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BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate quasilinear Schrödinger equations
Evan Smothers
June 2, 2015
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Table of Contents
1 Background
2 Degenerate Schrödinger Equations
3 MRS equation
4 Moving forward
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Outline
Goal: study degenerate dispersive equations
Consider two distinct types:
KdVSchrödinger
Specific problem: studying a degenerate nonlinear Schrödingerequation
Main questions:
Local and global existenceWavefront behavior
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Outline
Goal: study degenerate dispersive equations
Consider two distinct types:
KdVSchrödinger
Specific problem: studying a degenerate nonlinear Schrödingerequation
Main questions:
Local and global existenceWavefront behavior
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Outline
Goal: study degenerate dispersive equations
Consider two distinct types:
KdVSchrödinger
Specific problem: studying a degenerate nonlinear Schrödingerequation
Main questions:
Local and global existenceWavefront behavior
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Outline
Goal: study degenerate dispersive equations
Consider two distinct types:
KdVSchrödinger
Specific problem: studying a degenerate nonlinear Schrödingerequation
Main questions:
Local and global existenceWavefront behavior
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Outline
Goal: study degenerate dispersive equations
Consider two distinct types:
KdVSchrödinger
Specific problem: studying a degenerate nonlinear Schrödingerequation
Main questions:
Local and global existenceWavefront behavior
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Outline
Goal: study degenerate dispersive equations
Consider two distinct types:
KdVSchrödinger
Specific problem: studying a degenerate nonlinear Schrödingerequation
Main questions:
Local and global existenceWavefront behavior
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Dispersive Equations
Equations where different wavenumbers propagate withdifferent group velocities
Plane wave solutions Ae i(kx−ωt) give frequency in terms ofwavenumber: ω = ω(k) (dispersion relation)
Phase velocity ωk and group velocitydωdk differ
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Dispersive Equations
Equations where different wavenumbers propagate withdifferent group velocities
Plane wave solutions Ae i(kx−ωt) give frequency in terms ofwavenumber: ω = ω(k) (dispersion relation)
Phase velocity ωk and group velocitydωdk differ
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Dispersive Equations
Equations where different wavenumbers propagate withdifferent group velocities
Plane wave solutions Ae i(kx−ωt) give frequency in terms ofwavenumber: ω = ω(k) (dispersion relation)
Phase velocity ωk and group velocitydωdk differ
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
KdV
Asymptotic equation for unidirectional shallow water waves:
The KdV equation
ut + uxxx + 6uux = 0
Completely integrable
Existence of soliton solutions
Soliton: A localized wave of constant speed that emerges fromcollisions with other solitons unchanged but for a phase shift.
KdV soliton
c
2sech2
(√c
2(x − ct − a)
)Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
KdV
Asymptotic equation for unidirectional shallow water waves:
The KdV equation
ut + uxxx + 6uux = 0
Completely integrable
Existence of soliton solutions
Soliton: A localized wave of constant speed that emerges fromcollisions with other solitons unchanged but for a phase shift.
KdV soliton
c
2sech2
(√c
2(x − ct − a)
)Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
KdV
Asymptotic equation for unidirectional shallow water waves:
The KdV equation
ut + uxxx + 6uux = 0
Completely integrable
Existence of soliton solutions
Soliton: A localized wave of constant speed that emerges fromcollisions with other solitons unchanged but for a phase shift.
KdV soliton
c
2sech2
(√c
2(x − ct − a)
)Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
KdV
Asymptotic equation for unidirectional shallow water waves:
The KdV equation
ut + uxxx + 6uux = 0
Completely integrable
Existence of soliton solutions
Soliton: A localized wave of constant speed that emerges fromcollisions with other solitons unchanged but for a phase shift.
KdV soliton
c
2sech2
(√c
2(x − ct − a)
)Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
KdV
Asymptotic equation for unidirectional shallow water waves:
The KdV equation
ut + uxxx + 6uux = 0
Completely integrable
Existence of soliton solutions
Soliton: A localized wave of constant speed that emerges fromcollisions with other solitons unchanged but for a phase shift.
KdV soliton
c
2sech2
(√c
2(x − ct − a)
)Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Well-posedness of quasilinear equations of KdV type
Craig, Kappeler, and Strauss considered
Quasilinear KdV
ut + f (uxxx , uxx , ux , u, x , t) = 0
and showed well-posedness as long as ∂uxx f ≤ 0 and∂uxxx f ≥ � > 0.
∂uxx f ≤ 0 prevents behavior associated with backwards heatequation
Uniform bound on ∂uxxx f ensures dispersive effects dominate,leading to smoothing.
What if there’s no such uniform bound on ∂uxxx f ?
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Well-posedness of quasilinear equations of KdV type
Craig, Kappeler, and Strauss considered
Quasilinear KdV
ut + f (uxxx , uxx , ux , u, x , t) = 0
and showed well-posedness as long as ∂uxx f ≤ 0 and∂uxxx f ≥ � > 0.
∂uxx f ≤ 0 prevents behavior associated with backwards heatequation
Uniform bound on ∂uxxx f ensures dispersive effects dominate,leading to smoothing.
What if there’s no such uniform bound on ∂uxxx f ?
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Well-posedness of quasilinear equations of KdV type
Craig, Kappeler, and Strauss considered
Quasilinear KdV
ut + f (uxxx , uxx , ux , u, x , t) = 0
and showed well-posedness as long as ∂uxx f ≤ 0 and∂uxxx f ≥ � > 0.
∂uxx f ≤ 0 prevents behavior associated with backwards heatequation
Uniform bound on ∂uxxx f ensures dispersive effects dominate,leading to smoothing.
What if there’s no such uniform bound on ∂uxxx f ?
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Well-posedness of quasilinear equations of KdV type
Craig, Kappeler, and Strauss considered
Quasilinear KdV
ut + f (uxxx , uxx , ux , u, x , t) = 0
and showed well-posedness as long as ∂uxx f ≤ 0 and∂uxxx f ≥ � > 0.
∂uxx f ≤ 0 prevents behavior associated with backwards heatequation
Uniform bound on ∂uxxx f ensures dispersive effects dominate,leading to smoothing.
What if there’s no such uniform bound on ∂uxxx f ?
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
A family of degenerate dispersive equations
Rosenau and Hyman proposed:
The K (m, n) equations
ut + (um)x + (u
n)xxx = 0
When u → 0, dispersive effects due to uxxx vanishThis makes a local existence result harder (can’t use localsmoothing due to dispersion)
Numerical evidence indicates that K (2, 2) is ill-posed in H2
(Ambrose-Simpson-Wright-Yang)
Why H2?
