[PPT] - Wave motion around 2-D viscous shock profile Shih-Hsien Yu PowerPoint Presentation

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Wave motion around 2-D viscous shock profile

Shih-Hsien Yu

Department of Mathematics, National University of Singapore 24th Annual Workshop on Differential Equations 24th, National Sun Yat-sen University, Kaoshiong January 22-23, 2016

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Viscous shock layer stability An example:

A compressible Navier-Stokes equation in 2-D,

x ∈ R2.

ρt + ∇ ·

u = 0,

mt + ∇ ·

u ⊗ m + ∇p(ρ) − ∆ u = 0 ,

ρ(

x, t) : density at ( x, t)

u(

x, t) ∈ R2 : fluid velocity at ( x, t)

m ≡ ρ

u : momentum. p(ρ) = ργ : pressure, γ ∈ [1, 5/3)

A viscous shock layer Ψ(x − st): A travelling wave soluton.

limx→±∞ Ψ(x) = ρ±

m±
,
(ρ−, ρ+), (

m−, m+) : end states of a shock wave s : Speed of the shock wave

Time-asymptotic stability of viscous shock layer

To study (ρ, m)t( x, t) − Ψ(x − st) as t → ∞.

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Physics and Solution of PDE

Relationship in Classical PDE-Physics

Physics =

⇒ PDE (model of physics)

Solution of PDE =

⇒ Realization of Physics.

Practically no relationship in Modern PDE-Physics

Introduction of Real analysis = ⇒ Solutions of PDE(PDE not necessary physical) = ⇒ Realization of Physics

Viscous shock layer stablity in 2-D :

A platform for exploring new tools for studying PDE

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Illustrations

Sound Wave Fluid velocity Fluid velocity Shock wave front Sound wave refmection Sound wave refmection Subsonic Region Supersonic Region

A Simplifed Hyperbolic Diagram

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Illustrations

Sound Wave Fluid velocity Fluid velocity Shock wave front Sound wave refmection Sound wave refmection Subsonic Region Supersonic Region

Realistic Diagram

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Matheimatics-Physics natures in viscous conservation laws

When space-dimension=1, modern PDE is sufficient to show nonlinear time-asymptotic

stability of a viscous shock profile.

When space-dimension ≥ 2, perturbations in subsonic region progress after the light-cone in

space-time domain within a light-cone, and the perturbations with a light-cone decay with a slower

algebraic rate. This slowly decaying property causes modern PDE not sufficient for studying nonlinear coupling and nonlinear time-asymptotic stablity.

“

” of classical PDE shall remain and evolve with development of

Modern PDE.

To construct the solutions of PDE with further physics natures of the solutions such as the pointwise in the space-time domain, etc.. The key ingredient should be a better realization in physics domain and in transform variables of

“Green’s function”

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Model nonlinear problem in 2-D

Ut + UUx + Vy = ∆U

Vt + Vx + Uy = ∆V

shock wave

U V

=

H(x)

, H(x) ≡
1 for x < 0

− for x > 0

shock profile

U V

=

ϕ(x)

, ϕ(x) ≡ − tanh(x/2)

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

An illustration

Sound Wave Fluid velocity Fluid velocity Shock wave front Sound wave refmection Sound wave refmection Subsonic Region Supersonic Region

A simplifed diagram for basic elements

t

2

U 0

t x 2 y 2 U

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Five basic problems PLUS auxiliary problem

I. For two far fields

          

∂t + ∂x − ∆ +
∂y

∂y

U−(x, y, t) = 0

U−(x, y, 0) = δ(x)δ(y)

1

1



         

∂t − ∆ +
−∂x

∂y ∂y ∂x

U+(x, y, t) = 0

U+(x, y, 0) = δ(x)δ(y)

1

1

II. For interactions between two ends of a shock wave

          

∂t − ∆ +
∂x H(x)

∂y ∂y ∂x

G(x, y, t; x∗) = 0

G(x, y, 0; x∗) = δ(x − x∗)δ(y)

1

1



         

∂t − ∆ +
H(x)∂x

∂y ∂y ∂x

G(x, y, t; x∗) = 0

G(x, y, 0; x∗) = δ(x − x∗)δ(y)

1

1

III. For interactions between shock profile and a shock wave
(∂t − ∆ + ϕ(x)∂x )g(x, y, t; x∗) = 0

g(x, y, 0; x∗) = δ(x − x∗)δ(y) g(x, y, t; x∗) = cosh(x∗/2)

cosh(x/2) e− (x−x∗)2+y2 4t − t 4 4πt

IV. Auxilary problem

          

∂t − ∆ +
ϕ(x)∂x

∂y ∂y ∂x

G(x, y, t; x∗) = 0

G(x, y, 0; x∗) = δ(x − x∗)δ(y)

1

1

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Interactions at shock front

∂t and ∂y are tangential to shock front
∂t and ∂y can be replaced by transform variables s and η
∂x is normal to shock front and local
Laplace-Fourier transform:

L[f](x, iη, s) ≡

R

∞ e−iηy−stf(x, y, t)dtdy.

I.           

s + η2 + ∂x − ∂2

x +

iη

iη

L[U−] = δ(x)
1

1

s + η2 − ∂2

x +

−∂x

iη iη ∂x

L[U+] = δ(x)
1

1

II.

