Degenerate Schr odinger Equations MRS equation Moving forward · Background Degenerate Schr odinger...

Transcript of Degenerate Schr odinger Equations MRS equation Moving forward · Background Degenerate Schr odinger...

BackgroundDegenerate Schrödinger Equations

MRS equationMoving forward

Degenerate quasilinear Schrödinger equations

Evan Smothers

June 2, 2015

Evan Smothers Degenerate quasilinear Schrödinger equations


Table of Contents

1 Background

2 Degenerate Schrödinger Equations

3 MRS equation

4 Moving 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




Outline



KdVSchrödinger


Main questions:





Outline



KdVSchrödinger


Main questions:





Outline



KdVSchrödinger


Main questions:





Outline



KdVSchrödinger


Main questions:





Outline



KdVSchrödinger


Main questions:





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




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



KdV


The KdV equation





KdV soliton

c

2sech2

(√c

2(x − ct − a)




KdV


The KdV equation





KdV soliton

c

2sech2

(√c

2(x − ct − a)




KdV


The KdV equation





KdV soliton

c

2sech2

(√c

2(x − ct − a)




KdV


The KdV equation





KdV soliton

c

2sech2

(√c

2(x − ct − a)




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 ?






Quasilinear KdV











Quasilinear KdV











Quasilinear KdV









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?







ut + (um)x + (u

n)xxx = 0




Why H2?







ut + (um)x + (u

n)xxx = 0




Why H2?







ut + (um)x + (u

n)xxx = 0




Why H2?




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)




Compactons


K(2,2) compacton

u(x , t) =4c

3cos2

(x − ct

4

)χ[−2π,2π](x − ct)






Compactons


K(2,2) compacton

u(x , t) =4c

3cos2

(x − ct

4

)χ[−2π,2π](x − ct)






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




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




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)







iut +1

2∆u + µ|u|α−1u = 0









iut +1

2∆u + µ|u|α−1u = 0






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.




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.




A wild ∇u appears







A wild ∇u appears







A wild ∇u appears







A wild ∇u appears







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.




A simple example


Toy NLS









A simple example


Toy NLS









A simple example


Toy NLS









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

∣∣∣∣





φ(x , t) = − i2

∫ x0

u(x ′, t)dx ′




supx∈R

∣∣∣∣∫ x0

Imb(t)dt

∣∣∣∣





φ(x , t) = − i2

∫ x0

u(x ′, t)dx ′




supx∈R

∣∣∣∣∫ x0

Imb(t)dt

∣∣∣∣



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




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



Local ExistenceWavefront behavior

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.





DNLS Equation

iAt +(|A|2Ax

)x

= 0







DNLS Equation

iAt +(|A|2Ax

)x

= 0






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






DNLS II

iAt +(ρ(|A|2)Ax

)x

= 0


DNLS: n dimensions

iAt +∇ ·(|A|2∇A

)= 0






DNLS II

iAt +(ρ(|A|2)Ax

)x

= 0


DNLS: n dimensions

iAt +∇ ·(|A|2∇A

)= 0






DNLS II

iAt +(ρ(|A|2)Ax

)x

= 0


DNLS: n dimensions

iAt +∇ ·(|A|2∇A

)= 0




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






)x


H(A,A∗) = 14i


At + ∂x

[δHδA∗

]= 0


S = 12i

∫(A∂−1x A

∗ − A∗∂−1x A)dx

P = 12

∫AA∗dx






)x


H(A,A∗) = 14i


At + ∂x

[δHδA∗

]= 0


S = 12i

∫(A∂−1x A

∗ − A∗∂−1x A)dx

P = 12

∫AA∗dx




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?







iAt +∇ · (|A|2∇A)− A|∇A|2 = 0


H(A,A∗) =∫


iAt =δHδA∗








iAt +∇ · (|A|2∇A)− A|∇A|2 = 0


H(A,A∗) =∫


iAt =δHδA∗








iAt +∇ · (|A|2∇A)− A|∇A|2 = 0


H(A,A∗) =∫


iAt =δHδA∗





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)




Approach



iAt +(ρ(|A|2)Ax

)x

= 0






Approach



iAt +(ρ(|A|2)Ax

)x

= 0






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)




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 .




Proof Sketch






Proof Sketch






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)




Degenerate case


|A|7|Axxx |2 . 1






Degenerate case


|A|7|Axxx |2 . 1






Numerical Simulations




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)




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






S(v , v∗) =∫−1

2(v∗x vt + vxv

∗t ) +

1

4ivxv










S(v , v∗) =∫−1

2(v∗x vt + vxv

∗t ) +

1

4ivxv










S(v , v∗) =∫−1

2(v∗x vt + vxv

∗t ) +

1

4ivxv








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




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




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.







kt + (k4ρ2)x = 0

(kρ2)t +3

2(k4ρ4)x = 0



2








kt + (k4ρ2)x = 0

(kρ2)t +3

2(k4ρ4)x = 0



2








kt + (k4ρ2)x = 0

(kρ2)t +3

2(k4ρ4)x = 0



2





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






ut = ∇ · (un∇u)









ut = ∇ · (un∇u)









ut = ∇ · (un∇u)







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.







− i4ξf + |f |2f ′ = C


Similarity solution

A(x , t) = A0t−1/4 exp

(ix2

8√t|A0|








− i4ξf + |f |2f ′ = C


Similarity solution

A(x , t) = A0t−1/4 exp

(ix2

8√t|A0|








− i4ξf + |f |2f ′ = C


Similarity solution

A(x , t) = A0t−1/4 exp

(ix2

8√t|A0|








− i4ξf + |f |2f ′ = C


Similarity solution

A(x , t) = A0t−1/4 exp

(ix2

8√t|A0|




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



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



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









DNLS

iAτ +(|A|2Ax

)x

= 0









DNLS

iAτ +(|A|2Ax

)x

= 0



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



Returning to MRS






Returning to MRS






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?





ut + uux = H[u]








ut + uux = H[u]








ut + uux = H[u]






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





u 7−→ u + B(u, u)



u 7−→ u + H [Hu ·Hux ]






u 7−→ u + B(u, u)



u 7−→ u + H [Hu ·Hux ]




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





1 Ek(u) ∼ ||∂kx u||2L22



Ek =1

2||∂kx u||2L2 + 〈∂

kx u, ∂

kxH[Hu ·Hux ]〉






1 Ek(u) ∼ ||∂kx u||2L22



Ek =1

2||∂kx u||2L2 + 〈∂

kx u, ∂

kxH[Hu ·Hux ]〉






1 Ek(u) ∼ ||∂kx u||2L22



Ek =1

2||∂kx u||2L2 + 〈∂

kx u, ∂

kxH[Hu ·Hux ]〉




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.



What about MRS?



MRS normal form

u 7−→ u + 13∂x

(u2 − uv + 1

2v2)

v 7−→ v − 13∂x

(1

2u2 − uv + v2




What about MRS?



MRS normal form

u 7−→ u + 13∂x

(u2 − uv + 1

2v2)

v 7−→ v − 13∂x

(1

2u2 − uv + v2




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.





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)〉






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)〉




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


BackgroundDegenerate dispersive equationsNonlinear Schrödinger equations

Degenerate Schrödinger EquationsLocal ExistenceWavefront behavior


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