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
A family of degenerate dispersive equations
Rosenau and Hyman proposed:
The K (m, n) equations
ut + (um)x + (u
n)xxx = 0
When u → 0, dispersive effects due to uxxx vanishThis makes a local existence result harder (can’t use localsmoothing due to dispersion)
Numerical evidence indicates that K (2, 2) is ill-posed in H2
(Ambrose-Simpson-Wright-Yang)
Why H2?
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
A family of degenerate dispersive equations
Rosenau and Hyman proposed:
The K (m, n) equations
ut + (um)x + (u
n)xxx = 0
When u → 0, dispersive effects due to uxxx vanishThis makes a local existence result harder (can’t use localsmoothing due to dispersion)
Numerical evidence indicates that K (2, 2) is ill-posed in H2
(Ambrose-Simpson-Wright-Yang)
Why H2?
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
A family of degenerate dispersive equations
Rosenau and Hyman proposed:
The K (m, n) equations
ut + (um)x + (u
n)xxx = 0
When u → 0, dispersive effects due to uxxx vanishThis makes a local existence result harder (can’t use localsmoothing due to dispersion)
Numerical evidence indicates that K (2, 2) is ill-posed in H2
(Ambrose-Simpson-Wright-Yang)
Why H2?
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Compactons
Searching for travelling wave solutions to K (2, 2) yields
K(2,2) compacton
u(x , t) =4c
3cos2
(x − ct
4
)χ[−2π,2π](x − ct)
The K (2, 2) compacton lives in H2, making it a naturalfunction space for local existence questions
Width of compacton does not depend on speed ofpropagation (unlike KdV soliton)
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Compactons
Searching for travelling wave solutions to K (2, 2) yields
K(2,2) compacton
u(x , t) =4c
3cos2
(x − ct
4
)χ[−2π,2π](x − ct)
The K (2, 2) compacton lives in H2, making it a naturalfunction space for local existence questions
Width of compacton does not depend on speed ofpropagation (unlike KdV soliton)
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Compactons
Searching for travelling wave solutions to K (2, 2) yields
K(2,2) compacton
u(x , t) =4c
3cos2
(x − ct
4
)χ[−2π,2π](x − ct)
The K (2, 2) compacton lives in H2, making it a naturalfunction space for local existence questions
Width of compacton does not depend on speed ofpropagation (unlike KdV soliton)
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Local existence
When can we get local existence results for degenerate dispersiveequations?
Theorem (Ambrose and Wright)
For k ∈ ±1, L0 > 0, and φ ∈ H2([−L0, L0]), there existsT ∗(L0, k , φ) > 0 such that for all 0 < T < T
∗ there is a weaksolution u ∈ L2(0,T ,H7/4) of the Cauchy problem for the equation
ut = 2uuxxx − uxuxx + 2kuux
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Local existence
When can we get local existence results for degenerate dispersiveequations?
Theorem (Ambrose and Wright)
For k ∈ ±1, L0 > 0, and φ ∈ H2([−L0, L0]), there existsT ∗(L0, k , φ) > 0 such that for all 0 < T < T
∗ there is a weaksolution u ∈ L2(0,T ,H7/4) of the Cauchy problem for the equation
ut = 2uuxxx − uxuxx + 2kuux
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
A semilinear Schrödinger equation
Another example of nonlinear dispersion:
Semilinear Schrödinger Equation
iut +1
2∆u + µ|u|α−1u = 0
u is complex-valued, in contrast to the KdV-type equations
µ ∈ {−1, 1} (focusing versus defocusing), α > 1 (mostcommonly α = 3)
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
A semilinear Schrödinger equation
Another example of nonlinear dispersion:
Semilinear Schrödinger Equation
iut +1
2∆u + µ|u|α−1u = 0
u is complex-valued, in contrast to the KdV-type equations
µ ∈ {−1, 1} (focusing versus defocusing), α > 1 (mostcommonly α = 3)
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
A semilinear Schrödinger equation
Another example of nonlinear dispersion:
Semilinear Schrödinger Equation
iut +1
2∆u + µ|u|α−1u = 0
u is complex-valued, in contrast to the KdV-type equations
µ ∈ {−1, 1} (focusing versus defocusing), α > 1 (mostcommonly α = 3)
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
A generalization
If we instead consider
Semilinear Schrödinger II
ut = i∆u + F (u, ū,∇ū)
then standard energy estimates will yield local well-posedness fors > n/2 + 1 under modest assumptions on the boundedness of Fand its derivatives.
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
A wild ∇u appears
Why didn’t F depend on ∇u?Unless the coefficient of ∇u is real, it can’t be eliminated in astraightforward manner during energy estimates.
If v = ∂αu then estimating ∂t |v |2 requires dealing with termslike v̄∇v − v∇v̄ , so integration by parts won’t work.Alternatively, we can think of ut = iux (for u ∈ C) as acomplex realization of the Cauchy-Riemann equations, whichare ill-posed as evolution equations.
Hayashi-Ozawa: deal with this by making a substitution andobtaining estimates for an equivalent energy.
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
A wild ∇u appears
Why didn’t F depend on ∇u?Unless the coefficient of ∇u is real, it can’t be eliminated in astraightforward manner during energy estimates.
If v = ∂αu then estimating ∂t |v |2 requires dealing with termslike v̄∇v − v∇v̄ , so integration by parts won’t work.Alternatively, we can think of ut = iux (for u ∈ C) as acomplex realization of the Cauchy-Riemann equations, whichare ill-posed as evolution equations.
Hayashi-Ozawa: deal with this by making a substitution andobtaining estimates for an equivalent energy.
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
A wild ∇u appears
Why didn’t F depend on ∇u?Unless the coefficient of ∇u is real, it can’t be eliminated in astraightforward manner during energy estimates.
If v = ∂αu then estimating ∂t |v |2 requires dealing with termslike v̄∇v − v∇v̄ , so integration by parts won’t work.Alternatively, we can think of ut = iux (for u ∈ C) as acomplex realization of the Cauchy-Riemann equations, whichare ill-posed as evolution equations.
Hayashi-Ozawa: deal with this by making a substitution andobtaining estimates for an equivalent energy.
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
A wild ∇u appears
Why didn’t F depend on ∇u?Unless the coefficient of ∇u is real, it can’t be eliminated in astraightforward manner during energy estimates.
If v = ∂αu then estimating ∂t |v |2 requires dealing with termslike v̄∇v − v∇v̄ , so integration by parts won’t work.Alternatively, we can think of ut = iux (for u ∈ C) as acomplex realization of the Cauchy-Riemann equations, whichare ill-posed as evolution equations.
