Blow-up for the nonlinear Schrodinger equationwith point nonlinearity

Justin Holmer

Brown University

November 1, 2016

We consider the nonlinear Schrodinger equation (NLS) with pointnonlinearity in 1D

i∂tψ + ∂2xψ + δ|ψ|p−1ψ = 0

where δ = δ(x) is the Dirac mass at x = 0.

It can be interpreted as the free linear Schrodinger equation

i∂tψ + ∂2xψ = 0 , for x 6= 0

together with the jump conditions at x = 0

ψ(0, t)def= ψ(0−, t) = ψ(0+, t)

∂xψ(0+, t)− ∂xψ(0−, t) = −|ψ(0, t)|p−1ψ(0, t)

Physical context

It models (p = 3) a wave propagating in a 1D linear medium whichcontains a narrow strip of nonlinear Kerr-type material. Thisnonlinear strip is assumed to be much smaller than the typicalwavelength.

In fact, periodic and quasiperiodic arrays of nonlinear strips havebeen considered by a generalization of this equation in order tomodel wave propagation in some nonlinear superlattices:

D. Hennig, G.P Tsironis, M.I. Molina, and H. Gabriel, A nonlinearquasiperiodic Kronig-Penney model, Physics Letters A 190 (1994),259–263.

Mathematically, it has been shown to be a limiting case of NLSwith concentrated nonlinearity

ε−1f (ε−1x)|ψ|p−1ψ as ε→ 0

C. Cacciapuoti, D. Finco, D. Noja, A. Teta, The NLS equation indimension one with spatially concentrated nonlinearities: thepointlike limit, Lett. Math. Phys. 104 (2014), no. 12, 1557–1570.

Point nonlinearity NLS is related to NLS with a delta potential

i∂tψ + ∂2xψ − qδψ + |ψ|p−1ψ = 0

which has been studied by (among many others)

R. Fukuizumi, M. Ohta, T. Ozawa, Nonlinear Schrodinger equationwith a point defect, Ann. Inst. H. Poincar Anal. Non Linaire 25(2008), no. 5, pp. 837–845.

S. Le Coz, R. Fukuizumi, G. Fibich, B. Ksherim, Y. Sivan,Instability of bound states of a nonlinear Schrodinger equation witha Dirac potential, Phys. D 237 (2008), no. 8, pp. 1103–1128.

who considered wellposedness, stability, instability of solitarywaves, and

J. Holmer, J. Marzuola, M. Zworski, Fast soliton scattering bydelta impurities, Comm. Math. Phys. 274 (2007), no. 1, 187–216.

who considered soliton splitting.

With C. Liu, we have undertaken a study of blow-up for 1D pointNLS

i∂tψ + ∂2xψ + δ|ψ|p−1ψ = 0

to draw comparison with blow-up for the 1D standard NLS

i∂tψ + ∂2xψ + |ψ|p−1ψ = 0

We find that many known results for standard NLS carry over topoint NLS with either strengthened conclusion, simplified proof, ormore explicit solution.

Our long-term goal is to attempt some unsolved problems forstandard NLS in the context of point NLS, such as

I higher order asymptotics for L2 critical log-log blow-up, see P.Lushnikov, S. Dyachenko, N.Vladimirova, Beyondleading-order logarithmic scaling in the catastrophicself-focusing of a laser beam in Kerr media, Physical Review A88 (2013), pp. 013845

I finite codimensional stability of Bourgain-Wang L2 criticalblow-up, see J. Krieger, W. Schlag, Non-generic blow-upsolutions for the critical focusing NLS in 1-D, J. Eur. Math.Soc. (JEMS) 11 (2009), no. 1, pp. 1–125.

I construction of blow-up for 4th order NLS , see G. Baruch, G.Fibich, Singular solutions of the L2-supercritical biharmonicnonlinear Schrodinger equation, Nonlinearity 24 (2011), no. 6,pp. 1843–1859.

I soliton resolution-type results, see T. Duyckaerts, C. Kenig, F.Merle, Classification of radial solutions of the focusing,energy-critical wave equation, Camb. J. Math. 1 (2013), no.1, pp. 75–144.

For now, we have

I Reproved classical results on L2 critical minimal mass blow upand the L2 critical/supercritical blow-up threshold.

I Constructed the self-similar blow-up profiles in the full L2

supercritical regime.

