solitons on manifoldsmarzuola/karlsruhe/soliton_manifold-talk.pdf · solitons on manifolds jeremy...

Solitons onManifolds

Jeremy L.Marzuola

The theory ofnonlinear boundstates on Rd

ConcentrationCompactness

Nonlinear boundstates onHd

Nonlinearquasimodes onmanifolds withperiodic ellipticgeodesic orbitsA toy model

Conclusions andFuture Directions

Solitons on ManifoldsJoint work (on various projects) with Pierre Albin

(UIUC), Hans Christianson (UNC), Jason Metcalfe(UNC), Michael Taylor (UNC), Laurent Thomann

(Nantes)

Jeremy L. Marzuola

Department of MathematicsUniversity of North Carolina, Chapel Hill

Bielefeld Course on Nonlinear Waves - June 29, 2012

Jeremy L.Marzuola

Outline

The theory of nonlinear bound states on Rd

Concentration Compactness

Nonlinear bound states on Hd

Nonlinear quasimodes on manifolds with periodic ellipticgeodesic orbits

A toy model

Conclusions and Future Directions

Jeremy L.Marzuola

Ideas...

I We see a very nice trichotomy:Hd : dispersion is so strong that only localnonlinearity dominates ([Christianson-M 2010,Mancini-Sandeep 2008]),Rd : balance of dispersion and nonlinearity globally([Strauss 1977,....]),(M,g) : Locally geometry dominates overnonlinearity ([Albin-Christianson-M-Thomann 2011]).See also work using Mountain Pass Theorems byseveral authors, for instance [del Piño, Pistoia,...].

I Recent progress allows us to further explore theexistence of global bound states on WeaklyHomogeneous Manifolds:[Christianson-M-Metcalfe-Taylor (2012)].

Jeremy L.Marzuola

Ideas...

I We see a very nice trichotomy:Hd : dispersion is so strong that only localnonlinearity dominates ([Christianson-M 2010,Mancini-Sandeep 2008]),Rd : balance of dispersion and nonlinearity globally([Strauss 1977,....]),(M,g) : Locally geometry dominates overnonlinearity ([Albin-Christianson-M-Thomann 2011]).See also work using Mountain Pass Theorems byseveral authors, for instance [del Piño, Pistoia,...].

I Recent progress allows us to further explore theexistence of global bound states on WeaklyHomogeneous Manifolds:[Christianson-M-Metcalfe-Taylor (2012)].

Jeremy L.Marzuola

Nonlinear Optics

Figure: Some "Gaussian Beams" in the work on nonlinearoptics in the group of Ulf Peschel at MPI Science of Light.

Jeremy L.Marzuola

The Technical Formulation

I We wish to explore existence of stationary solutionsto the nonlinear Schrödinger equation on a manifold(M,g). Let −∆g be the Laplace-Beltrami operator onM with respect to the metric g. Consider thenonlinear Schrödinger equation (NLS − g) on M:

iut + ∆gu + |u|pu = 0, x ∈ Mu(0, x) = u0(x).

I A nonlinear bound state is a choice of initial conditionRλ(x) such that

u(t , x) = eiλtRλ(x)

satisfies (NLS − g) with initial data u(0, x) = Rλ(x).

Jeremy L.Marzuola

I We wish to explore existence of stationary solutionsto the nonlinear Schrödinger equation on a manifold(M,g). Let −∆g be the Laplace-Beltrami operator onM with respect to the metric g. Consider thenonlinear Schrödinger equation (NLS − g) on M:

iut + ∆gu + |u|pu = 0, x ∈ Mu(0, x) = u0(x).

I A nonlinear bound state is a choice of initial conditionRλ(x) such that

u(t , x) = eiλtRλ(x)

satisfies (NLS − g) with initial data u(0, x) = Rλ(x).

Jeremy L.Marzuola

I Plugging in the ansatz yields the following stationaryelliptic equation for Rλ:

−∆gRλ + λRλ − |Rλ|pRλ = 0.

Jeremy L.Marzuola

Existence in Rd

I Existence of solitary waves for a wide variety ofnonlinearities is proved in Berestycki-Lions byminimizing the quantity

T (u) =

∫Rd|∇u|2dx

with respect to the constraint

V (u) := −λ2

∫Rd|u|2dx +

F (|u|)dx = 1.