Hayashi-Ozawa: deal with this by making a substitution andobtaining estimates for an equivalent energy.
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
A wild ∇u appears
Why didn’t F depend on ∇u?Unless the coefficient of ∇u is real, it can’t be eliminated in astraightforward manner during energy estimates.
If v = ∂αu then estimating ∂t |v |2 requires dealing with termslike v̄∇v − v∇v̄ , so integration by parts won’t work.Alternatively, we can think of ut = iux (for u ∈ C) as acomplex realization of the Cauchy-Riemann equations, whichare ill-posed as evolution equations.
Hayashi-Ozawa: deal with this by making a substitution andobtaining estimates for an equivalent energy.
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
A simple example
Consider the case n = 1 and the equation
Toy NLS
ut = iuxx + uux + uūx
Taking three derivatives, we linearize in v = ∂3xu
vt = ivxx + uvx + uv̄x + l .o.t.
Multiply this equation by eφ and rearrange, letting w = eφv
wt = iwxx + (u − 2iφx) eφvx + ueφv̄x + l .o.t.
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
A simple example
Consider the case n = 1 and the equation
Toy NLS
ut = iuxx + uux + uūx
Taking three derivatives, we linearize in v = ∂3xu
vt = ivxx + uvx + uv̄x + l .o.t.
Multiply this equation by eφ and rearrange, letting w = eφv
wt = iwxx + (u − 2iφx) eφvx + ueφv̄x + l .o.t.
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
A simple example
Consider the case n = 1 and the equation
Toy NLS
ut = iuxx + uux + uūx
Taking three derivatives, we linearize in v = ∂3xu
vt = ivxx + uvx + uv̄x + l .o.t.
Multiply this equation by eφ and rearrange, letting w = eφv
wt = iwxx + (u − 2iφx) eφvx + ueφv̄x + l .o.t.
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
A simple example
Consider the case n = 1 and the equation
Toy NLS
ut = iuxx + uux + uūx
Taking three derivatives, we linearize in v = ∂3xu
vt = ivxx + uvx + uv̄x + l .o.t.
Multiply this equation by eφ and rearrange, letting w = eφv
wt = iwxx + (u − 2iφx) eφvx + ueφv̄x + l .o.t.
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Choice of substitution
To get a system in w that we can perform estimates on we shouldchoose
φ(x , t) = − i2
∫ x0
u(x ′, t)dx ′
The equation for w now has no wx terms, so the standardestimates will work.
In fact, we only need to eliminate the imaginary part of the uxterm: if b(x) is our ux term then
Integrability Condition (Hayashi-Ozawa 1994)
supx∈R
∣∣∣∣∫ x0
Imb(t)dt
∣∣∣∣
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Choice of substitution
To get a system in w that we can perform estimates on we shouldchoose
φ(x , t) = − i2
∫ x0
u(x ′, t)dx ′
The equation for w now has no wx terms, so the standardestimates will work.
In fact, we only need to eliminate the imaginary part of the uxterm: if b(x) is our ux term then
Integrability Condition (Hayashi-Ozawa 1994)
supx∈R
∣∣∣∣∫ x0
Imb(t)dt
∣∣∣∣
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Choice of substitution
To get a system in w that we can perform estimates on we shouldchoose
φ(x , t) = − i2
∫ x0
u(x ′, t)dx ′
The equation for w now has no wx terms, so the standardestimates will work.
In fact, we only need to eliminate the imaginary part of the uxterm: if b(x) is our ux term then
Integrability Condition (Hayashi-Ozawa 1994)
supx∈R
∣∣∣∣∫ x0
Imb(t)dt
∣∣∣∣
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
The general quasilinear framework
Kenig, Ponce, and Vega considered the equation
Quasilinear Schrödinger
ut = iajk(x , t, u, ū,∇u,∇ū)uxjxk+ ~b1(x , t, u, ū,∇u,∇ū) · ∇u + ~b2(x , t, u, ū,∇u,∇ū) · ∇ū+ c1(x , t, u, ū)u + c2(x , t, u, ū)ū + f (x , t)
Hs local well-posedness was established under lots of assumptions.
Ellipticity of ajkRegularity and decay at infinity of coefficients and theirderivatives
The Hamiltonian flow associated to the symbol of the initialdata is nontrapping
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
The general quasilinear framework
Kenig, Ponce, and Vega considered the equation
Quasilinear Schrödinger
ut = iajk(x , t, u, ū,∇u,∇ū)uxjxk+ ~b1(x , t, u, ū,∇u,∇ū) · ∇u + ~b2(x , t, u, ū,∇u,∇ū) · ∇ū+ c1(x , t, u, ū)u + c2(x , t, u, ū)ū + f (x , t)
Hs local well-posedness was established under lots of assumptions.
Ellipticity of ajkRegularity and decay at infinity of coefficients and theirderivatives
The Hamiltonian flow associated to the symbol of the initialdata is nontrapping
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
The general quasilinear framework
Kenig, Ponce, and Vega considered the equation
Quasilinear Schrödinger
ut = iajk(x , t, u, ū,∇u,∇ū)uxjxk+ ~b1(x , t, u, ū,∇u,∇ū) · ∇u + ~b2(x , t, u, ū,∇u,∇ū) · ∇ū+ c1(x , t, u, ū)u + c2(x , t, u, ū)ū + f (x , t)
Hs local well-posedness was established under lots of assumptions.
Ellipticity of ajkRegularity and decay at infinity of coefficients and theirderivatives
The Hamiltonian flow associated to the symbol of the initialdata is nontrapping
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
The general quasilinear framework
Kenig, Ponce, and Vega considered the equation
Quasilinear Schrödinger
ut = iajk(x , t, u, ū,∇u,∇ū)uxjxk+ ~b1(x , t, u, ū,∇u,∇ū) · ∇u + ~b2(x , t, u, ū,∇u,∇ū) · ∇ū+ c1(x , t, u, ū)u + c2(x , t, u, ū)ū + f (x , t)
Hs local well-posedness was established under lots of assumptions.
Ellipticity of ajkRegularity and decay at infinity of coefficients and theirderivatives
The Hamiltonian flow associated to the symbol of the initialdata is nontrapping
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
The general quasilinear framework
Kenig, Ponce, and Vega considered the equation
Quasilinear Schrödinger
ut = iajk(x , t, u, ū,∇u,∇ū)uxjxk+ ~b1(x , t, u, ū,∇u,∇ū) · ∇u + ~b2(x , t, u, ū,∇u,∇ū) · ∇ū+ c1(x , t, u, ū)u + c2(x , t, u, ū)ū + f (x , t)
Hs local well-posedness was established under lots of assumptions.
