-
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
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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
Simpson (Drexel) DNLSIMA November 1, 2016 2 / 47
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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.
Simpson (Drexel) DNLSIMA November 1, 2016 3 / 47
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Background & Overview
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
Simpson (Drexel) DNLSIMA November 1, 2016 4 / 47
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Background & Overview
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)
Simpson (Drexel) DNLSIMA November 1, 2016 5 / 47
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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)
Simpson (Drexel) DNLSIMA November 1, 2016 6 / 47
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Background & Overview Structural Properties
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
Simpson (Drexel) DNLSIMA November 1, 2016 7 / 47
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Background & Overview Structural Properties
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)
Simpson (Drexel) DNLSIMA November 1, 2016 8 / 47
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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) .
Simpson (Drexel) DNLSIMA November 1, 2016 20 / 47
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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);
Length scale goes to zero.
By our choice of L,
L(t) = u(, 0)qL2x(, t)q, q > 0,
if L 0, x(, t) .
Simpson (Drexel) DNLSIMA November 1, 2016 20 / 47
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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
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) .
Simpson (Drexel) DNLSIMA November 1, 2016 20 / 47
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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
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) .
Simpson (Drexel) DNLSIMA November 1, 2016 20 / 47
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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
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) .
Simpson (Drexel) DNLSIMA November 1, 2016 20 / 47
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Time Dependent Simulations of Finite Time Singularities Dynamic
Rescaling
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)
Simpson (Drexel) DNLSIMA November 1, 2016 21 / 47
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Time Dependent Simulations of Finite Time Singularities Dynamic
Rescaling
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 behavior
u(, ) S()e iC , a fixed profile. After rescaling S Q,Q Q + i
(14Q + Q
) iQ + i |Q|4Q = 0 (21)
Simpson (Drexel) DNLSIMA November 1, 2016 21 / 47
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Time Dependent Simulations of Finite Time Singularities Dynamic
Rescaling
Dynamic Rescaling Results = 2, Quintic Case
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)
Simpson (Drexel) DNLSIMA November 1, 2016 21 / 47
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Time Dependent Simulations of Finite Time Singularities Dynamic
Rescaling
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)
Simpson (Drexel) DNLSIMA November 1, 2016 21 / 47
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Time Dependent Simulations of Finite Time Singularities Dynamic
Rescaling
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 .
Simpson (Drexel) DNLSIMA November 1, 2016 22 / 47
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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
Simpson (Drexel) DNLSIMA November 1, 2016 23 / 47
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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
Simpson (Drexel) DNLSIMA November 1, 2016 24 / 47
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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
Simpson (Drexel) DNLSIMA November 1, 2016 24 / 47
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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
Simpson (Drexel) DNLSIMA November 1, 2016 24 / 47
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Inferences from Computation of Blowup Profiles Computed
Profiles
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
Simpson (Drexel) DNLSIMA November 1, 2016 25 / 47
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Inferences from Computation of Blowup Profiles Computed
Profiles
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
Simpson (Drexel) DNLSIMA November 1, 2016 25 / 47
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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
Simpson (Drexel) DNLSIMA November 1, 2016 26 / 47
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Inferences from Computation of Blowup Profiles Refined Local
Analysis
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)
Simpson (Drexel) DNLSIMA November 1, 2016 27 / 47
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Inferences from Computation of Blowup Profiles Refined Local
Analysis
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
Simpson (Drexel) DNLSIMA November 1, 2016 28 / 47
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Inferences from Computation of Blowup Profiles Refined Local
Analysis
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
Simpson (Drexel) DNLSIMA November 1, 2016 29 / 47
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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
Simpson (Drexel) DNLSIMA November 1, 2016 30 / 47
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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)
Simpson (Drexel) DNLSIMA November 1, 2016 31 / 47
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Time Dependent Simulations Revisited Adaptive Methods
Computational Challenge
Boundary Conditions and Long Time Integration
10 5 0 5 10
4
3
2
1
0
1
2
3
4s = 10.0, t = 0.00085254088085
Re. v
Im. v
|v|
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)
Simpson (Drexel) DNLSIMA November 1, 2016 31 / 47
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Time Dependent Simulations Revisited Adaptive Methods
Computational Challenge
Boundary Conditions and Long Time Integration
3 2 1 0 1 2 3x
0
2
4
6
8
s = 1.0, t = 0.00083115941778
Re. u
Im. u
|u|
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)
Simpson (Drexel) DNLSIMA November 1, 2016 31 / 47
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Time Dependent Simulations Revisited Adaptive Methods
Computational Challenge
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
Simpson (Drexel) DNLSIMA November 1, 2016 32 / 47
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Time Dependent Simulations Revisited Adaptive Methods
Computational Challenge
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
Simpson (Drexel) DNLSIMA November 1, 2016 33 / 47
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Time Dependent Simulations Revisited Adaptive Methods
Computational Challenge
gKdV Simulations from Klein & Peter (2015)n = 5,
Supercritical
Trailing edge has slower decay
Complicates boundary conditions for dynamic rescaling method
Simpson (Drexel) DNLSIMA November 1, 2016 34 / 47
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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),...