          

s + η2 − ∂2

x +

∂x H(x)

iη iη ∂x

L[G] = δ(x − x∗)
1

1

s + η2 − ∂2

x +

H(x)∂x

iη iη ∂x

L[G] = δ(x − x∗)
1

1

Interaction by II ⇒

     Both L[G] and

−∂x +
H(x)

1

L[G] are continuous in x.

Both L[G] and ∂xL[G] are continuous in x.

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Interaction ⇒ Scattering data

s + η2 − ∂2

x +

∂xH(x) iη iη ∂x

L[G] = δ(x − x∗)

1 1

x=0

x >0

*

x=0 x <0

*

L[G](x, iη, s; x∗) = L[U+](x − x∗, iη, s)

2

k,l=1

T +

kl (iη, s)e−λ+,k x∗+λ−,l x 2

k,l=1

R+

kl (iη, s)e−λ+,k x∗−λ+,l x 2

k,l=1

R−

kl (iη, s)eλ−,k x∗+λ−,l x 2

k,l=1

T −

kl (iη, s)eλ−,k x∗−λ+,l x

L[U−](x − x∗, iη, s)

R±

kl : Reflective matrix

T ±

kl

: Transmissive matrix

L[G](x, iη, s; x∗) =

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Laplace wave numbers

Characteristic Polynomials for ODE.

P±(ξ; iη, s) = det s + η2 − ξ2∓ξ iη iη s + η2 − ξ2 + ξ

Eight basic “Laplace” wave numbers λ±,j(iη, s): P±(λ±,j; iη, s) = 0.

(λ±,j is a complex wave number and P±(λ; iη, s) = 0 is an implicit dispersive relationship between the Laplace wave number and s complex frequency.)

                 {λ+,1, λ+,2, −λ+,1, −λ+,2}, {λ−,1, λ−,2, λ−,3, λ−,4} = 1/2 + {σ+, σ−, −σ+, −σ−}, λ+,1 =

(
(s + 1/4) − 1/2)2 + η2,

λ+,2 =

(
(s + 1/4) + 1/2)2 + η2,

σ± =

(1/4 + s + η2 ± iη).
Singular wave number: Λ ≡ λ+,1 −
s + 1/4 + 1/2 .
Four “Laplace” wave trains. (In contrast to wave train e−iκx+iω(κ)t)
e−λ+,1(iη,s)x, e−λ+,2(iη,s)x for x > 0,

eλ−,1(iη,s)x, eλ−,2(iη,s)x for x < 0.

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Laplace wave trains and I.

The symbols of L[U±](x, iη, s)

L[U+](x, iη, s) =

U+

11

U+

12

U+

21

U+

22

,

                           U+

11 ≡

−2λ+,1−

√ 4s+1+1

e−λ+,1|x|

−4λ+,1 √ 4s+1

+

−2λ+,2−

√ 4s+1+1

e−λ+,2|x|

−4λ+,2 √ 4s+1

U+

12 ≡ − iηe−λ+,1|x| 2λ+,1 √ 4s+1 − iηe−λ+,2|x| 2λ+,2 √ 4s+1

U+

21 ≡ − iηe−λ+,1|x| 2λ+,1 √ 4s+1 − iηe−λ+,2|x| 2λ+,2 √ 4s+1

U+

22 ≡ (2λ+,1+ √ 4s+1+1)e−λ+,1|x| 4λ+,1 √ 4s+1

+

2λ+,2+

√ 4s+1+1

e−λ+,2|x|

4λ+,2 √ 4s+1

L[U−](x, iη, s) =     − eλ−,3x

σ−,1

− eλ−,4x

σ−,2 eλ−,3x σ−,1

− eλ−,4x

σ−,2 eλ−,3x σ−,1

− eλ−,4x

σ−,2

− eλ−,3x

σ−,1

− eλ−,4x

σ−,2

    for x > 0, L[U−](x, iη, s) =     − eλ−,1x

σ−,1

− eλ−,2x

σ−,2 eλ−,1x σ−,1

− eλ−,2x

σ−,2 eλ−,1x σ−,1

− eλ−,2x

σ−,2

− eλ−,1x

σ−,1

− eλ−,2x

σ−,2

    for x < 0.

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Inverse of “Laplace” wave trains.

L−1[e−λ+,l x](y, t) and L−1[eλ−,l x](y, t).
{λ−,1,λ−,2,λ−,3,λ−,4}=1/2+{σ+,σ−,−σ+,−σ−},

σ± =

(1/4 + s + η2 ± iη),
λ+,1=
(√

(s+1/4)−1/2)2+η2, λ+,2=

(√

(s+1/4)+1/2)2+η2.

Cauchy Integral formula and Inverse Fourier transform = ⇒ L−1[e−λ+,2x](y, t) = 1 4π2i

R
Re(s)=0

eiyη+ste−λ+,2xdsdη, L−1[e−λ+,1x](y, t) = 1 4π2i

R
Re(s)=0

eiyη+ste−λ+,1xdsdη. |L−1[e−λ+,2x](y, t)| ≤ O(1)|xt|−(1−σ0)e−(|x|+|y|+t)/κ0 for x > 0.

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Inverse of the characteristic Laplace wave train

L−1[e−λ+,1x](y, t) = −L−1[e−λ+,2x](y, t) + 1 4π2

R2 e−ixζ+iηy−(ζ2+η2)ti2ζ
2 cos(
ζ2 + η2t) + sin(
ζ2 + η2t)
ζ2 + η2
dζdη.

Hadamard’s solution

Re(s) = 0 < 0 > 0 < 0 > 0 = {

2 2 + i 2 + 2 |

R} = {(

2 2

i

2 + 2 )|

R} = {

2

1/ 4 | R}

+ −

x y t

x2 + y2 = t

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Realization of Problem II

F ±

kl and R± kl are 2 × 2 matrice with (i, j)-entry in the form:

αkl;ij(iη, s) + βkl;ij(iη, s) λ+,1(iη, s) + Akl;ij(s)Bkl;ij(iη) + akl;ij(s)bkl;ij(iη) λ+,1(iη, s) . All coefficients αkl;ij and βkl;ij are analytic functions in Re(s) > −1/8 and |η| ≪ 1; and lim

Re(s)>−1/8 s→∞

| √ s|(|αkl;ij(iη, s)| + |βkl;ij(iη, s)|) < ∞ for |η| ≪ 1.

A11;ij B11;ij +

a11;ij b11;ij λ+,1

2×2

≡      − 1 2 − 1− √ 4s+1 4λ+,1 i(−4s+ √ 4s+1−1) 2 √ 4s+1η + i(2s( √ 4s+1−2)+ √ 4s+1−1) 2η √ 4s+1λ+,1 − 1 2 − 1− √ 4s+1 4λ+,1     ,

A12;ij B12;ij +

a1,2;ij lb2,1;ij λ+,1

2×2

≡   

− 1 2 √ 4s+1 + 4s+ √ 4s+1+1 (16s+4)λ+,1 − i 2η − i(−4s+ √ 4s+1−1) 4η √ 4s+1λ+,1 1 2 √ 4s+1+2 − 4s− √ 4s+1+1

(16s+4

√ 4s+1+4)λ+,1

   ,

A21;ij =A22;ij =B21;ij =B22;ij =a21;ij =a22;ij =b21;ij =b22;ij =0.

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

The primary wave structure in a finite Mach number region

(x − x∗)2 + (y − y∗)2 < Kt
2
k,l=1

L−1

Akl;ij (s)Bkl;ij (iη) +

akl;ij (s)bkl;ij (iη) λ+,1(iη, s)

e−λ+,k x∗−λ+,l x
(y, t) +

2

k,l=1

1 4π2

|η|≪1

eiηy

Re(s)=0

est

αkl;ij (iη, s) +

βkl;ij (iη, s) λ+,1(iη, s)

e−λ+,k x∗−λ+,l x ds
dη

x x * t 1/t D e c a y r a t e : 1/t D e c a y r a t e :

3/4

x * x *

(x − x∗)2 + (y − y∗)2 = Kt

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Problem IV: Auxiliary problem

Transform of the auxiliary problem:

x u

u=1

x

u=1 u=-1

y

u= u u= u= = u=H(x) u=-1

s + η2 + ϕ(x)∂x − ∂2

x

iη iη iη s + η2 + ∂x − ∂2

x

L[G]

= δ(x − x∗) 1 1

s + H(x)∂x − ∂2

x + η2

iη iη s + ∂x − ∂2

x + η2

L[G] =

(H(x) − ϕ(x))∂x

L[G]

+δ(x − x∗)I.

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

A sequence of interaction for auxiliary problem IV

s + H(x)∂x − ∂2

x + η2

iη iη s + ∂x − ∂2

x + η2

L[G h

0 ] = δ(x − x∗)I,

s + ϕ∂x − ∂2

x + η2

s + ∂x − ∂2

x + η2

L[G b

0 ] =

(H(x) − ϕ(x))∂x

L[G h

0 ]

i ≥ 1

s + H(x)∂x − ∂2

x + η2

iη iη s + ∂x − ∂2

x + η2

L[G h

i ] = −

iη iη

L[G b

i−1]

s + ϕ∂x − ∂2

x + η2

s + ∂x − ∂2

x + η2

L[G b

i ] =

(H(x) − ϕ(x))∂x

L[G h

i ]

⇓

s + ϕ(x)∂x − ∂2

x + η2

iη iη s + ∂x − ∂2

x + η2

∞

i=0

(L[G h

i ] + L[G b i ]) = δ(x − x∗)I,

if ∞

i=0(L[G h i ] + L[G b i ]) exists and (SOME PROPERTIES etc.)

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Integral formula for sequential iterations

Basic integredients

               L[U±](x; iη, s) : Fundamental solutions at far fields. T ±

kl (iη, s),

R±

kl (iη, s) : Scattering data.