Ellipticity of ajkRegularity and decay at infinity of coefficients and theirderivatives
The Hamiltonian flow associated to the symbol of the initialdata is nontrapping
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Comments on the KPV result
Nontrapping condition: implies an analogous integrabilitycondition for variable coefficient operators
Proof uses artificial viscosity method: insert −�∆2u term toregularize the problem
Pseudodifferential operators give a local smoothing estimateon linearized solutions, allowing an �-independent existencetime
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Comments on the KPV result
Nontrapping condition: implies an analogous integrabilitycondition for variable coefficient operators
Proof uses artificial viscosity method: insert −�∆2u term toregularize the problem
Pseudodifferential operators give a local smoothing estimateon linearized solutions, allowing an �-independent existencetime
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Degenerate dispersive equationsNonlinear Schrödinger equations
Comments on the KPV result
Nontrapping condition: implies an analogous integrabilitycondition for variable coefficient operators
Proof uses artificial viscosity method: insert −�∆2u term toregularize the problem
Pseudodifferential operators give a local smoothing estimateon linearized solutions, allowing an �-independent existencetime
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A degenerate quasilinear Schrödinger equation
DNLS Equation
iAt +(|A|2Ax
)x
= 0
Arises as an asymptotic equation for a two-wave system incompressible gas dynamics
Quantum-mechanical interpretation: a free particle with massinversely proportional to probability density.
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A degenerate quasilinear Schrödinger equation
DNLS Equation
iAt +(|A|2Ax
)x
= 0
Arises as an asymptotic equation for a two-wave system incompressible gas dynamics
Quantum-mechanical interpretation: a free particle with massinversely proportional to probability density.
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A degenerate quasilinear Schrödinger equation
DNLS Equation
iAt +(|A|2Ax
)x
= 0
Arises as an asymptotic equation for a two-wave system incompressible gas dynamics
Quantum-mechanical interpretation: a free particle with massinversely proportional to probability density.
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Two natural generalizations
First, we generalize to arbitrary functions of |A|2:
DNLS II
iAt +(ρ(|A|2)Ax
)x
= 0
We can also consider the equation in more than one spatialdimension:
DNLS: n dimensions
iAt +∇ ·(|A|2∇A
)= 0
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Two natural generalizations
First, we generalize to arbitrary functions of |A|2:
DNLS II
iAt +(ρ(|A|2)Ax
)x
= 0
We can also consider the equation in more than one spatialdimension:
DNLS: n dimensions
iAt +∇ ·(|A|2∇A
)= 0
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Two natural generalizations
First, we generalize to arbitrary functions of |A|2:
DNLS II
iAt +(ρ(|A|2)Ax
)x
= 0
We can also consider the equation in more than one spatialdimension:
DNLS: n dimensions
iAt +∇ ·(|A|2∇A
)= 0
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Two natural generalizations
First, we generalize to arbitrary functions of |A|2:
DNLS II
iAt +(ρ(|A|2)Ax
)x
= 0
We can also consider the equation in more than one spatialdimension:
DNLS: n dimensions
iAt +∇ ·(|A|2∇A
)= 0
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Hamiltonian formulation
The original equation iAt +(|A|2Ax
)x
= 0 can be written in theHamiltonian form
H(A,A∗) = 14i
∫AA∗(AA∗x − A∗Ax)dx
At + ∂x
[δHδA∗
]= 0
Conserved quantities in addition to H include the action andthe momentum:
S = 12i
∫(A∂−1x A
∗ − A∗∂−1x A)dx
P = 12
∫AA∗dx
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Hamiltonian formulation
The original equation iAt +(|A|2Ax
)x
= 0 can be written in theHamiltonian form
H(A,A∗) = 14i
∫AA∗(AA∗x − A∗Ax)dx
At + ∂x
[δHδA∗
]= 0
Conserved quantities in addition to H include the action andthe momentum:
S = 12i
∫(A∂−1x A
∗ − A∗∂−1x A)dx
P = 12
∫AA∗dx
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Hamiltonian formulation
The original equation iAt +(|A|2Ax
)x
= 0 can be written in theHamiltonian form
H(A,A∗) = 14i
∫AA∗(AA∗x − A∗Ax)dx
At + ∂x
[δHδA∗
]= 0
Conserved quantities in addition to H include the action andthe momentum:
S = 12i
∫(A∂−1x A
∗ − A∗∂−1x A)dx
P = 12
∫AA∗dx
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Hamiltonian in n dimensions
It appears that the n-dimensional equation cannot be put inHamiltonian form for n > 1. A modified equation that isHamiltonian:
Modified Hamiltonian DNLS
iAt +∇ · (|A|2∇A)− A|∇A|2 = 0
The Hamiltonian form of this equation is
H(A,A∗) =∫
AA∗∇A · ∇A∗dx
iAt =δHδA∗
What about local existence for this equation?
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Hamiltonian in n dimensions
It appears that the n-dimensional equation cannot be put inHamiltonian form for n > 1. A modified equation that isHamiltonian:
Modified Hamiltonian DNLS
iAt +∇ · (|A|2∇A)− A|∇A|2 = 0
The Hamiltonian form of this equation is
H(A,A∗) =∫
AA∗∇A · ∇A∗dx
iAt =δHδA∗
What about local existence for this equation?
Evan Smothers Degenerate quasilinear Schrödinger equations
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Hamiltonian in n dimensions
It appears that the n-dimensional equation cannot be put inHamiltonian form for n > 1. A modified equation that isHamiltonian:
Modified Hamiltonian DNLS
iAt +∇ · (|A|2∇A)− A|∇A|2 = 0
The Hamiltonian form of this equation is
H(A,A∗) =∫
AA∗∇A · ∇A∗dx
iAt =δHδA∗
What about local existence for this equation?
Evan Smothers Degenerate quasilinear Schrödinger equations
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Hamiltonian in n dimensions
It appears that the n-dimensional equation cannot be put inHamiltonian form for n > 1. A modified equation that isHamiltonian:
Modified Hamiltonian DNLS
iAt +∇ · (|A|2∇A)− A|∇A|2 = 0
The Hamiltonian form of this equation is
H(A,A∗) =∫
AA∗∇A · ∇A∗dx
iAt =δHδA∗
What about local existence for this equation?