Jeremy L.Marzuola

Existence

I Then, using a minimizing sequence and Schwarzsymmetrization, one sees the existence of thenonnegative, spherically symmetric, decreasingsoliton solution.

I For uniqueness, see McLeod, where a shootingmethod is implemented to show that the desiredsoliton behavior only occurs for one particular initialvalue.

Jeremy L.Marzuola

Existence

I Then, using a minimizing sequence and Schwarzsymmetrization, one sees the existence of thenonnegative, spherically symmetric, decreasingsoliton solution.

I For uniqueness, see McLeod, where a shootingmethod is implemented to show that the desiredsoliton behavior only occurs for one particular initialvalue.

Jeremy L.Marzuola

I From the work of P.L. Lions, we have the followingmeans of finding constrained minimizers:

I Let (ρn)n≥1 be a sequence in L1(Rd ) satisfying:

ρn ≥ 0 in Rd ,

∫Rdρndx = λ

where λ > 0 is fixed. Then there exists asubsequence (ρnk )k≥1 satisfying one of the threefollowing possibilities:

Jeremy L.Marzuola

I From the work of P.L. Lions, we have the followingmeans of finding constrained minimizers:

I Let (ρn)n≥1 be a sequence in L1(Rd ) satisfying:

ρn ≥ 0 in Rd ,

∫Rdρndx = λ

where λ > 0 is fixed. Then there exists asubsequence (ρnk )k≥1 satisfying one of the threefollowing possibilities:

Jeremy L.Marzuola

I i. (compactness) there exists yk ∈ Rd such thatρnk (·+ ynk ) is tight, i.e.:

∀ε > 0, ∃R <∞,∫

yk +BR

ρnk (x)dx ≥ λ− ε;

I ii. (vanishing) limk→∞ supy∈Rd

∫y+BR

ρnk (x)dx = 0,for all R <∞;

Jeremy L.Marzuola

I i. (compactness) there exists yk ∈ Rd such thatρnk (·+ ynk ) is tight, i.e.:

∀ε > 0, ∃R <∞,∫

yk +BR

ρnk (x)dx ≥ λ− ε;

I ii. (vanishing) limk→∞ supy∈Rd

∫y+BR

ρnk (x)dx = 0,for all R <∞;

Jeremy L.Marzuola

I iii. (dichotomy) there exists α ∈ [0, λ] such that for allε > 0, there exists k0 ≥ 1 and ρ1

k , ρ2k ∈ L1

+(Rd )satisfying for k ≥ k0:

‖ρnk − (ρ1k + ρ2

k )‖L1 ≤ ε,|∫

Rd ρ1kdx − α| ≤ ε,

Rd ρ2kdx − (λ− α)| ≤ ε,

d(Supp(ρ1k ),Supp(ρ2

k ))→∞.

Jeremy L.Marzuola

Existence

I The idea for Hd from [Christianson-M 2010] relies onconjugating ∆Hd into an operator on Euclideanspace, and then finding minimizers for the energyfunctional. See also [Mancini-Sandeep 2008].

I The problem of minimizing the functional is greatlysimplified assuming the functions involved dependonly on the radius r = |x |, as then the minimizationtheory in Rd may be used.

I Let us define a space H1r to be the space of all

spherically symmetric functions in H1.

Jeremy L.Marzuola

Existence

Jeremy L.Marzuola

Existence

Jeremy L.Marzuola

Radial Symmetry

I LemmaLet u be a solution to Hyperbolic NLS with initial datau0 ∈ H1

r and the nonlinearity f (|u|)u = |u|pu with4

d−2 > p > 0. Then u ∈ H1r .

I The proof of this lemma is by uniqueness, whichfollows from the implicit local uniqueness followingfrom the Strichartz estimates in Ionescu-Staffilani.

Jeremy L.Marzuola

Radial Symmetry

I LemmaLet u be a solution to Hyperbolic NLS with initial datau0 ∈ H1

r and the nonlinearity f (|u|)u = |u|pu with4

d−2 > p > 0. Then u ∈ H1r .