Solution of (s + η2 + ϕ(x)∂x − ∂2

x )L[g] = δ(x − x∗):

cosh(x∗/2) cosh(x/2) e−σB|x−x∗| σB ; σB ≡

s + 1/4 + η2
The sequential interations

L[G h

0 ](x, iη, s; x∗) =

R

L[G](x, iη, s; z)δ(z − x∗)dz, L[G b

0 ](x, iη, s; x∗) =

R

 

cosh(z/2)e−σB|x−z| cosh(x/2) H(z)−ϕ(z) σB

∂z   L[G h

0 ](z, iη, s; x∗)dz

i ≥ 1 L[G h

i ](x, iη, s; x∗) = −

R

L[G](x, iη, s; z) iη iη

L[G b

i−1](x, iη, s; x∗)dz,

L[G b

i ](x, iη, s; x∗) =

R

 

cosh(z/2)e−σB|x−z| cosh(x/2) H(z)−ϕ(z) σB

∂z   L[G h

i ](z, iη, s; x∗)dz.

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

A particle-collision like algorithm

cosh(z/2)

cosh(x/2) (H(z) − tanh(z/2)) = sgn(z) e−(|x|+|z|)/2 1 + e−|x| = sgn(z)e−(|x|+|z|)/2

∞

j=0

(−1)j e−j|x|

L[G h

i ](x, iη, s; x∗) = −

R

L[G](x, iη, s; z) iη iη

L[G b

i−1](x, iη, s; x∗)dz,

L[G b

i ](x, iη, s; x∗) =

e−|x|/2 1 + e−|x|

R
sgn(z)e−σB|x−z|e−|z|/2∂z

σB

L[G h

i ](z, iη, s; x∗)dz

≡Qi (x,x∗)

. Definition Let f(x, x∗) and g(x, x∗) be functions in x and x∗. f♯g(x, x∗) ≡

R

f(x, z)e−|z|/2g(z, x∗)dz.

The collision operator I

Qi+1(x, x∗) = −          

e−σB|x−x∗| σB

sgn(x∗)   ♯

∂x [G] ·

iη iη

≡H

♯ 1 1 + e−|x| · Qi        

≡ I[Qi ]

(x, x∗)

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

The extended Laplace wave trains

χ(x) = 1 if x > 0 else χ(x) = 0.

x x*

                                               Aα1,α2

j,σ

= e

−α1x−α2x∗−(−1)σm+ j (x−x∗)χ(x∗)χ(x)

1 + (−1)σ

2 χ(x − x∗) + 1 − (−1)σ 2 (1 − χ(x − x∗))

,

aα1,α2

j,k

= χ(x)(1 − χ(x∗))e

−α1x+(α2+ 1 2 )x∗−m+ j x+m− k x∗ ,

Eα1,α2

j,l

= e

−α1x−α2x∗−m+ j x−m+ l x∗ χ(x)χ(x∗),

Bα1,α2

k,σ

= e(α1+

1−δk 1 2 )x+(α2+ 1−δl 1 2 )x∗−(−1)σm− k (x−x∗)

·(1 − χ(x∗))(1 − χ(x)) 1+(−1)σ

2

χ(x − x∗) + 1−(−1)σ

2

(1 − χ(x − x∗))

,

bα1,α2

k,l

= (1 − χ(x))χ(x∗)e(α1+ 1

2 )x−α2x∗+m− k x−m+ l x∗ ,

Fα1,α2

j,l

= e

(α1+ 1−δk 1 2 )x+(α2+ 1−δl 1 2 )x∗+m− j x+m− l x∗ ,

where      σ = 0, 1; j, k, l = 1, 2, 3, αi ∈ 1

2 Z+ for i = 1, 2, 3;

     m+

1 = σB(iη, s),

m+

2 = λ+,1(iη, s),

m+

3 = λ+,2(iη, s),

     m−

1

= σB(iη, s), m−

2

= σ−,1(iη, s), m−

3

= σ−,2(iη, s).

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CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Laplace Wave Train for H

¯ A+

i,k ≡ Ak,0 i,0 ,

¯ a+

i,k ≡ Ak,0 i;1 ,

¯ A−

i,k ≡ Bk,0 i,0 ,

¯ a−

i,k ≡ Bk,0 i;1 ,

           A+

1,k(x, x∗) ≡ Ak,1/2 1,0

, A+

j,k(x, x∗) ≡ Ak+1/2,0 j,0

for j = 2, 3, a+

1,k(x, x∗) ≡ Ak,1/2 1,1

, a+

j,k(x, x∗) ≡ Ak+1/2,0 j,1

for j = 2, 3,            A−

1,k(x, x∗) ≡ Bk,1/2 1,0

, A−

j,k(x, x∗) ≡ Bk+1/2,0 j,0

for j = 2, 3, a−

1,k(x, x∗) ≡ Bk,1/2 1,1

, a−

j,k(x, x∗) ≡ Bk+1/2,0 j,1

for j = 2, 3.

SLIDE 24

CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

H (x, x∗; iη, s) = η

2

O0(iη, s)¯

A+

1,0♯¯

a+

2,0 + O0(iη, s)¯

A+

1,0♯¯

A+

2,0 + O0(iη, s)¯

a+

1,0♯¯

a+

3,0

+ η

2

  O0(iη, s)(A1,0 − A3,0) + O0(iη, s) a1,0 − a2,0

1 2 + m+ 1

− m+

2

+ O0(iη, s)

A1,0 + a3,0 − E0,0

1,3 1 2 − m+ 1

− m+

3

   + η

2 3

j=2

O0(iη, s)

a1,0 + Aj,0

1 2 + m+ 1

+ m+

j

+ ηO0(iη, s)

   A−

1;0 + a− 2;0 − F0,0 1,2

1 + m−

1

+ m−

2

−

A−

1;0 + a− 3;0 − F0,0 1,3

1 + m−

1

+ m−

3

   + ηO0(iη, s)    A−

1;0 − A− 2;0

−1 − m−

1

+ m−

2

−

A−

1;0 − A− 3;0

−1 − m−

1

+ m−

3

  

+ ηO0(iη, s)¯

a−

1;0 ♯(¯

A−

2;0 − ¯

A−

3;0 )

+ ηO0(iη, s)

  

1

2 − σ−,1

−σB + σ−,1 + 1

(a−

1;0 − a− 2;0 ) −

1

2 − σ−,2

−σB + σ−,2 + 1

(a−

1;0 − a− 3;0 )

  

+ ηO0(iη, s)

   A−

1;0 + a− 2;0 − F0,0 1,2

1 + m−

1

+ m−

2

−

A−

1;0 + a− 3;0 − F0,0 1,3

1 + m−

1

+ m−

3

   + ηO0(iη, s)    A−

1;0 − A− 2;0

−1 − m−

1

+ m−

2

−

A−

1;0 − A− 3;0

−1 − m−

1

+ m−

3

  

+ ηO0(iη, s)¯

a−

1;0 ♯(¯

A−

2;0 − ¯

A−

3;0 )

+ ηO0(iη, s)

   

1

2 − m− 2

1 − m−

1

+ m−

2

(a−

1;0 − a− 2;0 ) −

1

2 − m− 3

−1 − m−

1

+ m−

3

(a−

1;0 − a− 3;0 )

   

2

j,i=1

 

O1(iη, s) + O0(η, s)λ+,1
E

1/2,0 i,j

+

O1(iη, s) + O0(η, s)λ+,1
F

1/2,0 i,j

+

O1(iη, s) + O0(η, s)λ+,1
a

1/2,0 i,j

+

O1(iη, s) + O0(η, s)λ+,1
b

1/2,0 i,j

+

2

i=1

O1(iη, s) + O0(η, s)λ+,1

E

0,0 1,i +

O1(iη, s) + O0(η, s)λ+,1
F

0,0 1,i

+

O1(iη, s) + O0(η, s)λ+,1
a

0,0 1,i +

O1(iη, s) + O0(η, s)λ+,1
b

0,0 1,i

.

SLIDE 25

CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Interaction of Laplace wave trains

u♯v = (a1u1 + a2u2 + a3u3)/C(u, v), a1, a2, a3 ∈ {0, −1, 1}

                                                                           Aα1,α2

j,0

♯Aβ1,β2

k,1

= Eα1,β1

j,k

− A

α1,α2+( 1 2 +β1+β2) j,0

− A

β1+( 1 2 +α1+α2),β2 k,1 1 2 + α2 + β1 − m+ j

− m+

k

, Aα1,α2

j,0

♯Aβ1,β2

k,0

= A

α1,α2+( 1 2 +β1+β2) j,0

− A

β1+( 1 2 +α1+α2),β2 k,0 1 2 + α2 + β1 − m+ j

+ m+

k

, Aα1,α2

j,1

♯Aβ1,β2

k,1

= −A

α1,α2+( 1 2 +β1+β2) j,1

+ A

β1+( 1 2 +α1+α2),β2 k,1 1 2 + α2 + β1 + m+ j

− m+

k

, Aα1,α2

j,1

♯Aβ1,β2

k,0

= A

α1,α2+( 1 2 +β1+β2) j,1

+ A

β1+( 1 2 +α1+α2),β2 k,0 1 2 + α2 + β1 + m+ j

+ m+

k

, Aα1,α2

i,0

♯Eβ1,β2

j,k

= Eα1,β2

i,k

− E

( 1 2 +α1+α2)+β1,β2 j,k 1 2 + α2 + β1 − m+ i

+ m+

j

, Eα1,α2

i,j

♯Aβ1,β2

k,0

= E

α1,α2+( 1 2 +β1+β2) i,j 1 2 + β1 + α2 + m+ k + m+ j

, Aα1,α2

i,1

♯Eβ1,β2

j,k

= E

( 1 2 +α1+α2)+β1,β2 j,k 1 2 + α2 + β1 + m+ i

+ m+

j

, Eα1,α2

i,j

♯Aβ1,β2

k,1

= Eα1,β2

i,k

− E

α1,α2+( 1 2 +β1+β2) i,j 1 2 + β1 + α2 − m+ k + m+ j

,

SLIDE 26

CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D                  Aα1,α2

i,0

♯aβ1,β2

j,k

= aα1,β2

i,k

− a

( 1 2 +α1+α2)+β1,β2 j,k 1 2 + α2 + β1 − m+ i

+ m+

j

, bα1,α2

i,j

♯Aβ1,β2

k,0

= b

α1,α2+( 1 2 +β1+β2) i,j 1 2 + β1 + α2 + m+ k + m+ j

, Aα1,α2

i,1

♯aβ1,β2

j,k

= a

( 1 2 +α1+α2)+β1,β2 j,k 1 2 + α2 + β1 + m+ i

+ m+

j

, bα1,α2

i,j

♯Aβ1,β2

k,1

= Eα1,β2

i,k

− E

α1,α2+( 1 2 +β1+β2) i,j 1 2 + β1 + α2 − m+ k + m+ j

,    Aα1,α2

j,σ1

♯Bβ1,β2

k,σ2

= 0, Aα1,α2

j,σ1

♯bβ1,β2

k,l

= 0, aα1,α2

k,l

♯Aβ1,β2

j,σ1

= 0, Bα1,α2

j,σ1

♯Aβ1,β2

k,σ2

= 0, Bα1,α2

j,σ1

♯aβ1,β2

k,l

= 0, bα1,α2

k,l

♯Bβ1,β2

j,σ1

= 0,                                                  Bα1,α2

i,0

♯bβ1,β2

j,k

= bα1,β2

i,k

− b

( 3 2 −δi 1+α1+α2)+β1,β2 j,k

(3 − δi

1 − δj 1)/2 + α2 + β1 − m− i

+ m−

j

, aα1,α2

i,j

♯Bβ1,β2

k,1

= a

α1,α2+( 3 2 −δk 1 +β1+β2) i,j

(3 − δj

1 − δk 1 )/2 + β1 + α2 + m− k

+ m−

j

, Bα1,α2

i,0

♯bβ1,β2

j,k

= b

( 3 2 −δi 1+α1+α2)+β1,β2 j,k

(3 − δj

1 − δi 1)/2 + α2 + β1 + m− i

+ m−

j

, aα1,α2

i,j

♯Bβ1,β2

k,0

= aα1,β2

i,k

− a

α1,α2+( 3 2 −δk 1 +β1+β2) i,j

(3 − δj

1 − δk 1 )/2 + β1 + α2 − m− k

+ m−

j

.

SLIDE 27

CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D                                                                                                              Bα1,α2

j,1

♯Bβ1,β2

k,0

= Fα1,β1

j,k

− B

α1,α2+(3/2−δk 1 +β1+β2) j,1

− B

β1+(3/2−δj 1+α1+α2),β2 k,0

(3 − δj

1 − δk 1 )/2 + α2 + β1 − m− j

− m−

k

, Bα1,α2

j,1

♯Bβ1,β2

k,1

= B

α1,α2+(3/2−δk 1 +β1+β2) j,1

− B

β1+(3/2−δj 1+α1+α2),β2 k,1

(3 − δj

1 − δk 1 )/2 + α2 + β1 − m− j

+ m−

k

, Bα1,α2

j,0

♯Bβ1,β2

k,0

= −B

α1,α2+( 3 2 −δk 1 +β1+β2) j,0

+ B

β1+( 3 2 −δj 1+α1+α2),β2 k,0

(3 − δj

1 − δk 1 )/2 + α2 + β1 + m− j

− m−

k

, Bα1,α2

j,0

♯Bβ1,β2

k,1

= B

α1,α2+( 3 2 −δk 1 +β1+β2) j,0

+ B

β1+( 3 2 −δj 1+α1+α2),β2 k,1

(3 − δj

1 − δk 1 )/2 + α2 + β1 + m− j

+ m−

k

, Bα1,α2

i,0

♯Fβ1,β2

j,k

= Fα1,β2

i,k

− F

( 3 2 −δi 1+α1+α2)+β1,β2 j,k

(3 − δi

1 − δj 1)/2 + α2 + β1 − m− i

+ m−

j

, Fα1,α2

i,j

♯Bβ1,β2

k,1

= F

α1,α2+( 3 2 −δk 1 +β1+β2) i,j

(3 − δj

1 − δk 1 )/2 + β1 + α2 + m− k

+ m−

j

, Bα1,α2

i,0

♯Fβ1,β2

j,k

= F

( 3 2 −δi 1+α1+α2)+β1,β2 j,k

(3 − δj

1 − δi 1)/2 + α2 + β1 + m− i

+ m−

j

, Fα1,α2

i,j

♯Bβ1,β2

k,0

= Fα1,β2

i,k

− F

α1,α2+( 3 2 −δk 1 +β1+β2) i,j

(3 − δj

1 − δk 1 )/2 + β1 + α2 − m− k

+ m−

j

,

SLIDE 28

CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Equilibrium ♯-product

Definition

Let u and v be the Laplace wave trains. u♯v is an equilibrium sharp product iff

C(u, v)|η=0 = 0 for some s ∈ |Re(s)| < 1/8

Let ai are the basic Laplace wave train. The sequence

a1♯a2♯ · · · ♯an is an equilibrium compond ♯-product iff ak and ak+1 are equilibrium ♯-product for all k = 1, · · · , n − 1.

SLIDE 29

CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

The diagram of equilibrium ♯-product Laplace wave train for H

¯ A+

1,0]¯

A+

2,0



◆

¯ A+

1,0]¯

A+

2,0

◆





◆ 

¯ A+

1,0]¯

A+

2,0



◆

¯ A+

1,0]¯

A+

2,0



◆ 

¯ A+

1,0]¯

A+

2,0

¯ A+

1,0]¯

a+

2,0

' 

¯

A+

1,0]¯

a+

2,0

' 

'



¯

A+

1,0]¯

a+

2,0

' 

¯

A+

1,0]¯

a+

2,0

' 

¯

A+

1,0]¯

a+

2,0

¯ a+

1,0]¯

a+

3,0

¯ a+

1,0]¯

a+

3,0

¯ a+

1,0]¯

a+

3,0

¯ a+

1,0]¯

a+

3,0

¯ a+

1,0]¯

a+

3,0

A+

1,0

7 ⇡

A+

1,0

7 ⇡

7

⇡

A+

1,0

7 ⇡

A+

1,0

7 ⇡

A+

1,0

A+

2,0

?

⌫

A+

2,0

?

⌫

?

⌫

A+

2,0

?

⌫

A+

2,0

?

⌫

A+

2,0

A+

3,0

 ✏⌫ ✏⌫

⌘⇡

I G ? 7

⌫

I G ? 7

A+

3,0

 ✏⌫ ✏⌫

⌘⇡

I G ? 7

⌫

A+

3,0

 ✏⌫ ✏⌫

⌘⇡

I G ? 7

⌫

A+

3,0

 ✏⌫ ✏⌫

⌘⇡

I G ? 7

⌫

A+

3,0 a+

1,0

a+

1,0

a+

1,0

a+

1,0

a+

1,0

a+

2,0

I 7 '

a+

2,0

I 7 ' I 7 '

a+

2,0

I 7 '

a+

2,0

I 7 '

a+

2,0

a+

3,0

a+

3,0

a+

3,0

a+

3,0

a+

3,0

e−xa+

1,0

e−xa+

1,0

e−xa+

1,0

e−xa+

1,0

e−xa+

1,0

Eα,β

i,j

?

Eα,β

i,j

?

Eα,β

i,j

?

Eα,β

i,j

?

Eα,β

i,j

¯ A+

1,0]E0,0 2,j

D

¯ A+

1,0]E0,0 2,j

D

¯ A+

1,0]E0,0 2,j

D

¯ A+

1,0]E0,0 2,j

D

¯ A+

1,0]E0,0 2,j

A+

3,0]¯

A+

1,0]E0,0 2,j

G

A+

3,0]¯

A+

1,0]E0,0 2,j

G

A+

3,0]¯

A+

1,0]E0,0 2,j

G

A+

3,0]¯

A+

1,0]E0,0 2,j

G

A+

3,0]¯

A+

1,0]E0,0 2,j

Eα,0

i,1 ]¯

a+

3,0

Eα,0

i,1 ]¯

a+

3,0

Eα,0

i,1 ]¯

a+

3,0

Eα,0

i,1 ]¯

a+

3,0

Eα,0

i,1 ]¯

a+

3,0

Eα,0

i,2 ]a+ 1,0

Eα,0

i,2 ]a+ 1,0

Eα,0

i,2 ]a+ 1,0

Eα,0

i,2 ]a+ 1,0

Eα,0

i,2 ]a+ 1,0

Eα,0

i,2 ]¯

a+

1,0

Eα,0

i,2 ]¯

a+

1,0

Eα,0

i,2 ]¯

a+

1,0

Eα,0

i,2 ]¯

a+

1,0

Eα,0

i,2 ]¯

a+

1,0

SLIDE 30

CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Full scattering matrix

Γ ≡ Γ1 ∪ E1 ∪ Γ2 ∪ E2 ∪ Γ3 ∪ E3,

                                    

Γ1 ≡ {¯

A+

1,0♯¯

A+

2,0, ¯

A+

1,0♯¯

a+

2,0, ¯

a+

1,0♯¯

a+

3,0, A+ 1 , A+ 2 , A+ 3 , a+ 1 , a+ 2 , a+ 3,0, e−x a+ 1 },

E1 ≡

1≤i,j≤3

{E

0,0 i,j ,

E

δ

i 2+δ j 3 2

, δ

j 1 2

i,j

} ∪

3

j=1

{¯

A+

1,0♯E 0,0 2,j ,

A+

3,0♯¯

A+

1,0♯E 0,0 2,j } ∪ 3

i=1

{Eα,0

i,1

♯¯

a+

3,0, Eα,0 i,2

♯a+

1,0, Eα,0 i,2

♯¯

a+

1,0 : α = 0,

δi

2 + δi 3

2

}, Γ2 ≡ {(¯

A+

1,0♯¯

A+

2,0)♯(¯

a+

1,0♯¯

a+

3,0), (¯

A+

1,0♯¯

A+

2,0)♯a+ 1,0, (¯

A+

1,0♯¯

A+

2,0)♯a+ 3,0, (¯

A+

1,0♯¯

a+

2,0)♯(a1,0♯a3,0), · · · },

E2 ≡ {E0,0

i,2 ♯a+ 3,0|i = 1, 2, 3} ∪ {A+ 3,0♯E0,0 1,j |j = 1, 2, 3},

Γ3 ≡ {A+

3,0♯(¯

A+

1,0♯¯

A+

2,0)♯(¯

a+

1,0♯¯

a+

3,0), A+ 3,0♯(¯

A+

1,0♯¯

A+

2,0)♯a+ 1,0, A+ 3,0♯(¯

A+

1,0♯¯

A+

2,0)♯a+ 3,0, · · · },

E3 = {A+

3,0♯E0,0 1,2♯a+ 3,0}.

