing NLS can be defined analogously in dimensions 3and higher.

I Our approach: work with a discretized system withoutworrying about taking continuum limits. Gibbs measures canalways be defined for discrete systems.

I Try to understand the fine properties of the system and makeinferences about the dynamics.

I One long-term motivation: the soliton resolution conjecture.

Sourav Chatterjee Probability with discrete NLS

The soliton resolution conjecture

I The linear Schrodinger is a dispersive system: while∫Rd |ψ|2dx is conserved over time, for any compact set K ,

limt→∞

∫K|ψ|2dx = 0.

I In its simplest form, the soliton conjecture for the nonlinearNLS says that there is a compact set K such that the aboveequation does not hold for K , but [see below].

I One can find initial data that disperses even for the nonlinearNLS. But this is the worst-case scenario.

I The conjecture says that complete dispersion does not happenin the typical case, although it is not clear what ‘typical’means. [Readable reference: Terry Tao’s blog on this.]

I The conjecture has been proved in 1D (cubic), where thesystem is completely integrable. All other cases are unknown.

Sourav Chatterjee Probability with discrete NLS

Formulation via the invariant measure approach

I One way to quantify ‘typical’: a set of probability 1 under aninvariant probability measure whose support is the space of allinitial data with a given mass and energy (say).

I Characteristics of a random pick from the invariant measuredo not change as it evolves over time according to the NLSflow.

I Implication: If the random initial data has positive mass in aneighborhood of the origin with probability 1, then it willcontinue to have this characteristic for all time, satisfying thesoliton resolution conjecture.

I Difficulty: We do not know such invariant measures for thecontinuum system (focusing cubic) in 3D and higher.

I Note: The existence of soliton solutions, e.g. ground statesolitons, is known. The difficulty is in proving that suchbehavior is typical.

Sourav Chatterjee Probability with discrete NLS

The discrete cubic NLS

I Since we cannot handle the continuum system, one alternativeis to discretize space.

I Let G = (V ,E ) be the lattice approximation of thed-dimensional unit torus [0, 1]d , whereV = {0, 1/L, 2/L, . . . , (L− 1)/L}d and E consists ofnearest-neighbor edges.

I Let h = 1/L denote the lattice spacing. Let n = Ld be thesize of the graph.

I The discrete nearest-neighbor Laplacian on G is defined as

∆f (x) :=1

h2

∑y : (x ,y)∈E

(f (y)− f (x)).

I The discrete cubic NLS on G is a family of coupled ODEswith fx(t) := f (x , t), satisfying for all x ∈ V

id

dtfx = −∆fx + κ|fx |2fx .

Sourav Chatterjee Probability with discrete NLS

The discrete Gibbs measure

I The discrete Hamiltonian associated with the focusing NLS is:

H(f ) :=2

n

∑(x ,y)∈E

∣∣∣∣ fx − fyh

∣∣∣∣2 − 1

n

∑x∈V

|fx |4.

I Note analogy with continuum Hamiltonian:

H(f ) = 2

∫|∇f |2dx −

∫|f |4dx .

I Let N(f ) = 1n

∑x∈V |fx |2 denote the mass.

I Given β,B > 0, the probability measure µ on Cn defined as

d µ :=1

Ze−β

eH(f )1{eN(f )≤B}

∏x∈V

dfx

is an invariant probability measure for the discrete focusingcubic NLS. Here Z = Z (β,B) is the normalizing constant.

Sourav Chatterjee Probability with discrete NLS

The limiting free energy

I Let m : [2,∞)→ R be the function

m(θ) :=θ

2− 1

2+θ

2

√1− 2

θ+ log

(1

2− 1

2

√1− 2

θ

).

I Let θc be the unique real zero of m. Numerically,θc ≈ 2.455407.

Theorem (Chatterjee & Kirkpatrick, 2010)

Suppose d ≥ 3. As the lattice size n→∞ (and hence the latticespacing h→ 0),

log Z (β,B)

n→

{log(Bπe) if βB2 ≤ θc ,log(Bπe) + m(βB2) if βB2 > θc .

Sourav Chatterjee Probability with discrete NLS

Features of the Gibbs measure

I Let ψ be a random initial data from the Gibbs measure.I Let M1(ψ) and M2(ψ) denote the largest and second largest

components of the vector (|ψx |2)x∈V .I When βB2 > θc , there is high probability that M1(ψ) ≈ an

and M2(ψ) = o(n), where

a = a(β,B) :=B

2+

B

2

√1− 2

βB2.

I Moreover,∑

x |ψx |2 ≈ Bn with high probability.I Largest component carries more than half of the total mass:

maxx

|ψx |2∑y |ψy |2

≈ a

B>

1

2.

I On the other hand, when βB2 < θc , then M1(ψ) = o(n) andmaxx |ψx |2/

∑y |ψy |2 ≈ 0.

Sourav Chatterjee Probability with discrete NLS

Fraction of mass at the heaviest site

β

cθ

1

a

Figure: The fraction of mass at the heaviest site jumps from roughly zerofor small inverse temperature, to roughly .71 at the critical threshold.(Here B = 1.)