I The proof of this lemma is by uniqueness, whichfollows from the implicit local uniqueness followingfrom the Strichartz estimates in Ionescu-Staffilani.

Jeremy L.Marzuola

Radial SymmetryI We show that any constrained minimizer in

hyperbolic space may be replaced by one that isspherically symmetric, so that we may neglect theangular derivative.

I To do this, we modify the standard argument ofLieb-Loss in Rd , using heat kernel arguments toshow symmetric decreasing rearrangement orSchwarz symmetrization lowers the kinetic energy inHd .

I The symmetric decreasing rearrangement on Hd isgiven by

f ∗(Ω) = inft : λf (t) ≤ µ(B(dist (Ω,0))),

where µ is the natural measure on Hd , dist is thehyperbolic distance function on Hd and

λf (t) = µ(|f | > t).

Jeremy L.Marzuola

λf (t) = µ(|f | > t).

Jeremy L.Marzuola

λf (t) = µ(|f | > t).

Jeremy L.Marzuola

Radial Symmetry

I First of all, it is clear f ∗ is spherically symmetric,nonincreasing, lower semicontinuous and

‖f ∗‖Lp(Hd ) = ‖f‖Lp(Hd )

for any 1 ≤ p ≤ ∞.

Jeremy L.Marzuola

Radial Symmetry

I LemmaSuppose f ∈ H1(Hd ), and f ∗ is the symmetric decreasingrearrangement of f . Then

‖∇f ∗‖L2(Hd ) ≤ ‖∇f‖L2(Hd ).

Jeremy L.Marzuola

Radial Symmetry

I We use standard Hilbert space theory as inLieb-Loss. Namely, we observe that the kineticenergy satisfies

‖∇f‖L2(Hd ) = limt→0

I t (f ),

I t (f ) = t−1[(f , f )Hd − (f ,e∆Hd t f )Hd ]

and (·, ·)Hd is the natural L2 inner-product on Hd .

Jeremy L.Marzuola

Radial Symmetry

I As (f , f ) = (f ∗, f ∗) by construction, we need

(f ∗,e∆Hd t f ∗)Hd ≥ (f ,e∆Hd t f )Hd

in order to see that symmetrization decreases thekinetic energy.

Jeremy L.Marzuola

Radial Symmetry

I In Rd , this is done using convolution operators andthe Riesz rearrangement inequality, which we do nothave here. Instead, we use heat kernel decay and ana rearrangement theorem of Draghici.

Jeremy L.Marzuola

Radial Symmetry

I For each t > 0, the heat kernel on hyperbolic space,pd (ρ, t), is a decreasing function of the hyperbolicdistance ρ.

Jeremy L.Marzuola

Radial Symmetry

I From Draghici, we have used the following theorem:

Theorem (Draghici)Let X = Hd , fi : X → R+ be m nonnegative functions,Ψ ∈ AL2(Rm

+) be continuous and Kij : [0,∞)→ [0,∞),i < j , j ∈ 1, . . . ,m be decreasing functions. We define

I[f1, . . . , fm] =

∫X m

Ψ(f1(Ω1), . . . , fm(Ωm))

×Πi<jKij(d(Ωi ,Ωj))dΩ1 . . . dΩm.

Then, the following inequality holds:

I[f1, . . . , fm] ≤ I[f ∗1 , . . . , f∗m].

Jeremy L.Marzuola

Existence

I In this section we begin to analyze HNLS in the casef (|u|)u is a so-called “focusing” nonlinearity.

I From the polar form of ∆Hd , we approach theproblem by comparison to the standard Laplacian onRd . In this direction, let us recall that the metric forRd in polar coordinates is given by

ds2 = dr2 + r2dω2,

so that the Jacobian is rd−1.

Jeremy L.Marzuola

Existence

I In this section we begin to analyze HNLS in the casef (|u|)u is a so-called “focusing” nonlinearity.

I From the polar form of ∆Hd , we approach theproblem by comparison to the standard Laplacian onRd . In this direction, let us recall that the metric forRd in polar coordinates is given by

ds2 = dr2 + r2dω2,

so that the Jacobian is rd−1.

