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• On Nonlinear Dispersive Equations in PeriodicStructures: Semiclassical Limits and Numerical Methods

Peter A. Markowich

Department of Applied Mathematics and Theoretical Physics, University of Cambridge

June 6, 2008

P. A. Markowich (DAMTP,U. of Cambridge) Nonlinear Dispersive Equations June 6, 2008 1 / 38

• The linear Schrodinger Equation 26

{it =

2

2 + V (x) , x Rd , t R

(x , t = 0) = I (=I (x) exp

(iS(x)

)... complex-valued wave function

> 0...semiclassical parameter,

• The linear Schrodinger Equation 26

{it =

2

2 + V (x) , x Rd , t R

(x , t = 0) = I (=I (x) exp

(iS(x)

)... complex-valued wave function

> 0...semiclassical parameter,

• Free Schrodinger Equation: V 0

{it =

2

2 , x Rd , t R

(x , t = 0) = exp(

ikx

)plane wave, wave-vector k

(x , t) = exp(

i(

k x |k |

2

2

t

)

space-timeO()-wavelengthoscillations

)

E (k) =|k |2

2...dispersion relation

V 6 0 no explicit computation!

P. A. Markowich (DAMTP,U. of Cambridge) Nonlinear Dispersive Equations June 6, 2008 3 / 38

• Free Schrodinger Equation: V 0

{it =

2

2 , x Rd , t R

(x , t = 0) = exp(

ikx

)plane wave, wave-vector k

(x , t) = exp(

i(

k x |k |

2

2

t

)

space-timeO()-wavelengthoscillations

)

E (k) =|k |2

2...dispersion relation

V 6 0 no explicit computation!

P. A. Markowich (DAMTP,U. of Cambridge) Nonlinear Dispersive Equations June 6, 2008 3 / 38

• Free Schrodinger Equation: V 0

{it =

2

2 , x Rd , t R

(x , t = 0) = exp(

ikx

)plane wave, wave-vector k

(x , t) = exp(

i(

k x |k |

2

2

t

)

space-timeO()-wavelengthoscillations

)

E (k) =|k |2

2...dispersion relation

V 6 0 no explicit computation!

P. A. Markowich (DAMTP,U. of Cambridge) Nonlinear Dispersive Equations June 6, 2008 3 / 38

• W(entzel)-K(ramers)-B(rillouin)-ansatz

= e i

S{

t + div(S) = 0 transport equationSt +

12 |S |

2 + V (x) = 2

2 phase equation

equivalently, with v := S :{t + div(v) = 0

vt +(|v |2

2 + V (x))

= 2

2 (

)quantum hydrodynamic equations,dispersively regularized irrotational compressible Euler system, with

external pressure V (x) and internal quantum pressure 22 (

).

P. A. Markowich (DAMTP,U. of Cambridge) Nonlinear Dispersive Equations June 6, 2008 4 / 38

• Formal semiclassical (zero-dispersion) limit 0

transport equation

0t + div(

0S0) = 0S0t +

12 |S

0|2 + V (x) = 0

}WKB-system

Hamilton-Jacobi equation

0t + div(0v 0) = 0

v 0t +(|v0|2

2 + V (x))

= 0

} irrotational compressibleEuler-system withexternal pressure

problem: the solution S0 of the HJ-equation generally develops finite-timesingularities!

P. A. Markowich (DAMTP,U. of Cambridge) Nonlinear Dispersive Equations June 6, 2008 5 / 38

• Theorem

(J. Keller, P. Lax, ..., 50): Let Tc > 0 be the caustic onset time of the

HJ-equation. Then 0 exp(i S0 )L((0,T );L2(Rd )) = O() if

0 < T < Tc .

beyond caustics:

V. Maslov 60: phase shifts

P. Gerard, P. Markowich, N. Mauser, F. Poupaud, P.L. Lions, T. Paul,C. Sparber 90-08: semiclassical (Wigner) measures

S. Jin, S. Osher 02-08; C. Sparber, P. Markowich, N. Mauser 01:multi-valued solutions of HJ-equations

P. A. Markowich (DAMTP,U. of Cambridge) Nonlinear Dispersive Equations June 6, 2008 6 / 38

• Theorem

(J. Keller, P. Lax, ..., 50): Let Tc > 0 be the caustic onset time of the

HJ-equation. Then 0 exp(i S0 )L((0,T );L2(Rd )) = O() if

0 < T < Tc .

beyond caustics:

V. Maslov 60: phase shifts

P. Gerard, P. Markowich, N. Mauser, F. Poupaud, P.L. Lions, T. Paul,C. Sparber 90-08: semiclassical (Wigner) measures

S. Jin, S. Osher 02-08; C. Sparber, P. Markowich, N. Mauser 01:multi-valued solutions of HJ-equations

P. A. Markowich (DAMTP,U. of Cambridge) Nonlinear Dispersive Equations June 6, 2008 6 / 38

• Semiclassical (Wigner) Measures

Definition

Let L2(Rd) be a sequence of wave functions and () 0 a scale.Then w M+(Rdx Rd ) is called a semiclassical measure of on thescale () if for all a S(Rdx Rd ), along a subsequence:

lim0

Rd(x)aw (x , D) dx =

RdxRd

a(x , )w(dx , d).

Theorem

(80 folklore) The semiclassical measure(s) w = w(x , , t) of the solution(t) of the IVP-problem for the Schrodinger equation satisfies(y) theLiouville equation{

wt + xw xV w = 0 on Rdx Rd Rtw(t = 0) = wI (a semiclassical measure of

I ).

P. A. Markowich (DAMTP,U. of Cambridge) Nonlinear Dispersive Equations June 6, 2008 7 / 38

• Semiclassical (Wigner) Measures

Definition

Let L2(Rd) be a sequence of wave functions and () 0 a scale.Then w M+(Rdx Rd ) is called a semiclassical measure of on thescale () if for all a S(Rdx Rd ), along a subsequence:

lim0

Rd(x)aw (x , D) dx =

RdxRd

a(x , )w(dx , d).

Theorem

(80 folklore) The semiclassical measure(s) w = w(x , , t) of the solution(t) of the IVP-problem for the Schrodinger equation satisfies(y) theLiouville equation{

wt + xw xV w = 0 on Rdx Rd Rtw(t = 0) = wI (a semiclassical measure of

I ).

P. A. Markowich (DAMTP,U. of Cambridge) Nonlinear Dispersive Equations June 6, 2008 7 / 38

• Connection to WKB-Asymptotics:

If I (x) =I (x) exp

(SI (x)

), then wI (x , ) = I (x)( SI (x)).

The solution of the Liouville equation stays monokinetic

w(x , , t) = 0(x , t)( S0(x , t))

as long as S0 is the smooth solution of the HJ-equation{S0t +

12 |S

0|2 + V (x) = 0S0(x , t = 0) = SI (x) .

After caustic onset: multi-valued solution theory (C. Sparber, P.Markowich, N. Mauser 02; S. Jin, S. Osher 04)!

P. A. Markowich (DAMTP,U. of Cambridge) Nonlinear Dispersive Equations June 6, 2008 8 / 38

• Nonlinear Schrodinger Equations: V = f ()

{it =

2

2 + f (||2) , x Rd , t > 0 ,

(t = 0) =I exp

(i SI

), I , SI smooth

formal (compressible, isentropic, irrotational) Euler limit as 0:{0t + div(

0v 0) = 0 , 0(t = 0) = I ,v 0t +

(12 |v

0|2 + f (0))

= 0 , v 0(t = 0) = SI

Theorem

(E. Grenier 98; R. Carles 07): f > 0 on R+. Let T > 0 be smaller than the maximalexistence time (of smooth solutions) of the irrotational isentropic Euler system. Then p0 expi S0

L((0,T );Hs (Rd ))

0 0 for some s > 0 .

Proof: Theory of symmetric hyperbolic systems, energy estimates.

P. A. Markowich (DAMTP,U. of Cambridge) Nonlinear Dispersive Equations June 6, 2008 9 / 38

• Nonlinear Schrodinger Equations: V = f ()

{it =

2

2 + f (||2) , x Rd , t > 0 ,

(t = 0) =I exp

(i SI

), I , SI smooth

formal (compressible, isentropic, irrotational) Euler limit as 0:{0t + div(

0v 0) = 0 , 0(t = 0) = I ,v 0t +

(12 |v

0|2 + f (0))

= 0 , v 0(t = 0) = SI

Theorem

(E. Grenier 98; R. Carles 07): f > 0 on R+. Let T > 0 be smaller than the maximalexistence time (of smooth solutions) of the irrotational isentropic Euler system. Then p0 expi S0

L((0,T );Hs (Rd ))

0 0 for some s > 0 .

Proof: Theory of symmetric hyperbolic systems, energy estimates.

P. A. Markowich (DAMTP,U. of Cambridge) Nonlinear Dispersive Equations June 6, 2008 9 / 38

• Semiclassical Limits in Periodic Structures

Figure: Periodic crystal lattice, = Zd , fundamental domainC

Figure: Electrons moving in aperiodic lattice potential,V = V(x/).

Assumption: lattice spacing =semiclassical parameter

NLS: it = 2

2 + V

(x

) + V (x)

slow-fast coupling

+ ()||2 binary interaction

P. A. Markowich (DAMTP,U. of Cambridge) Nonlinear Dispersive Equations June 6, 2008 13 / 38

• Semiclassical Limits in Periodic Structures

Figure: Periodic crystal lattice, = Zd , fundamental domainC

Figure: Electrons moving in aperiodic lattice potential,V = V(x/).

Assumption: lattice spacing =semiclassical parameter

NLS: it = 2

2 + V

(x

) + V (x)

slow-fast coupling

+ ()||2 binary interaction

P. A. Markowich (DAMTP,U. of Cambridge) Nonlinear Dispersive Equations June 6, 2008 13 / 38

• The Failure of the Standard WKB-Method

Linear case = 0:

it = 2

2 + V

(x

) + V (x) , =

e i

S{

t + div(S) = 0St +

12 |S |

2 + V(

x

)+ V (x) =

2

2

m drop the O(2)-pressure term

St +1

2|S |2 + V

(x

)+ V (x)

H(x , x,S)

= 0 caustic onset time=O(2)!H...Hamiltonian -periodic in y = x

Homogenisation theory based on viscosity solutions:

S0t + H(x ,S0) = 0

P. A. Markowich (DAMTP,U. of Cambridge) Nonlinear Dispersive Equations June 6, 2008 14 / 38

• The Failure of the Standard WKB-Method

Linear case = 0:

it = 2

2 + V

(x

) + V (x) , =

e i

S{

t + div(S) = 0St +

12 |S |

2 + V(

x

)+ V (x) =

2

2

m drop the O(2)-pressure term

St +1

2|S |2 + V

(x

)+ V (x)

H(x , x,S)

= 0 caustic onset time=O(2)!H...Hamiltonian -periodic i