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Approach
Goal: Obtain local existence of solutions to the DNLS equation.Two steps:
1 Get local existence for the equation
iAt +(ρ(|A|2)Ax
)x
= 0
in the case that ρ is bounded away from zero.
2 See if it’s possible to get an existence time independent of thebound on ρ (or find some way to pass to a limit in a suitablesense)
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Approach
Goal: Obtain local existence of solutions to the DNLS equation.Two steps:
1 Get local existence for the equation
iAt +(ρ(|A|2)Ax
)x
= 0
in the case that ρ is bounded away from zero.
2 See if it’s possible to get an existence time independent of thebound on ρ (or find some way to pass to a limit in a suitablesense)
Evan Smothers Degenerate quasilinear Schrödinger equations
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Approach
Goal: Obtain local existence of solutions to the DNLS equation.Two steps:
1 Get local existence for the equation
iAt +(ρ(|A|2)Ax
)x
= 0
in the case that ρ is bounded away from zero.
2 See if it’s possible to get an existence time independent of thebound on ρ (or find some way to pass to a limit in a suitablesense)
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Non-degenerate case
Proposition (Local Existence)
Suppose A0 ∈ H3(T), ρ ∈ C 4(R) and there exist ρ0,C > 0 suchthat ρ0 ≤ ρ(α) ≤ C |α|. Then there exists T > 0 depending onρ, ρ0,C and A ∈ L2(0,T ;H3(T)) ∩ C (0,T ;H2(T)) satisfying{
iAt +(ρ(|A|2)Ax
)x
= 0
A(x , 0) = A0(x)
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Proof Sketch
Use parabolic regularization to obtain solutions on a timeinterval of O(�)L2 estimate on ρ7/4Axxx gives a uniform existence time andallows us to pass to a limit
The assumption that ρ is bounded away from zero is critical:otherwise this estimate is not equivalent to an estimate on||Axxx ||L2 .
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Proof Sketch
Use parabolic regularization to obtain solutions on a timeinterval of O(�)L2 estimate on ρ7/4Axxx gives a uniform existence time andallows us to pass to a limit
The assumption that ρ is bounded away from zero is critical:otherwise this estimate is not equivalent to an estimate on||Axxx ||L2 .
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Proof Sketch
Use parabolic regularization to obtain solutions on a timeinterval of O(�)L2 estimate on ρ7/4Axxx gives a uniform existence time andallows us to pass to a limit
The assumption that ρ is bounded away from zero is critical:otherwise this estimate is not equivalent to an estimate on||Axxx ||L2 .
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Degenerate case
In the degenerate case we have ρ(|A|2) = |A|2, so ourestimate is ∫
|A|7|Axxx |2 . 1
However, this does not translate to an H3 estimate
We can better understand behavior of solutions to thedegenerate equation by considering some numericalsimulations (courtesy of John Hunter)
Evan Smothers Degenerate quasilinear Schrödinger equations
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Degenerate case
In the degenerate case we have ρ(|A|2) = |A|2, so ourestimate is ∫
|A|7|Axxx |2 . 1
However, this does not translate to an H3 estimate
We can better understand behavior of solutions to thedegenerate equation by considering some numericalsimulations (courtesy of John Hunter)
Evan Smothers Degenerate quasilinear Schrödinger equations
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Degenerate case
In the degenerate case we have ρ(|A|2) = |A|2, so ourestimate is ∫
|A|7|Axxx |2 . 1
However, this does not translate to an H3 estimate
We can better understand behavior of solutions to thedegenerate equation by considering some numericalsimulations (courtesy of John Hunter)
Evan Smothers Degenerate quasilinear Schrödinger equations
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Local ExistenceWavefront behavior
Numerical Simulations
Evan Smothers Degenerate quasilinear Schrödinger equations
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Numerical observations
The numerics demonstrate a few important properties of solutions:
Finite speed of propagation for compactly supported data
A waiting time prior to dispersing outward from initial data
Well-behaved modulus with rapid oscillations near theboundary
These observations lead us to consider local behavior near thewavefront of a solution. Two methods for doing this:
1 Whitham’s averaged Lagrangian method
2 Similarity solutions (motivated by study of porous mediumequation)
Evan Smothers Degenerate quasilinear Schrödinger equations
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Local ExistenceWavefront behavior
Numerical observations
The numerics demonstrate a few important properties of solutions:
Finite speed of propagation for compactly supported data
A waiting time prior to dispersing outward from initial data
Well-behaved modulus with rapid oscillations near theboundary
These observations lead us to consider local behavior near thewavefront of a solution. Two methods for doing this:
1 Whitham’s averaged Lagrangian method
2 Similarity solutions (motivated by study of porous mediumequation)
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
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Local ExistenceWavefront behavior
Numerical observations
The numerics demonstrate a few important properties of solutions:
Finite speed of propagation for compactly supported data
A waiting time prior to dispersing outward from initial data
Well-behaved modulus with rapid oscillations near theboundary
These observations lead us to consider local behavior near thewavefront of a solution. Two methods for doing this:
1 Whitham’s averaged Lagrangian method
2 Similarity solutions (motivated by study of porous mediumequation)
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Local ExistenceWavefront behavior
Numerical observations
The numerics demonstrate a few important properties of solutions:
Finite speed of propagation for compactly supported data
A waiting time prior to dispersing outward from initial data
Well-behaved modulus with rapid oscillations near theboundary
These observations lead us to consider local behavior near thewavefront of a solution. Two methods for doing this:
1 Whitham’s averaged Lagrangian method
2 Similarity solutions (motivated by study of porous mediumequation)
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Local ExistenceWavefront behavior
Numerical observations
The numerics demonstrate a few important properties of solutions:
Finite speed of propagation for compactly supported data
A waiting time prior to dispersing outward from initial data
Well-behaved modulus with rapid oscillations near theboundary
These observations lead us to consider local behavior near thewavefront of a solution. Two methods for doing this:
1 Whitham’s averaged Lagrangian method
2 Similarity solutions (motivated by study of porous mediumequation)
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Local ExistenceWavefront behavior
Numerical observations
The numerics demonstrate a few important properties of solutions:
Finite speed of propagation for compactly supported data
A waiting time prior to dispersing outward from initial data
Well-behaved modulus with rapid oscillations near theboundary
These observations lead us to consider local behavior near thewavefront of a solution. Two methods for doing this:
1 Whitham’s averaged Lagrangian method
2 Similarity solutions (motivated by study of porous mediumequation)
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Local ExistenceWavefront behavior
Numerical observations
The numerics demonstrate a few important properties of solutions:
Finite speed of propagation for compactly supported data
A waiting time prior to dispersing outward from initial data
Well-behaved modulus with rapid oscillations near theboundary
These observations lead us to consider local behavior near thewavefront of a solution. Two methods for doing this:
1 Whitham’s averaged Lagrangian method
2 Similarity solutions (motivated by study of porous mediumequation)
Evan Smothers Degenerate quasilinear Schrödinger equations
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Lagrangian formulation
The degenerate Schrödinger equation has Lagrangian
S(v , v∗) =∫−1
2(v∗x vt + vxv
∗t ) +
1
4ivxv
∗x (v∗x vxx − vxv∗xx)dxdt
where v = ∂−1x A.Because of this, we can apply the averaged Lagrangian method ofWhitham:
Write v = ρ(x , t)eiφ(x,t)� , substitute into Lagrangian
Average over a phase in φ to obtain the averaged Lagrangian
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Lagrangian formulation
The degenerate Schrödinger equation has Lagrangian
S(v , v∗) =∫−1
2(v∗x vt + vxv
∗t ) +
1
4ivxv
∗x (v∗x vxx − vxv∗xx)dxdt
where v = ∂−1x A.Because of this, we can apply the averaged Lagrangian method ofWhitham:
Write v = ρ(x , t)eiφ(x,t)� , substitute into Lagrangian
Average over a phase in φ to obtain the averaged Lagrangian
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Lagrangian formulation
The degenerate Schrödinger equation has Lagrangian
S(v , v∗) =∫−1
2(v∗x vt + vxv
∗t ) +
1
4ivxv
∗x (v∗x vxx − vxv∗xx)dxdt
where v = ∂−1x A.Because of this, we can apply the averaged Lagrangian method ofWhitham:
Write v = ρ(x , t)eiφ(x,t)� , substitute into Lagrangian
Average over a phase in φ to obtain the averaged Lagrangian
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Lagrangian formulation
The degenerate Schrödinger equation has Lagrangian
S(v , v∗) =∫−1
2(v∗x vt + vxv
∗t ) +
1
4ivxv
∗x (v∗x vxx − vxv∗xx)dxdt
where v = ∂−1x A.Because of this, we can apply the averaged Lagrangian method ofWhitham:
Write v = ρ(x , t)eiφ(x,t)� , substitute into Lagrangian
Average over a phase in φ to obtain the averaged Lagrangian
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Averaged Lagrangian
Letting k = φx , ω = −φt , our averaged Lagrangian is
S̄(ρ, φ) =∫
kωρ2 − 12k5ρ4dxdt
S̄ should solve the Euler-Lagrange equations
δS̄δρ
=δS̄δφ
= 0
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Averaged Lagrangian
Letting k = φx , ω = −φt , our averaged Lagrangian is
S̄(ρ, φ) =∫
kωρ2 − 12k5ρ4dxdt
S̄ should solve the Euler-Lagrange equations
δS̄δρ
=δS̄δφ
= 0
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Results of Euler-Lagrange equations
Adding in the compatibility condition kt + ωx = 0 yields the 2-Dsystem:
Averaged Lagrangian system
kt + (k4ρ2)x = 0
(kρ2)t +3
2(k4ρ4)x = 0
This system is hyperbolic in k and η = kρ2 (solve by Riemanninvariants)
What happens as |A| → 0? Analysis suggests ρk1−√3
2
approaches a time-independent state.
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Results of Euler-Lagrange equations
Adding in the compatibility condition kt + ωx = 0 yields the 2-Dsystem:
Averaged Lagrangian system
kt + (k4ρ2)x = 0
(kρ2)t +3
2(k4ρ4)x = 0
This system is hyperbolic in k and η = kρ2 (solve by Riemanninvariants)
What happens as |A| → 0? Analysis suggests ρk1−√3
2
approaches a time-independent state.
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Results of Euler-Lagrange equations
Adding in the compatibility condition kt + ωx = 0 yields the 2-Dsystem:
Averaged Lagrangian system
kt + (k4ρ2)x = 0
(kρ2)t +3
2(k4ρ4)x = 0
This system is hyperbolic in k and η = kρ2 (solve by Riemanninvariants)
What happens as |A| → 0? Analysis suggests ρk1−√3
2
approaches a time-independent state.
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Results of Euler-Lagrange equations
Adding in the compatibility condition kt + ωx = 0 yields the 2-Dsystem:
Averaged Lagrangian system
kt + (k4ρ2)x = 0
(kρ2)t +3
2(k4ρ4)x = 0
This system is hyperbolic in k and η = kρ2 (solve by Riemanninvariants)
What happens as |A| → 0? Analysis suggests ρk1−√3
2
approaches a time-independent state.
Evan Smothers Degenerate quasilinear Schrödinger equations
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Porous Medium Equation
Finite propagation speed of solutions, waiting time prior to movingwavefront invites comparison to PME and other nonlinear diffusionequations. For instance:
ut = ∇ · (un∇u)
Multiple families of similarity solutions
Existence of a family of waiting-time solutions (comes fromsimilarity solutions in one space variable)
Idea: derive analogous similarity solutions
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Porous Medium Equation
Finite propagation speed of solutions, waiting time prior to movingwavefront invites comparison to PME and other nonlinear diffusionequations. For instance:
ut = ∇ · (un∇u)
Multiple families of similarity solutions
Existence of a family of waiting-time solutions (comes fromsimilarity solutions in one space variable)
Idea: derive analogous similarity solutions
Evan Smothers Degenerate quasilinear Schrödinger equations
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Local ExistenceWavefront behavior
Porous Medium Equation
Finite propagation speed of solutions, waiting time prior to movingwavefront invites comparison to PME and other nonlinear diffusionequations. For instance:
ut = ∇ · (un∇u)
Multiple families of similarity solutions
Existence of a family of waiting-time solutions (comes fromsimilarity solutions in one space variable)
Idea: derive analogous similarity solutions
Evan Smothers Degenerate quasilinear Schrödinger equations
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Local ExistenceWavefront behavior
Porous Medium Equation
Finite propagation speed of solutions, waiting time prior to movingwavefront invites comparison to PME and other nonlinear diffusionequations. For instance:
ut = ∇ · (un∇u)
Multiple families of similarity solutions
Existence of a family of waiting-time solutions (comes fromsimilarity solutions in one space variable)
Idea: derive analogous similarity solutions
Evan Smothers Degenerate quasilinear Schrödinger equations
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Similarity solutions to DNLS
Search for similarity solutions of the form A(x , t) = t−1/4f (ξ)for ξ = x
t1/4and f ∈ C. This yields the ODE
− i4ξf + |f |2f ′ = C
If we choose C = 0 we obtain the one-parameter family ofnon-degenerate solutions
Similarity solution
A(x , t) = A0t−1/4 exp
(ix2
8√t|A0|
)For C 6= 0 the ODE is harder to solve, but we would like tofind other solutions that represent wavefront behavior.