Jeremy L.Marzuola

Existence

I Similarly, the Jacobian from the polar coordinaterepresentation of Hd is sinhd−1 r . We employ anisometry T taking L2(rd−1drdω) to L2(sinhd−1 rdrdω),so that T−1(−∆Hd )T is a non-negative, unbounded,essentially self-adjoint operator on L2(Rd ).

Jeremy L.Marzuola

Existence

I We define

φ(r) =( r

sinh r

) d−12,

φ−1(r) =

(sinh r

) d−12

and take Tu = φu.I Conjugating −∆Hd by φ, we have a second order

differential operator on Rd with the leading orderterm almost the Laplacian on Rd .

Jeremy L.Marzuola

Existence

I We define

φ(r) =( r

sinh r

) d−12,

φ−1(r) =

(sinh r

) d−12

and take Tu = φu.I Conjugating −∆Hd by φ, we have a second order

differential operator on Rd with the leading orderterm almost the Laplacian on Rd .

Jeremy L.Marzuola

Existence

I Indeed, we have

φ−1(−∆Hd )(φu) = −∆u +

[Vd (r) +

(d − 1

Jeremy L.Marzuola

Existence

I Here

V0(r) =1− d

Vd (r) =(d − 1)(d − 3)

(r2 − sinh2 rr2 sinh2 r

so that

−∆ = −∆Rd −r2 − sinh2 rr2 sinh2 r

∆Sd−1 .

Jeremy L.Marzuola

ExistenceI For completeness, we record the following simple

lemma.

LemmaThe function

V =sinh2 r − r2

r2 sinh2 r

satisfies the following properties:

(i) V ∈ C∞(R),

(ii) V ≥ 0,

(iii) V (0) =13,

(iv) V = O(r−2), r →∞, and

(v) V ′(r) = 0 only at r = 0.

Jeremy L.Marzuola

Existence

I RemarkNote that the potential V3 = 0, and the lemma impliesV2 ≥ 0 has a “bump” at 0, while for d ≥ 4, the potentialVd ≤ 0 has a “well” at 0.

Jeremy L.Marzuola

Existence

I After this conjugation to Rd , Hyperbolic NLSbecomes−iut − ∆u + (d−1)2

4 u + Vd (x)u − f (x ,u) = 0, x ∈ Rd

u(0, x) = u0(x) ∈ H1,

where now the nonlinearity f takes the following formafter conjugation:

f (x ,u) = φ−1f (φu)(φu)

sinh r

) p(d−1)2 |u|pu.

Jeremy L.Marzuola

Existence

I We have the naturally defined conserved quantities

Q(u) = ‖u‖2L2

E(u) =

[12|∇u|2 +

a(x)|∇angu|2

(Vd (|x |) +

(d − 1)2

)|u|2 − F (x ,u)

Jeremy L.Marzuola

Existence

I Here

a(x) =|x |2 − sinh2 |x ||x |2 sinh2 |x |

is the “offset” of the spherical Laplacian in thedefinition of ∆,

F (x ,u) =

∫ u(x)

0f (x , s)ds

sinh |x |

)(d−1)p/2

|u|p+2

=: K (|x |)|u|p+2.

Jeremy L.Marzuola

Existence

I From the work of Banica, we have global existencefor p < 4

d and finite time blow-up for 4d ≤ p < 4

d−2 .

Jeremy L.Marzuola

Existence

I We make a soliton ansatz for Hyperbolic NLS in Rd :u(x , t) = eiλtRλ, for a function Rλ depending on areal parameter (the soliton parameter) λ > 0.

I Plugging this ansatz into the conjugated equationHyperbolic NLS we see we must have

−∆Rλ +

((d − 1)2

4+ λ+ Vd (r)

)Rλ − f (x ,Rλ) = 0.

I Hence, we seek a minimizer of the associatedenergy functional to this nonlinear elliptic equation for‖u‖L2 fixed.

Jeremy L.Marzuola

Existence

−∆Rλ +

((d − 1)2

4+ λ+ Vd (r)

)Rλ − f (x ,Rλ) = 0.

Jeremy L.Marzuola

Existence

−∆Rλ +

((d − 1)2

4+ λ+ Vd (r)

)Rλ − f (x ,Rλ) = 0.