      

γ ∈ Γ, γk ≡ e−kx γ,

k ∈ Z+,

γk ≡ span(γk ),

  

γ =

k∈Z+ γk ,

Γ ≡

γ∈Γ γ .

The coefficient ring is generated by e−|x∗| and all memorphic functions in (iη, s) in Re(s) > 0 and |η| ≪ 1. For each h, γ ∈ Γ, h♯

j≥0 cj γj = u∈Γ Mij (h♯; u, γ)cj ui =

⇒ h♯ ⇐ ⇒ [h♯] ≡       [M(h♯; u1, γ1)] [M(h♯; u1, γ2)] · · · [M(h♯; u1, γn)] [M(h♯; u2, γ1)] [M(h♯; u2, γ2)] · · · [M(h♯; u2, γn)] . . . [M(h♯; un, γ1)] [M(h♯; un, γ2)] · · · [M(h♯; un, γn)]      

(ui ,γj )∈Γ×Γ

, = ⇒ [I] =

h∈Γ

c(h)[h♯]       D · · · D · · · D · · · . . . . . . ...      

Γ×Γ

, D =         1 · · · −1 1 · · · 1 −1 1 · · · −1 1 −1 1 · · · . . .         D =

1

1+e−|x| ·

SLIDE 31

CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

Pointwise structure of the Green’s function

When |η| ≪ 1, ∞

n=1[I]n uniformly converges in (iη, s, Λ) in |Λ|, |η| < δ0 and

|Re(s)| < 1/16.

∞
n=1

[I]n =

∞

k=0

Ak(iη, s; x∗) + Bk(iη, s; x∗)Λk : Ak and Bk analytic in Re(s) > −δ0 and |η| < δ0.

x x * t x * x *

x = (x, y),

x′ = (x′, y), x∗ = (x∗, 0), x∗ > 0, t > 1

(x − x∗)2 + (y − y∗)2 = Kt
(x − x∗)2 + (y − y∗)2 = t

|G ( x, t; x∗)| ≤ O(1)χ(x)      χ(t − | x − x∗|) t √ t + t − | x − x∗| + χ(t − | x + x∗|) t √ t + t − | x + x∗| + e

− (t−| x− x∗|)2 C0t

t5/4      + O(1)(1 − χ(x))      χ(t − | x′ − x∗|) t √ t + t − | x′ − x∗| + χ(t − | x′ + x∗|) t √ t + t − | x′ + x∗| + e

− (t−| x′− x∗|)2 C0t

t5/4     

x′=0

ex/C0 + O(1)e

− | x− x∗|+t C0

.

SLIDE 32

CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

The final construction

The nonlinear problem
Ut + UUx + Vy − ∆U = 0,

Vt + Vx + Uy − ∆V = 0, uin(x, y) ≡ U(x, y, 0) − ϕ(x), vin(x, y) ≡ V(x, y, 0), |uin| + |vin| ≤ O(1)εe−(|x|+|y|), ε ≪ 1, (ϕ(x) = − tanh(x/2)).

The perturbation. (u, v) ≡ (U − ϕ(x), V).
ut + (ϕ(x)u)x + vy = ∆u − 1

2 (u2)/2,

vt + vx + uy = ∆v, (u, v)|t=0 = (uin, vin). x x * t 1/t D e c a y r a t e : 1/t D e c a y r a t e :

3/4

x * x * x∗ = 0

The leading pertubation. (u1, v1):
∂t u1 + ∂x (H(x)u1) + ∂y v1 = ∆u1,

∂t v1 + ∂x v1 + ∂y u1 = ∆v1, (u1, v1)|t=0 = (uin, vin). |u1|, |v1| ≤ O(1)εχ(x) χ(t − | x|) (t + 1) + e

− x C0 χ(t − |

x|) (t + 1)3/4 + e

− (t−| x|)2 C0t

(t + 1)

+ O(1)ε

(1 − χ(x))χ(t − |y|)e

x C0

(t + 1)3/4 + e

− |x| C0 − (|y|−t)2 C0t

(t + 1)3/4 + e

− | x− x∗|+t C0

.

SLIDE 33

CONSERVATION LAWS INTERACTION DYNAMICS IN MULTI-D

The secondary pertubation. (u2, v2) ≡ (u − u1, v − v1):

   ∂t u2 + ∂x (ϕ(x)u2) + ∂y v2 = ∆u2 + ∂x

(H − ϕ)u1 + (u1+u2)2

2

,

∂t v2 + ∂x v2 + ∂y u2 = ∆v2, (u2, v2)|t=0 = (0, 0).

Anti-derivative variable: (Ux , Vx ) = (u2, v2):

   ∂t U + ϕ(x)Ux + ∂y V = ∆U +

(H − ϕ)u1 + (u1+u2)2

2

,

∂t V + ∂x V + ∂y U = ∆V , (U , V )|t=0 = (0, 0).

The integral representation of (u2, v2) by Green’s function G (x, y, t; x∗).

u2 v2

= ∂x

t

R2 G (x, y − y∗, t − τ; x∗)
(H − ϕ)u1 + (u1+u2)2

2

(x∗, y∗, τ)dx∗dy∗dτ.
Ansatz assumption |u2| ≤ O(1)|(u1, v1)|.