Results we discuss today:

I J. Holmer, C. Liu, Blow-up for the 1D nonlinear Schrdingerequation with point nonlinearity I: Basic theory,arXiv:1510.03491

I J. Holmer, C. Liu, Blow-up for the 1D nonlinear Schrodingerequation with point nonlinearity II: Supercritical blow-upprofiles, to be posted in a few days

1D point NLS obeys the scaling law

ψ(x , t) solves pNLS =⇒ ψλ(x , t) = λ1/(p−1)ψ(λx , λ2t) solves pNLS

The scale-invariant Sobolev norm Hσc satisfying

‖ψ‖Hσc = ‖ψλ‖Hσc

is

σc =1

2− 1

p − 1

Thus

I p = 3 (cubic) is the L2 critical setting σc = 0

I 3 < p <∞ is the L2 supercritical setting 0 < σc <12

In contrast, p = 5 is the L2 critical case for 1D standard NLS.

The conservation laws take the form

M(ψ) = ‖ψ‖2L2

E (ψ) =1

2‖∂xψ‖2L2 −

1

p + 1|ψ(0, ·)|p+1

There is no conservation of momentum, since Noether’s theoremlinks this to translational invariance,

An H1 local well-posedness theory is available:

R. Adami, A. Teta, A class of nonlinear Schrodinger equations withconcentrated nonlinearity, J. Funct. Anal. 180 (2001), no. 1, pp.148–175.

from which it follows that if the maximal forward time T∗ > 0 ofexistence is finite (i.e. T∗ <∞) then necessarily

limt↗T∗

‖ψx(t)‖L2 = +∞

we say that the solution ψ(t) blows-up at time t = T∗ > 0.

Solitary wave solutions

ψ(x , t) = e itϕ0(x)

where ϕ0 solves the stationary equation

0 = ϕ0 − ∂2xϕ0 − δ|ϕ0|p−1ϕ0

It is straightforward that the unique solution is

ϕ0(x) = 21/(p−1)e−|x |

Rescalings of this are the only solitary waves for point NLS.

The Pohozhaev identities take the form

‖ϕ0‖2L2 = ‖∂xϕ0‖2L2 =1

2‖ϕ0‖p+1

L∞ = 22

p−1

and can be verified by direct computation. Moreover, the analogueof

M.I. Weinstein, Nonlinear Schrodinger equations and sharpinterpolation estimates, Comm. Math. Phys. 87 (1982/83), no. 4,pp. 567–576.

is the following.

Proposition (sharp Gagliardo-Nirenberg inequality)

For any ψ ∈ H1,|ψ(0)|2 ≤ ‖ψ‖L2‖ψ′‖L2

Equality is achieved if and only if there exists θ ∈ R, α > 0, andβ > 0 such that ψ(x) = e iθαφ0(βx).

This has the important application of sharp criteria for blow-up inthe L2 supercritical and L2 critical cases

Theorem (L2 critical global existence/blow-up dichotomy)

Suppose that ψ(t) is an H1x solution to 1D point NLS.

1. If M(ψ) < M(φ0) = 2, then E (ψ) > 0 and ψ(t) satisfies thebound

‖ψx(t)‖2L2 ≤2E (ψ)

1− 12M(ψ)

and is hence a global solution (no blow-up).

2. If E (ψ) < 0 then ψ(t) blows-up in finite time.

Theorem (L2 supercritical global existence/blow-up dichotomy)

Suppose that ψ(t) is an H1x solution of 1D point NLS for p > 3

satisfying

M(ψ)1−σcσc E (ψ) < M(ϕ0)

1−σcσc E (ϕ0)

Let

η(t) =‖ψ‖

1−σcσc

L2‖ψx(t)‖L2x

‖ϕ0‖1−σcσc

L2‖(ϕ0)x‖L2

Then

1. If η(0) < 1, then the solution ψ(t) is global in both timedirections and η(t) < 1 for all t ∈ R.

2. If η(0) > 1, then the solution ψ(t) blows-up in the negativetime direction at some T− < 0, blows-up in the positive timedirection at some T+ > 0, and η(t) > 1 for all t ∈ (T−,T+).

For standard NLS, these types of global existence/blow-up criteriawere given in the 1970s by

I S.N. Vlasov, V.A. Petrishchev, and V.I. Talanov

I V.E. Zakharov

I Glassey

Additional sufficient criteria for blow-up given by

P.M. Lushnikov, Dynamic criterion for collapse, Pisma Zh. Eksp.Teor. Fiz. 62 (1995) pp. 447–452.