Jeremy L.Marzuola

Existence

I We note that the continuous spectrum is “shifted”according to the term

d − 12

I In the end, this term does not alter the existenceargument for soliton solutions, however, it doesexpand the allowed range of soliton parameters fromλ ∈ (0,∞) to

λ ∈ (−(

d − 12

,∞).

Jeremy L.Marzuola

Existence

I We note that the continuous spectrum is “shifted”according to the term

d − 12

I In the end, this term does not alter the existenceargument for soliton solutions, however, it doesexpand the allowed range of soliton parameters fromλ ∈ (0,∞) to

λ ∈ (−(

d − 12

,∞).

Jeremy L.Marzuola

Existence

I Hence, we set

µd = λ+

(d − 1

Jeremy L.Marzuola

Quasimodes

I By separating variables in the t direction, we write

ψ(x , t) = e−iλtu(x),

from which we get the stationary equation

(λ−∆g)u = σ|u|pu.

I The construction in the proof finds a functionuλ(x) = λ(d−1)/8g(λ1/4x) such that g is rapidlydecaying away from Γ, C∞, g is normalized in L2, and

(λ−∆g)uλ = σ|uλ|puλ + E(uλ),

where the error E(uλ) is expressed by the atruncation of an asymptotic series similar to that inthe work of Thomann and is of lower order in λ.

Jeremy L.Marzuola

Quasimodes

I By separating variables in the t direction, we write

ψ(x , t) = e−iλtu(x),

from which we get the stationary equation

(λ−∆g)u = σ|u|pu.

I The construction in the proof finds a functionuλ(x) = λ(d−1)/8g(λ1/4x) such that g is rapidlydecaying away from Γ, C∞, g is normalized in L2, and

(λ−∆g)uλ = σ|uλ|puλ + E(uλ),

where the error E(uλ) is expressed by the atruncation of an asymptotic series similar to that inthe work of Thomann and is of lower order in λ.

Jeremy L.Marzuola

Quasimodes

I The result is an improvement over the trivialapproximate solution. It is well known that there existquasimodes for the linear equation localized near Γof the form

vλ(x) = λ(d−1)/8eisλ1/2f (s, λ1/4x), (λ > 0),

with f a function rapidly decaying away from Γ, and sa parametrization around Γ, so that vλ(x) satisfies

(λ−∆g)vλ = O(λ−∞)‖vλ‖

in any seminorm, see Ralston.

Jeremy L.Marzuola

Quasimodes

I Then(λ−∆g)vλ = σ|vλ|pvλ + E2(vλ),

where the error E2(vλ) = |vλ|pvλ satisfies

‖E2(vλ)‖Hs = O(λs/2+p(d−1)/8).

Jeremy L.Marzuola

Toy Model

I In this section we consider a toy model in twodimensions. As it is a toy model, we will not dwell onerror analysis, and instead make Taylorapproximations at will without remarking on the errorterms. Consider the manifold

M = Rx/2πZ× Rθ/2πZ,

equipped with a metric of the form

ds2 = dx2 + A2(x)dθ2,

where A ∈ C∞ is a smooth function, A ≥ ε > 0 forsome ε.

Jeremy L.Marzuola

Toy Model

I From this metric, we get the volume form

dVol = A(x)dxdθ,

and the Laplace-Beltrami operator acting on 0-forms

∆g f = (∂2x + A−2∂2

θ + A−1A′∂x )f .

Jeremy L.Marzuola

Quasimodes

I We observe that we can conjugate ∆g by anisometry of metric spaces and separate variables sothat spectral analysis of ∆g is equivalent to aone-variable semiclassical problem with potential.

I Let S : L2(X ,dVol)→ L2(X ,dxdθ) be the isometrygiven by

Su(x , θ) = A1/2(x)u(x , θ).

Jeremy L.Marzuola

Quasimodes

I We observe that we can conjugate ∆g by anisometry of metric spaces and separate variables sothat spectral analysis of ∆g is equivalent to aone-variable semiclassical problem with potential.

I Let S : L2(X ,dVol)→ L2(X ,dxdθ) be the isometrygiven by

Su(x , θ) = A1/2(x)u(x , θ).

