On Nonlinear Dispersive Equations in Periodic Structures ... Semiclassical Limits and Numerical ......

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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