Evan Smothers Degenerate quasilinear Schrödinger equations
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Similarity solutions to DNLS
Search for similarity solutions of the form A(x , t) = t−1/4f (ξ)for ξ = x
t1/4and f ∈ C. This yields the ODE
− i4ξf + |f |2f ′ = C
If we choose C = 0 we obtain the one-parameter family ofnon-degenerate solutions
Similarity solution
A(x , t) = A0t−1/4 exp
(ix2
8√t|A0|
)For C 6= 0 the ODE is harder to solve, but we would like tofind other solutions that represent wavefront behavior.
Evan Smothers Degenerate quasilinear Schrödinger equations
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Similarity solutions to DNLS
Search for similarity solutions of the form A(x , t) = t−1/4f (ξ)for ξ = x
t1/4and f ∈ C. This yields the ODE
− i4ξf + |f |2f ′ = C
If we choose C = 0 we obtain the one-parameter family ofnon-degenerate solutions
Similarity solution
A(x , t) = A0t−1/4 exp
(ix2
8√t|A0|
)For C 6= 0 the ODE is harder to solve, but we would like tofind other solutions that represent wavefront behavior.
Evan Smothers Degenerate quasilinear Schrödinger equations
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Local ExistenceWavefront behavior
Similarity solutions to DNLS
Search for similarity solutions of the form A(x , t) = t−1/4f (ξ)for ξ = x
t1/4and f ∈ C. This yields the ODE
− i4ξf + |f |2f ′ = C
If we choose C = 0 we obtain the one-parameter family ofnon-degenerate solutions
Similarity solution
A(x , t) = A0t−1/4 exp
(ix2
8√t|A0|
)For C 6= 0 the ODE is harder to solve, but we would like tofind other solutions that represent wavefront behavior.
Evan Smothers Degenerate quasilinear Schrödinger equations
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Local ExistenceWavefront behavior
Similarity solutions to DNLS
Search for similarity solutions of the form A(x , t) = t−1/4f (ξ)for ξ = x
t1/4and f ∈ C. This yields the ODE
− i4ξf + |f |2f ′ = C
If we choose C = 0 we obtain the one-parameter family ofnon-degenerate solutions
Similarity solution
A(x , t) = A0t−1/4 exp
(ix2
8√t|A0|
)For C 6= 0 the ODE is harder to solve, but we would like tofind other solutions that represent wavefront behavior.
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MRS equation
One way DNLS arises is as an asymptotic equation for the MRSequation. (Hunter 1995)
MRS equation {ut + uux = vvt + vvx = −u
Derived by Majda, Rosales, and Schonbeck in 1988
Models asymptotics of resonant reflection of sound waves offan entropy wave in compressible gas dynamics
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MRS equation
One way DNLS arises is as an asymptotic equation for the MRSequation. (Hunter 1995)
MRS equation {ut + uux = vvt + vvx = −u
Derived by Majda, Rosales, and Schonbeck in 1988
Models asymptotics of resonant reflection of sound waves offan entropy wave in compressible gas dynamics
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Deriving DNLS from MRS
Multiple scale expansion: let τ = �2t and write
u ∼ �u1(x , t; τ) + �2u2(x , t; τ) + ...v ∼ �v1(x , t; τ) + �2v2(x , t; τ) + ...
At O(�), we get the linearized solution:u = A(x , τ)e it + c.c .
v = iA(x , τ)e it + c.c .
At O(�3) we get the solvability condition
DNLS
iAτ +(|A|2Ax
)x
= 0
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Deriving DNLS from MRS
Multiple scale expansion: let τ = �2t and write
u ∼ �u1(x , t; τ) + �2u2(x , t; τ) + ...v ∼ �v1(x , t; τ) + �2v2(x , t; τ) + ...
At O(�), we get the linearized solution:u = A(x , τ)e it + c.c .
v = iA(x , τ)e it + c.c .
At O(�3) we get the solvability condition
DNLS
iAτ +(|A|2Ax
)x
= 0
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Deriving DNLS from MRS
Multiple scale expansion: let τ = �2t and write
u ∼ �u1(x , t; τ) + �2u2(x , t; τ) + ...v ∼ �v1(x , t; τ) + �2v2(x , t; τ) + ...
At O(�), we get the linearized solution:u = A(x , τ)e it + c.c .
v = iA(x , τ)e it + c.c .
At O(�3) we get the solvability condition
DNLS
iAτ +(|A|2Ax
)x
= 0
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Returning to MRS
We want to investigate the lifespan of small solutions to theMRS equation.
Notice: solutions to the linearized equation oscillate withconstant frequency in time and do not enjoy any dispersivedecay.
This property is similar to the inviscid Burgers-Hilbertequation
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Returning to MRS
We want to investigate the lifespan of small solutions to theMRS equation.
Notice: solutions to the linearized equation oscillate withconstant frequency in time and do not enjoy any dispersivedecay.
This property is similar to the inviscid Burgers-Hilbertequation
Evan Smothers Degenerate quasilinear Schrödinger equations
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Returning to MRS
We want to investigate the lifespan of small solutions to theMRS equation.
Notice: solutions to the linearized equation oscillate withconstant frequency in time and do not enjoy any dispersivedecay.
This property is similar to the inviscid Burgers-Hilbertequation
Evan Smothers Degenerate quasilinear Schrödinger equations
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Lifespan for Burgers-Hilbert
Inviscid Burgers-Hilbert
ut + uux = H[u]
Suppose our initial data is small, say ||u0||Hs ≤ �. What sort oflifespan can we expect for a solution?
Standard energy estimates yield a quadratic lifespan
Can we expect to do any better than this?
Evan Smothers Degenerate quasilinear Schrödinger equations
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Lifespan for Burgers-Hilbert
Inviscid Burgers-Hilbert
ut + uux = H[u]
Suppose our initial data is small, say ||u0||Hs ≤ �. What sort oflifespan can we expect for a solution?
Standard energy estimates yield a quadratic lifespan
Can we expect to do any better than this?
Evan Smothers Degenerate quasilinear Schrödinger equations
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Lifespan for Burgers-Hilbert
Inviscid Burgers-Hilbert
ut + uux = H[u]
Suppose our initial data is small, say ||u0||Hs ≤ �. What sort oflifespan can we expect for a solution?
