Singularity Formation in Derivative Nonlinear Schrödinger ... · Singularity Formation in...

Singularity Formation in Derivative NonlinearSchrodinger Equations

Gideon Simpson

School of MathematicsDrexel University

November 1, 2016

Joint withY. Cher (Toronto), X. Liu (Toronto), C. Sulem (Toronto)

Simpson (Drexel) DNLSIMA November 1, 2016 1 / 47

Outline

1 Background & OverviewStructural PropertiesPrevious ResultsChallenges & Results

2 Time Dependent Simulations of Finite Time SingularitiesDirect Evidence for Singularity FormationDynamic Rescaling

3 Inferences from Computation of Blowup ProfilesComputed ProfilesRefined Local Analysis

4 Time Dependent Simulations Revisited Adaptive MethodsComputational ChallengeAdaptive Meshing


Background & Overview

Derivative Nonlinear Schrodinger Equation (DNLS)

Characteristic Form: it + i ||2x + xx = 0 (1a)Conservation Form: it + i

(||2

)x

+ xx = 0 (1b)

(1a) becomes (1b) via the Gauge transformation

= exp

{ i

2

x||2dx

}. (2)

Generalized DNLS (gDNLS)

Characteristic Form: it + i ||2x + xx = 0 (3a)Conservation Form: it + i

(||2

)x

+ xx = 0 (3b)

No known analog of the Gauge transformation for 6= 1.



Physical Origin of DNLS

Weakly nonlinear Alfven waves in plasma physics, in a long wavelength approximation, are governed by

it + i(||2

)x

+ xx = 0

where = y + iz is the magnetic field in the transverse direction

Self-steepening optical pulses can be modeled by

it + i ||2x + xx = 0

where represents the dimensionless, complex valued, slowly varyingenvelope of the electric field (CLLE)

gDNLS has not (yet) appeared in a physical model



Integrability

Cubic DNLS equation is completely integrable, Kaup-Newell (1978)

Orbitally stable solitons in modulated H1, Colin-Ohta (2006)

An L2 critical integrable equation

Recent work on well-posedness via inverse scattering: Liu, Perry, &Sulem (2015), Pelinovksy & Shimabukuro (2016)


Background & Overview Structural Properties

gDNLS Scaling

Scaling

Given a solution (x , t)

(x , t) = 1

2(x , 2t) (4)

is also a solution

Norm Scaling

L2 based Sobolev norms scale as

Hs = s+ 1

2 (11)Hs (5)

The scale invariant Sobolev norm, Hsc = Hsc , is

sc =12

(1 1

), Critical Sobolev Exponent (6)



L2 Critical Equations and Singularities

Focusing NLSiut + u + |u|2u = 0 (7)

has finite time singularities for d 2; d = 2 is L2 criticalgkDV

ut + unux + uxxx = 0 (8)

has finite time singularities for n 4; n = 4 is L2 critical



gDNLS Invariants

Mass, L2, (Critical norm for = 1)

Q[] =

||2dx (9)

Momentum

P[] =

Dx, Dx 1i x (10)

Hamiltonian

E [] =

|x |2 + 12(+1)

+1Dx+1 (11a)

t = iE

(11b)


Background & Overview Previous Results

Well-Posedness Results

Cubic Nonlinearity (Hayashi, Hayashi & Ozawa (1992, 1993),. . . )

By a change of variables, cubic DNLS is equivalent to

iUt + Uxx = iU2V , iVt + Vxx = iV 2U,

GWP for small data 0L2 0, a constant, very rapidlyThen

L(t)2 2At + K

Take K = 2At? > 0 since L(0) > 0.

Hence,L(t) =

2A(t? t);

Length scale goes to zero.

By our choice of L,

L(t) = u(, 0)qL2x(, t)q, q > 0,

if L 0, x(, t) .


Time Dependent Simulations of Finite Time Singularities Dynamic Rescaling

Implications of a and L

a(t) = L(t)L(t)Assume that a A > 0, a constant, very rapidly

ThenL(t)2 2At + K

Take K = 2At? > 0 since L(0) > 0.

Hence,L(t) =

2A(t? t);


By our choice of L,

L(t) = u(, 0)qL2x(, t)q, q > 0,

if L 0, x(, t) .



Implications of a and L

a(t) = L(t)L(t)Assume that a A > 0, a constant, very rapidlyThen

L(t)2 2At + K

Take K = 2At? > 0 since L(0) > 0.

Hence,L(t) =

2A(t? t);


By our choice of L,

L(t) = u(, 0)qL2x(, t)q, q > 0,

if L 0, x(, t) .



Dynamic Rescaling Results = 2, Quintic Case

1000

100 02

46

0

1

2

|u|

0 2 4 60

1

2

3

4

a

Since a A > 0, we conclude a collapse occurs in finite time.

Appears to be generic; other initial conditions lead to similar behavioru(, ) S()e iC , a fixed profile. After rescaling S Q,

Q Q + i(

14Q + Q

) iQ + i |Q|4Q = 0 (21)




1000

100 02

46

0

1

2

|u|

0 2 4 60

1

2

3

4

a

Since a A > 0, we conclude a collapse occurs in finite time.Appears to be generic; other initial conditions lead to similar behavior

u(, ) S()e iC , a fixed profile. After rescaling S Q,Q Q + i

(14Q + Q

) iQ + i |Q|4Q = 0 (21)




1000

100 02

46

0

1

2

|u|

0 2 4 60

5

10

15

20

25

30

b

Since a A > 0, we conclude a collapse occurs in finite time.Appears to be generic; other initial conditions lead to similar behavior

u(, ) S()e iC , a fixed profile. After rescaling S Q,Q Q + i

(14Q + Q

) iQ + i |Q|4Q = 0 (21)




1000

100 02

46

0

1

2

|u|

0 2 4 60

1

2

3

4

a

Since a A > 0, we conclude a collapse occurs in finite time.Appears to be generic; other initial conditions lead to similar behavioru(, ) S()e iC , a fixed profile. After rescaling S Q,

Q Q + i(

14Q + Q

) iQ + i |Q|4Q = 0 (21)



Scaling Parameters for other Values of 1.1

0 0.2 0.4 0.6 0.8 110

2

101

100

101

102

/M

a()aM

= 1.1, aM = 0.13, M = 9 = 1.3, aM = 1.09, M = 4.6 = 1.5, aM = 3.77, M = 1.5 = 1.7, aM = 0.54, M = 6

0 0.2 0.4 0.6 0.8 1

100

101

/M

b()bM

= 1.1, bM = 3.18, M = 9 = 1.3, bM = 2.76, M = 4.6 = 1.5, bM = 2.66, M = 1.5 = 1.7, bM = 1.24, M = 6

In all cases, we find a A() > 0 and b B()As 1, the asymptotic is harder to resolveRecall the rescaled blowup profile:

Q Q + i(

12Q + Q

) iQ + i |Q|2Q = 0

with Q, and only depending on .


Inferences from Computation of Blowup Profiles

1 Background & Overview

2 Time Dependent Simulations of Finite Time Singularities

3 Inferences from Computation of Blowup ProfilesComputed ProfilesRefined Local Analysis

4 Time Dependent Simulations Revisited Adaptive Methods


Inferences from Computation of Blowup Profiles Computed Profiles

Blowup ProfileLiu, Simpson & Sulem, Physica D (2013)

Inverting coordinate transformations, as t t?

(x , t) [

12a(t?t)

] 14

Q

(xx?2a(t?t)

+ ba

)e i(+

12a

log t?t?t

)

Q, a and b are universal; they do not depend on 0; Q is an attractor

x?, t? and will depend on the data

Understanding Q gives insight into singularity formation

Existence/Uniqueness of (Q, a, b) is an open problem would providerigorous proof of finite time singularity, but would not be in energyspace



Numerically Computed Blowup ProfilesLiu, Simpson & Sulem, Physica D (2013), 1.08

20 10 0 10 201.2

1.4

1.6

1.8

2

0

0.5

1

1.5

2

2.5

|Q|

1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

Profiles appear to converge as 1 appears to go to zero; = 0 at = 1 would be inconclusive

tends to a finite, nonzero constant



Numerically Computed Blowup ProfilesLiu, Simpson & Sulem, Physica D (2013), 1.08

20 10 0 10 201.2

1.4

1.6

1.8

2

0

0.5

1

1.5

2

2.5

|Q|

1 1.2 1.4 1.6 1.8 21.5

1.6

1.7

1.8

1.9

2

2.1

2.2

Profiles appear to converge as 1 appears to go to zero; = 0 at = 1 would be inconclusive

tends to a finite, nonzero constant


Inferences from Computation of Blowup Profiles Refined Local Analysis

Profile Properties as 1Cher, Simpson, & Sulem, arXiv:1602.02381

-20 -15 -10 -5 0 5 10 15 20

|Q|

0

0.5

1

1.5

2

2.5

3 = 1.044

= 1.06

= 1.1

New code and refined asymptotics allow 1.044Time independent nonlinear solver Parallel (large mesh/largedomain) finite difference scheme



A as 1Cher, Simpson, & Sulem, arXiv:1602.02381

Large behavior of Q:

Q A||1

2

(1 b

2a||

)exp

{ ia

(log || b

a||

)}(22)

with

A

4( 1) (23)

A+ 43/4a1/2 exp

{a

+2

3

(2 b)3/2

a

}(24)



A as 1, ContinuedCher, Simpson, & Sulem, arXiv:1602.02381

10.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1A2

a11/(k + l)(1 1/)

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

10-300

10-200

10-100

100

1

A+3/4

aexp

(

a+ 2

3

3/2

a

)

NOTE: |Q| has a much larger prefactor for < 0



a and b ParametersCher, Simpson, & Sulem, arXiv:1602.02381

1

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

a

0

0.05

0.1

0.15

0.2

1

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Combining asymptotics and numerics, we predict

a ( 1)a , a 3.2 = 2 b ( 1)b , b = 2

Note: Blowup solutions will have zero momentum (and energy) used toclose the system of equations for the constants


Time Dependent Simulations Revisited Adaptive Methods

1 Background & Overview

2 Time Dependent Simulations of Finite Time Singularities

3 Inferences from Computation of Blowup Profiles

4 Time Dependent Simulations Revisited Adaptive MethodsComputational ChallengeAdaptive Meshing


Time Dependent Simulations Revisited Adaptive Methods Computational Challenge

Boundary Conditions and Long Time Integration

Dynamic Rescaling Region

To the left, u is large

Slow, ||1/(2) decay, and largeA

1 constant shows

up in the rescaled coordinates

Large domain needed for RobinBoundary Conditions

Original x coordinate allowssimple Dirichlet conditions, butresolution needed at singularity

Q Q + ia(

12Q + Q

) ibQ + i |Q|2Q = 0 (25)




10 5 0 5 10

4

3

2

1

0

1

2

3

4s = 10.0, t = 0.00085254088085

Re. v

Im. v

|v|



1 constant shows




Q Q + ia(

12Q + Q

) ibQ + i |Q|2Q = 0 (25)




3 2 1 0 1 2 3x

0

2

4

6

8

s = 1.0, t = 0.00083115941778

Re. u

Im. u

|u|



1 constant shows




Q Q + ia(

12Q + Q

) ibQ + i |Q|2Q = 0 (25)



Boundary Conditions

Time Independent Problem

Do not need Dirichlet conditions, but domain must be large enough thatRobin Boundary conditions can be developed for:

Q Q + ia(

12Q + Q

) ibQ +

XXXXXi |Q|2Q = 0 (26)

Time Dependent Problem

Moderate unscaled domain (periodic/Dirichlet BCs) with a uniformmesh on original problem cannot resolve singularity

Dynamic rescaling still requires a large domain to accomodate the BCs

Absorbing boundary conditions?

Unscaled problem on moderate domain with adaptive mesh



Aside, gKdV

ut + unux + uxxx = 0,

n 4 corresponds to L2 critical/supercriticalFinite time singularities known to occur

Mixture of hyperbolic/dispersive terms like gDNLS,

t + ||2x ixx = 0



gKdV Simulations from Klein & Peter (2015)n = 5, Supercritical

Trailing edge has slower decay

Complicates boundary conditions for dynamic rescaling method


Time Dependent Simulations Revisited Adaptive Methods Adaptive Meshing

Equal Arc Length Placement

Pick the location of the mesh,

xmin = x0 < x1 < x2 < . . . < xN < xN+1 = xmax (27)

such that xi+1xi

1 + |x |2dx = Constant (28)

Dynamic rescaling uniformly distributes mesh points near singularity

Boundary conditions are better handled no longer computing justnear the singularity

Winslow (1967), Budd, Huang, & Russell (2009), Ren & Wang(2000), Fibich, Ren & Wang (2003), Ditkowski & Gavish (2009),...



Modified Equation

Based on uH1 (t? t), with = (), take

(t) = u1/H1

, s =

t0

dt

(t )(29)

Modified equation

ius + i(s)|u|2ux + (s)uxx = 0 (30)

Singularity moved to s , but spatial scale unchanged.



Adaptive Algorithm

Strang splitting of hyperbolic & dispersive piece:

Hyperbolic: ius + i(s)|u|2ux = 0 (31)Dispersive: ius + (s)uxx = 0 (32)

Hyperbolic piece solved exactly in Lagrangian coordinates by methodfo characteristics

Dispersive piece solved by Crank-Nicolson time stepping with finiteelement discretization & simple Dirichlet conditions

Mesh adapted to satisfy equal arc length constraint



Adaptive Algorithm, Continued

Half Step of Dispersive Scheme

(M(x) is4 K (x))u(1) = (M(x) + is4 K (x))u (33)

Full Step of Lagrangian Hyperbolic Scheme

x 7 x + s|u(1)|2 = x(1) (34)

Remesh & Update Use equal arc length constraint to obtain(x,u(2))update K (x), M(x) and

Half Step of Dispersive Scheme

(M(x) is4 K (x))u = (M(x) +is

4 K (x))u(2) (35)



Adaptive Meshing, = 2, Real & Imaginary Parts



Adaptive Meshing, = 2, Mesh & Amplitude



Adaptive Meshing, = 2, Dynamic Rescaling Coordinates



Adaptive Meshing, = 2, Scalars

10-6 10-5 10-4 10-3

t

100

101

102

103

104

105

106

107

108

109

1010

uLuH11/L



Adaptive Meshing, = 1.05, Real & Imaginary Parts



Adaptive Meshing, = 1.05, Dynamic Rescaling



Adaptive Meshing, = 1.05, Scalars

10-5 10-4 10-3 10-2

t

100

101

102

103

104

105

uLuH11/L


Summary & Acknowledgements

Summary

Supercritical gDNLS appears to have finite time singularities with auniversal blowup profile

Ongoing work to study the 1 limit via adaptive methodsLarge data well posedness in the energy space in H1 remainsunresolved



Summary


Ongoing work to study the 1 limit via adaptive methods

Large data well posedness in the energy space in H1 remainsunresolved



Summary


Ongoing work to study the 1 limit via adaptive methodsLarge data well posedness in the energy space in H1 remainsunresolved



Acknowledgements

5 0 5 10 0

0.1

0.2

0.3

0.4

0

5

10

t

x

||

= 1 with large data

Collaborators Y. Cher(Toronto), X. Liu (Toronto) & C. Sulem (Toronto)

Publications Liu, Simpson, Sulem Phys. D (2013)Cher, Simpson, Sulem arXiv:1602.02381

Funding NSERC, NSF, DOE, NSF DMS-1409018

http://www.math.drexel.edu/~simpson/

http://www.math.drexel.edu/~simpson/Background & OverviewStructural PropertiesPrevious ResultsChallenges & ResultsTime Dependent Simulations of Finite Time SingularitiesDirect Evidence for Singularity FormationDynamic RescalingInferences from Computation of Blowup ProfilesComputed ProfilesRefined Local AnalysisTime Dependent Simulations Revisited Adaptive MethodsComputational ChallengeAdaptive Meshingfd@rm@0: fd@rm@1: fd@rm@2: fd@rm@3: fd@rm@4: fd@rm@5: fd@rm@6:

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