Our blow-up result for point NLS does not require a finite variancehypothesis since a localized virial identity suffices.

In the case of global existence for standard NLS, the question ofscattering has been addressed by

T. Duyckaerts, J. Holmer, S. Roudenko, Scattering for thenon-radial 3D cubic nonlinear Schrodinger equation, Math. Res.Lett. 15 (2008), no. 6, pp. 1233–1250.

Jason Murphy, The radial defocusing nonlinear Schrodingerequation in three space dimensions, Comm. PDE 40 (2015), pp.265–308

Benjamin Dodson, Global well-posedness and scattering for themass critical nonlinear Schrodinger equation with mass below themass of the ground state, arXiv:1104.1114

Threshold phenomena for standard NLS has been characterized by

T. Duyckaerts, S. Roudenko, Threshold solutions for the focusing3D cubic Schrodinger equation, Rev. Mat. Iberoam. 26 (2010),no. 1, pp. 1–56.

The problems of scattering and characterization of thresholdbehavior for 1D point NLS have not yet been studied.

For point NLS, in the case L2 critical case p = 3, there is anadditional symmetry, pseudoconformal transform, which is

ψ(x , t) solves pNLS

7→ ψ(x , t) =e ix

2/4t

t1/2ψ(

x

t,−1

t) solves pNLS

It is also valid for standard NLS in the L2 critical case, and wasobserved by Ginibre & Velo (1979).

Applying the pseudoconformal transformation to the ground statesolution e itϕ0(x), together with time reversal, time translation,and scaling symmetries, gives the explicit blow-up solution

Sλ,T∗(x , t) =e i/λ

2(T∗−t)e−ix2/4(T∗−t)

[λ(T∗ − t)]1/2ϕ0

(x

λ(T∗ − t)

)This is a solution with initial condition

Sλ,T∗(x , 0) =e i/λ

2T∗e−ix2/4T∗

(λT∗)1/2ϕ0

(x

λT∗

)that blows-up at forward time t = T∗ > 0. Moreover,‖Sλ,T∗‖L2x = ‖ϕ0‖L2 , and hence by a previous theorem, Sλ,T∗ is aminimal mass blow-up solution to point NLS.

Following a classical result for standard NLS

Frank Merle, Determination of blow-up solutions with minimalmass for nonlinear Schrodinger equations with critical power, DukeMathematical Journal 69 (1993), no. 2, 427–454.

Taoufik Hmidi and Sahbi Keraani, Blowup theory for the criticalnonlinear schrodinger equations revisited, InternationalMathematics Research Notices 2005 (2005), no. 46, 2815–2828.

we can show that a minimal mass blow-up solution of point NLS isnecessarily a pseudoconformal transformation of the ground state.

Theorem (characterization of minimal mass L2 critical blow-upsolutions)

Suppose that ψ(t) is an H1 solution to point NLS that blows-up attime T∗ > 0 and ‖ψ0‖L2 = ‖ϕ0‖L2 . Then there exists θ ∈ R andλ > 0 such that ψ = e iθSλ,T∗ .

Explicit form of blow-up. For the standard NLS

i∂tu + ∆u + |u|p−1u = 0

we have the scaling relationship for constant λ > 0

u(x , t) solves NLS =⇒ uλ(x , t) = λ2/(p−1)u(λx , λ2t) solves NLS

It is natural to take λ(t) unknown and transform u 7→ v where

u(x , t) = λ2/(p−1)v(λ(t)x , s(t)) ,ds

dt= λ2

u(x , t) solving NLS is then equivalent to v(z , s) solving

i∂sv + ∆v + ihΛv − ihσcv + |v |p−1v = 0

where

h = λsλ , Λ =d

2+ z · ∇

Then we can consider two cases

I Type I, or self-simliar, h(s) = h0 > 0 a constant

I Type II, h(s)↘ 0

In the case for which σc = d2 −

1p−1 satisfies 0 ≤ σc < 1, take

v(z , s) = e isη(y)

and obtain a profile equation

η −∆η − ihΛη + ihσcη − |η|p−1η = 0 (∗)

I Type I case, h = h0 > 0, and η solving (*) is called aself-similar profile. We seek such solutions in the case0 < σc < 1

I Type II case, take h = 0 in (*) which reduces to η = Q theground state solitary wave profile. We seek such blow-upsolutions in the case σc = 0.

In the σc = 0, Type II case, there are two main types,

I pseudoconformal and Bourgain-Wang family

J. Bourgain, W. Wang, Construction of blowup solutions forthe nonlinear Schrodinger equation with critical nonlinearity,Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. PisaCl. Sci. (4) 25 (1997), no. 1-2, 197–215.

I log-log blow-up solutions, heuristically derived by

G.M. Fraiman, Asymptotic stability of manifold of self-similarsolutions in self-focusing, Soviet Phys. JETP 61 (1985), no.2, pp. 228–233;

M.J. Landman, G.C. Papanicolaou, C. Sulem, P.-L. Sulem,Rate of blowup for solutions of the nonlinear Schrodingerequation at critical dimension, Phys. Rev. A (3) 38 (1988),no. 8, pp. 3837–3843.

. . . and rigorously constructed in 1D quintic in weighted space by

Galina Perelman, On the formation of singularities in solutions ofthe critical nonlinear Schrodinger equation, Ann. Henri Poincar 2(2001), no. 4, pp. 605–673.

F. Merle, P. Raphael, (5 papers), 2001–2005 in dimensions1 ≤ d ≤ 5 and in H1

In the slightly L2 supercritical case 0 < σc � 1, the Type Ifinite-energy self-similar blow-up profiles were constructed usingdynamical systems techniques by

Nancy Kopell and Michael Landman, Spatial structure of thefocusing singularity of the nonlinear Schrodinger equation: ageometrical analysis, SIAM J. Appl. Math. 55 (1995), no. 5,1297–1323

and corresponding H1 blow-up has been shown by

F. Merle, P. Raphael, J. Szeftel, Stable self-similar blow-updynamics for slightly L2 super-critical NLS equations., Geom.Funct. Anal. 20 (2010), no. 4, pp. 1028–1071.

Interestingly other self-similar profiles were later found: the multibump solutions

Chris J. Budd, Shaohua Chen, and Robert D. Russell, Newself-similar solutions of the nonlinear Schrodinger equation withmoving mesh computations, J. Comput. Phys. 152 (1999), no. 2,756–789.

Vivi Rottschafer and Tasso J. Kaper, Geometric theory formulti-bump, self-similar, blowup solutions of the cubic nonlinearSchrodinger equation, Nonlinearity 16 (2003), no. 3, 929–961.MR1398655

Textbook treatments of NLS blow-up by our conference organizers:

Catherine Sulem and Pierre-Louis Sulem, The nonlinearSchrodinger equation: self-focusing and wave collapse, AppliedMathematical Sciences, vol. 139, Springer-Verlag, New York, 1999.

Gadi Fibich, The nonlinear Schrodinger equation, AppliedMathematical Sciences, vol. 192, Springer, Cham, 2015, Singularsolutions and optical collapse.

We now look at 1D point NLS in the full supercritical range0 < σc <

12 . We start with a simple observation from substitution:

Lemma (structure of L2 supercritical self-similar blow-upsolutions)

The function

ψ(x , t) = λ(t)1/(p−1)e iτ(t)η(λ(t)x)

solves 1D point NLS for p > 3 with limt↗T∗ λ(t) = +∞ if andonly if there exists h > 0 and κ ∈ R such that

λ(t) =1√

2h(T∗ − t), τ(t) =

κ

2hln

(T∗

T∗ − t

)+ τ(0)

and η(z) solves the stationary equation

(κ+ ihσc)η − ihΛzη − ηzz − δ|η|p−1η = 0 , Λz =1

2+ z∂z

To better understand the blow-up profile, let us drop therelationship between σc and p (given by σc = 1

2 −1

p−1) and just

consider the equation for arbitrary h > 0, and 0 < σ < 12

−ihΛzη − ηzz − δ|η|p−1η = (−1− ihσ)η

When does this have a solution in the energy space? Taking

η(z) = e−14iz2hϕ(z), then η(z) solves the above if and only if ϕ(z)

solves

−ϕzz −1

4h2z2ϕ− δ|ϕ|p−1ϕ = (−1− ihσ)ϕ

Two independent solutions of the eigenvalue problem for theinverted harmonic well

−wzz −1

4h2z2w = hλw

are given by w(z , λ), w∗(z , λ) defined by certain integral formulaecalled the parabolic Weber functions

. . . with the asymptotic behavior

w(z) ∼ (h1/2z)iλ−12 e

14ihz2eπλ/4e iπ/8 as z → +∞

w∗(z) ∼ (h1/2z)−iλ−12 e−

14ihz2eπλ/4e−iπ/8 as z → +∞

Thus, the most general solution ϕ is given by

ϕ(z) =

{α+w(z) + α∗+w

∗(z) for z > 0

α−w(−z) + α∗−w∗(−z) for z < 0

for λ = −h−1− iσ, where the four complex constants α+, α∗+, α−,α∗− must be chosen to satisfy the jump conditions.

We will not address this general problem since we seek a finite

energy solution ϕ for which η(z) = e−14ihz2ϕ(z) ∈ L∞z ∩ H1

z , whichfurther constrains the problem to

α∗+ = 0 , α∗− = 0

Such a solution ϕ is called outgoing.

Since we must have continuity, we need α+ = α− and hence

ϕ(z) = αw(|z |)

for some α ∈ C. The derivative jump condition

∂xϕ(0+)− ∂xϕ(0−) = −|ϕ(0)|p−1ϕ(0)

becomes2wz(0, λ) = −|w(0, λ)|p−1w(0, λ)|α|p−1

A solution α exists if and only if

2h1/2A(λ)def= −2wz(0, λ)

w(0, λ)is real and positive

The explicit integral formula for the parabolic Weber function gives

A(λ) =e−iπ/4

√2Γ(34 −

12 iλ)

Γ(14 −12 iλ)

, λ = −h−1 − iσ

Thus the existence of finite energy outgoing solutions ϕ has beenreduced to the algebraic condition:

Given h > 0, find 0 < σ(h) < 12 such that A(λ) is real and positive

where

A(λ) =e−iπ/4

√2Γ(34 −

12 iλ)

Γ(14 −12 iλ)

, λ = −h−1 − iσ

We find it convenient to convert this to

f (σ, h−1) = Im logA(λ) ∈ 2πZ

Using the behavior for σ = 0 and σ = 1, we prove existence usingthe intermediate value theorem and uniqueness by showing∂σf > 0 from an identity for the digamma function.

TheoremFor each 0 < h <∞, there exists a unique 0 < σ(h) < 1 such thatA(λ) is real and positive, and thus a corresponding outgoingsolution ϕ.

From the Binet log gamma formula

log Γ(z) = (z − 1

2) log z − z + 1

2 ln(2π) + 2

∫ +∞

0

arctan(tz−1)

e2πt − 1dt

we derive a Stirling type estimate that captures exponentially smalldifferences and obtain

log Γ(z + σ)− log Γ(z) = σ log z − 12σ(1− σ)z−1 + O(σ|z |−2)

with z = 14 −

12σ −

12 ih−1 = −1

2 ih−1(1 + 1

2 ih − iσh)

TheoremFor all 0 < h� 1, the unique value σ(h) such that A(λ) is realand positive is

σ(h) = 2e−π/hh−1(1 + O(h))

We numerically solved for σ(h) using the MATLAB fzero

function.

h -10 0.5 1 1.5 2 2.5 3

sigm

a

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

The numerical solution shows that σ(h−1) is decreasing as afunction of h−1, starting from σ = 1

2 at the limit h−1 = 0. Thuswe have the numerical finding of unique inverse: for each0 < σ < 1

2 , there exists a unique 0 < h <∞ such that σ = σ(h).

h -10 0.5 1 1.5 2 2.5 3

log(

sigm

a)

-8

-7

-6

-5

-4

-3

-2

-1

0

Plot of numerical solution of log σ(h) versus h−1 in solid black,together with h−1 →∞ asymptotic

log σ(h) = log 2− πh−1 + log h−1

in dashed red, showing good agreement for h−1 > 1.

As h→ 0, we can also derive the asymptotic expansion

ϕ(x) ≈

{c−,h(1− 1

4h2x2)−1/4e−h

−1[arcsin( 12hx)+ 1

2hx(1− 1

4h2x2)1/2] for x � 2h−1

c+,he14ix2x−ih

−1+σ− 12 for x � 2h−1

where c−,h = 21/(p−1), c+,h = 21/(p−1)21/2eπi/4e−12ih−1

e−π/(2h).

This follows from the integral form the parabolic Weber function,contour deformation, and stationary phase. Thus the solutionbehaves as

I For x � 2h−1, initially like 21/(p−1)e−|x |, with a correction tothe decay rate that comes from the potential

I For x � 2h−1, has algebraic decay xσ−12 with the outgoing

quadratic phase e14ix2 .

This is typically proved using WKB, but we find it natural to usethe explicit integral solution and methods for computing integralasymptotics.