Jeremy L.Marzuola

Quasimodes

I A simple calculation gives

−∆f = (−∂2x − A−2(x)∂2

θ + V1(x))f ,

where the potential

V1(x) =12

A′′A−1 − 14

(A′)2A−2.

Jeremy L.Marzuola

Existence

I We are interested in the nonlinear Schrödingerequation, so we make a separated ansatz:

uλ(t , x , θ) = e−itλeikθψ(x),

where k ∈ Z and ψ is to be determined (dependingon both λ and k ).

Jeremy L.Marzuola

Existence

I Applying the Schrödinger operator (with ∆ replacing∆) to uλ yields the equation

(Dt + ∆)eitλeikθψ(x) = σ|ψ|peitλeikθψ(x)

with2 =

2d − 1

≤ p <4

d − 1= 4,

where we have used the standard notation D = −i∂.

Jeremy L.Marzuola

Existence

I We are interested in the behaviour of a solution orapproximate solution near an elliptic periodicgeodesic, which occurs at a maximum of the functionA.

I For simplicity, let

A(x) =√

(1 + cos2(x))/2,

so that in a neighbourhood of x = 0, A2 ∼ 1− x2 andA−2 ∼ 1 + x2.

Jeremy L.Marzuola

Existence

I We are interested in the behaviour of a solution orapproximate solution near an elliptic periodicgeodesic, which occurs at a maximum of the functionA.

I For simplicity, let

A(x) =√

(1 + cos2(x))/2,

so that in a neighbourhood of x = 0, A2 ∼ 1− x2 andA−2 ∼ 1 + x2.

Jeremy L.Marzuola

Existence

I The function V1(x) ∼ const. in a neighbourhood ofx = 0, so we will neglect V1. If we assume ψ(x) islocalized near x = 0, we get the stationary reducedequation

(−λ+ ∂2x − k2(1 + x2))ψ = −σ|ψ|pψ.

Jeremy L.Marzuola

Existence

I Let h = |k |−1 and use the rescaling operatorTψ(x) = Th,0ψ(x) = h−1/4ψ(h−1/2x) ( below withn = 1) to conjugate:

T−1(−λ+ ∂2x − k2(1 + x2))TT−1ψ = T−1(σ|ψ|pψ)

(−λ+ h−1∂2x − k2(1 + hx2)φ = σh−p/4|φ|pφ,

where φ = T−1ψ.

Jeremy L.Marzuola

Existence

I Let us now multiply by h:

(−∂2 + x2 − E)φ = σhq|φ|pφ,

E =1− λh2

q = 1− pd − 1

4= 1− p

Observe the range restriction on p is precisely so that

0 < q ≤ 1/2.

Jeremy L.Marzuola

Existence

I We make a WKB type ansatz, although in practicewe will only take two terms (more is possible if thenonlinearity is smooth):

φ = φ0 + hqφ1, E = E0 + hqE1.

I The first two equations are

h0 : (−∂2 + x2 − E0)φ0 = 0,

hq : (−∂2 + x2 − E0 − hqE1)φ1 = E1φ0 + σ|φ0|pφ0.

Observe we have included the hqE1φ1 term on theleft hand side.

Jeremy L.Marzuola

Existence

I We make a WKB type ansatz, although in practicewe will only take two terms (more is possible if thenonlinearity is smooth):

φ = φ0 + hqφ1, E = E0 + hqE1.

I The first two equations are

h0 : (−∂2 + x2 − E0)φ0 = 0,

hq : (−∂2 + x2 − E0 − hqE1)φ1 = E1φ0 + σ|φ0|pφ0.

Observe we have included the hqE1φ1 term on theleft hand side.

Jeremy L.Marzuola

Applications

I General theory of nonlinear bound states onsymmetric spaces - What carries over fromEuclidean Space?? Spectral Properties??Uniqueness?? Sharp inequality bounds??

I Stability Theory - Related to spectral properties ofthe linearized operator, uniqueness, ...

I Traveling Waves - Motion generate by symmetries ofthe manifold and killing vector fields that commutewith the Laplacian ... lots of symmetry required.

Jeremy L.Marzuola

Applications

Jeremy L.Marzuola

Applications

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