Standard energy estimates yield a quadratic lifespan
Can we expect to do any better than this?
Evan Smothers Degenerate quasilinear Schrödinger equations
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Lifespan for Burgers-Hilbert
Inviscid Burgers-Hilbert
ut + uux = H[u]
Suppose our initial data is small, say ||u0||Hs ≤ �. What sort oflifespan can we expect for a solution?
Standard energy estimates yield a quadratic lifespan
Can we expect to do any better than this?
Evan Smothers Degenerate quasilinear Schrödinger equations
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Normal form transformation
A first guess: make a normal form transformation (motivated byShatah). That is, make the substitution
u 7−→ u + B(u, u)
where B(u, u) is quadratic.
Normal form transformation: inviscid Burgers-Hilbert
u 7−→ u + H [Hu ·Hux ]
Problem: This normal form transformation leads to a loss ofderivatives
Evan Smothers Degenerate quasilinear Schrödinger equations
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Normal form transformation
A first guess: make a normal form transformation (motivated byShatah). That is, make the substitution
u 7−→ u + B(u, u)
where B(u, u) is quadratic.
Normal form transformation: inviscid Burgers-Hilbert
u 7−→ u + H [Hu ·Hux ]
Problem: This normal form transformation leads to a loss ofderivatives
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Normal form transformation
A first guess: make a normal form transformation (motivated byShatah). That is, make the substitution
u 7−→ u + B(u, u)
where B(u, u) is quadratic.
Normal form transformation: inviscid Burgers-Hilbert
u 7−→ u + H [Hu ·Hux ]
Problem: This normal form transformation leads to a loss ofderivatives
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
The modified energy method
Solution: Define a modified energy Ek . What do we want fromEk?
1 Ek(u) ∼ ||∂kx u||2L22
dEkdt contains no quadratic or cubic terms.
Modified energy: inviscid Burgers-Hilbert
Ek =1
2||∂kx u||2L2 + 〈∂
kx u, ∂
kxH[Hu ·Hux ]〉
Yields cubic lifespan for small solutions
Evan Smothers Degenerate quasilinear Schrödinger equations
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MRS equationMoving forward
The modified energy method
Solution: Define a modified energy Ek . What do we want fromEk?
1 Ek(u) ∼ ||∂kx u||2L22
dEkdt contains no quadratic or cubic terms.
Modified energy: inviscid Burgers-Hilbert
Ek =1
2||∂kx u||2L2 + 〈∂
kx u, ∂
kxH[Hu ·Hux ]〉
Yields cubic lifespan for small solutions
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
The modified energy method
Solution: Define a modified energy Ek . What do we want fromEk?
1 Ek(u) ∼ ||∂kx u||2L22
dEkdt contains no quadratic or cubic terms.
Modified energy: inviscid Burgers-Hilbert
Ek =1
2||∂kx u||2L2 + 〈∂
kx u, ∂
kxH[Hu ·Hux ]〉
Yields cubic lifespan for small solutions
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
The modified energy method
Solution: Define a modified energy Ek . What do we want fromEk?
1 Ek(u) ∼ ||∂kx u||2L22
dEkdt contains no quadratic or cubic terms.
Modified energy: inviscid Burgers-Hilbert
Ek =1
2||∂kx u||2L2 + 〈∂
kx u, ∂
kxH[Hu ·Hux ]〉
Yields cubic lifespan for small solutions
Evan Smothers Degenerate quasilinear Schrödinger equations
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What about MRS?
Wagenmaker (1994) showed that solutions have alogarithmically improved lifespan
Normal form transformation for MRS:
MRS normal form
u 7−→ u + 13∂x
(u2 − uv + 1
2v2)
v 7−→ v − 13∂x
(1
2u2 − uv + v2
)This transformation suffers the same loss of derivatives as inBurgers-Hilbert.
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
What about MRS?
Wagenmaker (1994) showed that solutions have alogarithmically improved lifespan
Normal form transformation for MRS:
MRS normal form
u 7−→ u + 13∂x
(u2 − uv + 1
2v2)
v 7−→ v − 13∂x
(1
2u2 − uv + v2
)This transformation suffers the same loss of derivatives as inBurgers-Hilbert.
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
What about MRS?
Wagenmaker (1994) showed that solutions have alogarithmically improved lifespan
Normal form transformation for MRS:
MRS normal form
u 7−→ u + 13∂x
(u2 − uv + 1
2v2)
v 7−→ v − 13∂x
(1
2u2 − uv + v2
)This transformation suffers the same loss of derivatives as inBurgers-Hilbert.
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Modified Energy for MRS
As in Burgers-Hilbert, throw out highest-order terms from normalform transformation.
MRS modified energy
Ek =1
2||∂kx u||2L2 +
1
2||∂kx v ||2L2
+1
3〈∂kx u, ∂k+1x (u2 − uv +
1
2v2)〉 − 1
3〈∂kx v , ∂k+1x (
1
2u2 − uv + v2)〉
Unfortunately, Ek is not equivalent to ||∂kx u||2 + ||∂kx v ||2, so themethod fails.
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Modified Energy for MRS
As in Burgers-Hilbert, throw out highest-order terms from normalform transformation.
MRS modified energy
Ek =1
2||∂kx u||2L2 +
1
2||∂kx v ||2L2
+1
3〈∂kx u, ∂k+1x (u2 − uv +
1
2v2)〉 − 1
3〈∂kx v , ∂k+1x (
1
2u2 − uv + v2)〉
Unfortunately, Ek is not equivalent to ||∂kx u||2 + ||∂kx v ||2, so themethod fails.
Evan Smothers Degenerate quasilinear Schrödinger equations
BackgroundDegenerate Schrödinger Equations
MRS equationMoving forward
Modified Energy for MRS
As in Burgers-Hilbert, throw out highest-order terms from normalform transformation.
MRS modified energy
Ek =1
2||∂kx u||2L2 +
1
2||∂kx v ||2L2
+1
3〈∂kx u, ∂k+1x (u2 − uv +
1
2v2)〉 − 1
3〈∂kx v , ∂k+1x (
1
2u2 − uv + v2)〉
Unfortunately, Ek is not equivalent to ||∂kx u||2 + ||∂kx v ||2, so themethod fails.
Evan Smothers Degenerate quasilinear Schrödinger equations
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Future Research
Long-term goal: local existence result for the DNLS equation
Analyze similarity solutions
Investigate wavefront behavior
Study modified versions of DNLS and other relateddegenerate dispersive equations
Understand normal form methods for these equations
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