Sourav Chatterjee Probability with discrete NLS

Typicality of soliton-like solutions

Theorem (Chatterjee & Kirkpatrick, 2010)

Suppose βB2 > θc and d ≥ 3. Suppose ψ(x , t) is a solution of thediscrete cubic NLS on lattice of size n with random initial dataψ(·, 0) from the Gibbs measure µ. Then for any q ∈ (0, 1) that issufficiently close to 1, with probability at least 1− exp(−nq) thereis a point x0 such that for all t ∈ [0, exp(nq)],∣∣∣∣ |ψ(x0, t)|2

n− a(β,B)

∣∣∣∣ ≤ n−(1−q)/4

andmaxx 6=x0 |ψ(x , t)|2

n≤ n−(1−q).

Sourav Chatterjee Probability with discrete NLS

Stability of the soliton: proof sketch

I If βB2 > θc , we show that with high probability, there is asingle point with abnormally large value of the wave functionat t = 0.

I As time progresses, the invariance of the Gibbs measureimplies that the above picture must hold for all but a verysmall proportion of times.

I We use estimates related to the NLS flow to show that if thesoliton has to jump from one site to another, that cannothappen in a time period as small as allowed by the precedingestimate. Hence the soliton cannot jump.

I The above argument is a simple combination of probabilisticand PDE methods.

Sourav Chatterjee Probability with discrete NLS

Analysis of the Gibbs measure

I Simplifying step: in d ≥ 3, the interaction term in theHamiltonian has almost no effect if n is large. In other words,the problem attains a mean-field character. Not obvious apriori. Not true in d = 1 or 2.

I Upon ignoring the interaction term, the problem boils down toanalyzing the uniform probability distribution on subsets of Cn

like

Γa,b =

{f ∈ Cn :

∑x∈V

|fx |2 ≈ bn,∑x∈V

|fx |4 ≈ a2n2

}.

Here ≈ means equal up to error of order smaller than themain term.

I To show: A uniformly chosen point from Γa,b has exactly oneabnormally large component with high probability.

I Also, need to compute the volume of Γa,b to connect uniformdistributions with Gibbs measures.

Sourav Chatterjee Probability with discrete NLS

A simple problem

I Suppose Z1,Z2 are i.i.d. complex Gaussian random variables,and we are given that |Z1|4 + |Z2|4 ≈ n, where n is a largenumber.

I What can you say about the pair (Z1,Z2), conditional on thisevent?

I Conditionally, (Z1,Z2) is distributed on the L4 annulus{(z1, z2) : |z1|4 + |z2|4 ≈ n} with probability density functionproportional to exp(−|z1|2 − |z2|2).

I Thus, the conditional distribution must concentrate on thepart of the annulus where |z1|2 + |z2|2 is minimized. Thishappens where the L4 and L2 balls touch.

I This implies that with high probability, either |Z1|4 ≈ n or|Z2|4 ≈ n.

Sourav Chatterjee Probability with discrete NLS

A slightly more complex problem

I The above logic continues to hold if we replace (Z1,Z2) by(Z1, . . . ,Zk), as long as k is sufficiently small compared to n.

I Precisely, if k � n/ log n, then the same argument shows thatconditional on |Z1|4 + · · ·+ |Zk |4 ≈ n, we can say that withhigh probability, there is exactly one i such that |Zi |4 ≈ n.

I However, we are interested in the case where k is of the sameorder as n. The previous argument breaks down becausevolumes compete with densities.

I But the result is still true, with a small modification.

Sourav Chatterjee Probability with discrete NLS

An even more complex problem

I Suppose Z1, . . . ,Zn are i.i.d. standard complex Gaussianrandom variables.

I Let A be the event |Z1|4 + · · · |Zn|4 ≈ bn where b is a constantgreater than E|Z1|4 = 8 and ≈ has some appropriate meaning.

I What does the vector (Z1, . . . ,Zn) look like, conditional on A?

Sourav Chatterjee Probability with discrete NLS

Analysis of the complex problem

I Note that

P(A) ≥ P(|Z1|4 ≈ (b − 8)n)P(|Z2|4 + · · ·+ |Zn|4 ≈ 8n)

≈ e−c√

n,

where c denotes a constant depending on b.I On the other hand, if B denotes the event that the histogram

of the data {|Z1|2, . . . , |Zn|2} deviates significantly from itsexpected shape, then P(B) ≤ e−dn, where d is a constantdepending on the deviation.

I Thus, P(B|A) ≤ P(B)/P(A) = very small.I Together, this implies that given A, there must exist a

relatively small set S ⊆ {1, . . . , n} such that∑i∈S |Zi |4 ≈ (b − 8)n.

I The argument for the simpler problem can be now applied toshow that there is exactly one i such that |Zi |2 ≈ (b − 8)n.

Sourav Chatterjee Probability with discrete NLS

From Gaussian measures to Gibbs measures

I When the standard Gaussian distribution is restricted to asubset of a thin L2 annulus (e.g. intersection of L2 and L4

annuli), it becomes the uniform distribution on this set.

I The Gibbs measures are approximately convex combinationsof uniform distributions on an infinite collection ofintersections of L2 and L4 annuli.

I Thus, our analysis proceeds as: Gaussian→ Uniform→ Gibbs.

Sourav Chatterjee Probability with discrete NLS

Continuum limit

I Unfortunately, the Gibbs measures do not tend to acontinuum limit at the grid size goes to zero.

I The reason is the energy levels attained by the Gibbs measureare too low to admit good behavior in the limit.

I In fact, we have a theorem proving convergence to white noiseplus one exceptional point.

I Future goal: to develop a continuum version. Requires newideas in addition to the existing ones.

Sourav Chatterjee Probability with discrete NLS