Simpson (Drexel) DNLSIMA November 1, 2016 35 / 47
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Time Dependent Simulations Revisited Adaptive Methods Adaptive
Meshing
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.
Simpson (Drexel) DNLSIMA November 1, 2016 36 / 47
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Time Dependent Simulations Revisited Adaptive Methods Adaptive
Meshing
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
Simpson (Drexel) DNLSIMA November 1, 2016 37 / 47
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Time Dependent Simulations Revisited Adaptive Methods Adaptive
Meshing
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)
Simpson (Drexel) DNLSIMA November 1, 2016 38 / 47
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Time Dependent Simulations Revisited Adaptive Methods Adaptive
Meshing
Adaptive Meshing, = 2, Real & Imaginary Parts
Simpson (Drexel) DNLSIMA November 1, 2016 39 / 47
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Time Dependent Simulations Revisited Adaptive Methods Adaptive
Meshing
Adaptive Meshing, = 2, Mesh & Amplitude
Simpson (Drexel) DNLSIMA November 1, 2016 40 / 47
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Time Dependent Simulations Revisited Adaptive Methods Adaptive
Meshing
Adaptive Meshing, = 2, Dynamic Rescaling Coordinates
Simpson (Drexel) DNLSIMA November 1, 2016 41 / 47
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Time Dependent Simulations Revisited Adaptive Methods Adaptive
Meshing
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
Simpson (Drexel) DNLSIMA November 1, 2016 42 / 47
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Time Dependent Simulations Revisited Adaptive Methods Adaptive
Meshing
Adaptive Meshing, = 1.05, Real & Imaginary Parts
Simpson (Drexel) DNLSIMA November 1, 2016 43 / 47
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Time Dependent Simulations Revisited Adaptive Methods Adaptive
Meshing
Adaptive Meshing, = 1.05, Dynamic Rescaling
Simpson (Drexel) DNLSIMA November 1, 2016 44 / 47
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Time Dependent Simulations Revisited Adaptive Methods Adaptive
Meshing
Adaptive Meshing, = 1.05, Scalars
10-5 10-4 10-3 10-2
t
100
101
102
103
104
105
uLuH11/L
Simpson (Drexel) DNLSIMA November 1, 2016 45 / 47
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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
Simpson (Drexel) DNLSIMA November 1, 2016 46 / 47
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Summary & Acknowledgements
Summary
Supercritical gDNLS appears to have finite time singularities
with auniversal blowup profile
Ongoing work to study the 1 limit via adaptive methods
Large data well posedness in the energy space in H1
remainsunresolved
Simpson (Drexel) DNLSIMA November 1, 2016 46 / 47
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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
Simpson (Drexel) DNLSIMA November 1, 2016 46 / 47
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Summary & Acknowledgements
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/
Simpson (Drexel) DNLSIMA November 1, 2016 47 / 47
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: