solitary waves in layered nonlinear mediafaculty.washington.edu/rjl/talks/maryland04.pdf ·...

Solitary Waves in Layered Nonlinear Media

Randall J. LeVequeDepartment of Applied Mathematics

University of Washington

−5 0 5 10 15 20

−0.1

−0.05

0

0.05

0.1

0.15

PSfrag replacements

One-dimensional Elasticity

Notation:

X(x, t) = location of particle indexed by x in thereference (unstrained) configuration

X(x, 0) = x if initially unstrained

ε(x, t) = Xx(x, t) − 1 = strain

u(x, t) = velocity of particle indexed by x

ρ(x) = density

Constitutive relation

stress = σ(ε, x), where ε = strain

Heterogeneous =⇒ explicit dependence on x.

Linear elasticity: (Hooke’s law)

σ(ε, x) = K(x)ε

where K is the bulk modulus of compressibility.

PSfrag replacementsε

σ

linear

nonlinear

System of two conservation laws:

εt − ux = 0

(ρu)t + σx = 0

or qt + f(q, x)x = 0 with

q =[

ερu

]

=

[

q1

q2

]

, f(q, x) =

[

q2/ρ(x)−σ(q1, x)

]

Quasi-linear hyperbolic form:

qt + fq(q, x)qx = −fx(q, x)

Jacobian: fq(q, x) =

[

0 −1/ρ(x)−σε(ε, x) 0

]

Eigenvalues (wave speeds): ±√

σε/ρ.

Wave-propagation algorithms for spatially-varying fluxes

qt + f(q, x)x = 0

Work with Derek Bale, Sorin Mitran, and James Rossmanith,

A wave-propagation method for conservation laws and balance laws withspatially varying flux functions, SISC 24 (2002), pp. 955-978

Connections to relaxation schemes (Jin and Xin):Work with Marica Pelanti,

A Class of Approximate Riemann Solvers and Their Relation to RelaxationSchemes, J. Comput. Phys., 172 (2001), pp. 572-591.

Applications:• Wave propagation in heterogeneous nonlinear media• Flow in heterogeneous porous media• Traffic flow with varying road conditions• Solving conservation laws on curved manifolds

CLAWPACK

http://www.amath.washington.edu/˜claw/

• Fortran codes with Matlab graphics routines.• Many examples and applications to run or modify.• 1d, 2d, and 3d.

User supplies:• Riemann solver, splitting data into waves and fluctuations

(Need not be in conservation form)

• Boundary condition routine to extend data to ghost cellsStandard bc1.f routine includes many standard BC’s

• Initial conditions — qinit.f

Linear case: σ = Kε

qt + Aqx = 0 with q =[

εm

]

, A =

[

0 −1/ρ−K 0

]

.

Diagonalize A: R−1AR = Λ where Λ =[ −c 0

0 c

]

, c =√

K/ρ,

R =[

1 1Z −Z

]

, R−1 = 12Z

[

Z 1Z −1

]

Z = impedance = ρc =√

Kρ.

System can be written as

R−1qt + R−1AR R−1qx = 0 or wt + Λwx = 0.

Linear case

With w = R−1q,wt + Λwx = 0.

This is a decoupled pair of advection equations for thecharacteristic variables:

w1t − cw1

x = 0 =⇒ w1(x, t) = w1(x + ct, 0)

w2t + cw2

x = 0 =⇒ w2(x, t) = w2(x − ct, 0)

q(x, t) is a linear comb. of left-going and right-going waves:

q(x, t) = Rw(x, t) = w1(x, t)[

1Z

]

+ w2(x, t)[

1−Z

]

.

Example: q(x, 0) =

[

H(x)

0

]

Then

w(x, 0) = R−1q(x, 0) =1

2

[

H(x)

H(x)

]

q(x, t) = Rw(x, t) = w1(x, t)

[

1

Z

]

+ w2(x, t)

[

1

−Z

]

=1

2H(x + ct)

[

1

Z

]

+1

2H(x − ct)

[

1

−Z

]

Example: q(x, 0) =

[

H(x)

0

]

Then

w(x, 0) = R−1q(x, 0) =1

2

[

H(x)

H(x)

]

q(x, t) = Rw(x, t) = w1(x, t)

[

1

Z

]

+ w2(x, t)

[

1

−Z

]

=1

2H(x + ct)

[

1

Z

]

+1

2H(x − ct)

[

1

−Z

]

Example: q(x, 0) =

[

H(x)

0

]

Then

w(x, 0) = R−1q(x, 0) =1

2

[

H(x)

H(x)

]

q(x, t) = Rw(x, t) = w1(x, t)

[

1

Z

]

+ w2(x, t)

[

1

−Z

]

=1

2H(x + ct)

[

1

Z

]

+1

2H(x − ct)

[

1

−Z

]

Example: q(x, 0) =

[

H(x)

0

]

Then

w(x, 0) = R−1q(x, 0) =1

2

[

H(x)

H(x)

]

q(x, t) = Rw(x, t) = w1(x, t)

[

1

Z

]

+ w2(x, t)

[

1

−Z

]

=1

2H(x + ct)

[

1

Z

]

+1

2H(x − ct)

[

1

−Z

]

Example: q(x, 0) =

[

H(x)

0

]

Then

w(x, 0) = R−1q(x, 0) =1

2

[

H(x)

H(x)

]

q(x, t) = Rw(x, t) = w1(x, t)

[

1

Z

]

+ w2(x, t)

[

1

−Z

]

=1

2H(x + ct)

[

1

Z

]

+1

2H(x − ct)

[

1

−Z

]

Waves in a homogeneous linear medium

0 5 10 15 20 25 30 35 40 45 50−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

PSfrag replacements

strain

0 5 10 15 20 25 30 35 40 45 50−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

PSfrag replacementsstrain

stress

0 5 10 15 20 25 30 35 40 45 50

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

PSfrag replacementsstrainstress

velocity

0 5 10 15 20 25 30 35 40 45 50−0.4

−0.3

−0.2

−0.1

0

0.1

PSfrag replacementsstrainstress

velocity

w1

c ≡ 1, Z ≡ 1

At t = 40 the leading edge of wave is at x = 40.

Waves in a layered elastic plate

−5 0 5 10 15 20

−0.1

−0.05

0

0.05

0.1

0.15time t = 10

−5 0 5 10 15 20

−0.1

−0.05

0

0.05

0.1

0.15time t = 15

−5 0 5 10 15 20

−0.1

−0.05

0

0.05

0.1

0.15time t = 20

Waves in a heterogeneous linear medium

0 5 10 15 20 25 30 35 40 45 50−0.1

0

0.1

0.2

0.3

0.4

0.5

PSfrag replacements

strain

0 5 10 15 20 25 30 35 40 45 50−0.1

0

0.1

0.2

0.3

0.4

0.5

PSfrag replacementsstrain

stress

0 5 10 15 20 25 30 35 40 45 50

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

PSfrag replacementsstrainstress

velocity

0 5 10 15 20 25 30 35 40 45 50−2

−1.5

−1

−0.5

0

0.5

PSfrag replacementsstrainstress

velocity

w1

c ≡ 1 but Z is different in the layers.

At t = 40 the leading edge of wave is at x ≈ 32.

Waves in a heterogeneous linear medium, t = 400

260 270 280 290 300 310 320 330−0.1

0

0.1

0.2

0.3

0.4

0.5

PSfrag replacements

strain

260 270 280 290 300 310 320 330−0.1

0

0.1

0.2

0.3

0.4

0.5

stress at t = 400

PSfrag replacementsstrainstress

260 270 280 290 300 310 320 330

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

PSfrag replacementsstrainstress

velocity

260 270 280 290 300 310 320 330−2

−1.5

−1

−0.5

0

0.5

PSfrag replacementsstrainstress

velocity

w1

Waves in layered medium

Note: Form of wave depends on impedance Z.

If Z varies with x then we expect reflection at interfaces

PSfrag replacements

c

1

Zl

α

1

−Zl

β

1

Zr

Waves in layered medium

Note: Form of wave depends on impedance Z.

If Z varies with x then we expect reflection at interfaces

5 10 15 20 25 300

2

4

6

8

10

12

14

16

18

20

Nonlinear materials

εt − ux = 0

(ρu)t + σx = 0

where

σ(ε, x) = K(x)ε + βK2(x)ε2

orσ(ε, x) = eK(x)ε − 1 ≈ K(x)ε + 1

2K2(x)ε2

Layered medium can be related to Toda lattice

The wave speed increases with ε:

c(ε, x) =

√

σε(ε, x)

ρ(x)≈√

K(x)(1 + ε)

ρ(x)

Leads to shock formation in a homogeneous medium.

Nonlinear materials

εt − ux = 0

(ρu)t + σx = 0

where

σ(ε, x) = K(x)ε + βK2(x)ε2

orσ(ε, x) = eK(x)ε − 1 ≈ K(x)ε + 1

2K2(x)ε2

Layered medium can be related to Toda lattice

The wave speed increases with ε:

c(ε, x) =

√

σε(ε, x)

ρ(x)≈√

K(x)(1 + ε)

ρ(x)

Leads to shock formation in a homogeneous medium.

Nonlinear homogeneous medium

0 20 40 60 80 100 120−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3stress at time t = 40

0 20 40 60 80 100 120−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3stress at time t = 80

20 40 60 80 100 120 140−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3stress at time t = 120

Nonlinear with constant linearized impedance

Z(x) = ρ(x)c(x) =√

ρ(x)K(x)(1 + ε)

→√

ρ(x)K(x) as ε → 0

Layered medium with ρ(x)K(x) ≡ constant:

75 80 85 90 95 100 105 110 115 120

0

0.1

0.2

0.3

0.4

strain at time t = 1000

Layered nonlinear medium with ZA 6= ZB

0 50 100 150−0.2

0

0.2

0.4

0.6

0.8stress at time t = 30

0 50 100 150−0.2

0

0.2

0.4

0.6

0.8stress at time t = 60

0 50 100 150−0.2

0

0.2

0.4

0.6

0.8stress at time t = 100

0 50 100 150−0.2

0

0.2

0.4

0.6

0.8stress at time t = 250

0 50 100 150−0.2

0

0.2

0.4

0.6

0.8stress at time t = 450

0 50 100 150−0.2

0

0.2

0.4

0.6

0.8stress at time t = 600

0 50 100 150−0.2

0

0.2

0.4

0.6

0.8stress at time t = 750

0 50 100 150−0.2

0

0.2

0.4

0.6

0.8stress at time t = 950

Layered nonlinear medium with ZA 6= ZB

0 50 100 150−0.2

0

0.2

0.4

0.6

0.8stress at time t = 30

0 50 100 150−0.2

0

0.2

0.4

0.6

0.8stress at time t = 60

0 50 100 150−0.2

0

0.2

0.4

0.6

0.8stress at time t = 100

0 50 100 150−0.2

0

0.2

0.4

0.6

0.8stress at time t = 250

0 50 100 150−0.2

0

0.2

0.4

0.6

0.8stress at time t = 450

0 50 100 150−0.2

0

0.2

0.4

0.6

0.8stress at time t = 600

0 50 100 150−0.2

0

0.2

0.4

0.6

0.8stress at time t = 750

0 50 100 150−0.2

0

0.2

0.4

0.6

0.8stress at time t = 950

Zoomed view of 3 “stegotons”

100 105 110 115 120 125 130 135 140−0.2

0

0.2

0.4

0.6

0.8

PSfrag replacements

Stress

100 105 110 115 120 125 130 135 140−0.2

0

0.2

0.4

0.6

PSfrag replacements

Strain

Propagation of a stegoton

0 5 10 15 20 25 30−0.1

0

0.1

0.2

0.3

0.4

0.5

PSfrag replacements

t = 20

215 220 225 230 235 240 245−0.1

0

0.1

0.2

0.3

0.4

0.5

PSfrag replacementst = 20

t = 270

500 505 510 515 520 525 530−0.1

0

0.1

0.2

0.3

0.4

0.5

PSfrag replacementst = 20

t = 270

t = 600

Collision of two stegotons

0 5 10 15 20 25 30−0.1

0

0.1

0.2

0.3

0.4

0.5

PSfrag replacements

t = 20

215 220 225 230 235 240 245−0.1

0

0.1

0.2

0.3

0.4

0.5

PSfrag replacementst = 20

t = 270

500 505 510 515 520 525 530−0.1

0

0.1

0.2

0.3

0.4

0.5

PSfrag replacementst = 20

t = 270

t = 600

Collision of two stegotons

Movie:

Solution at fixed point x0

660 680 700 720 740 760 780−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

sigma and u

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−1

−0.5

0

0.5

1

1.5

2

On the right we plot

1

aσ(x0, τ) and

1

au(x0, τ)

as functions ofτ =

√a(t − tm)

and tm is the time the velocity reaches its peak value −a.

Plot of speed vs. amplitude

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.75

0.8

0.85

0.9

amplitude

spee

d

v = 0.8 + 0.142a.

Soliton behavior?

• Relation of layered medium to Toda lattice.

Discrete nonlinear mass-spring system with soliton solutions.

• Nonlinear homogenization theory.

Reduction to higher order nonlinear PDE with constantcoefficients.

Lattice model for vibration of solids

Discrete particles coupled with springs:

Xk(t) = location of kth particle, with mass mk

Xk(0) = k∆x undisturbed configuration (∆x = 1)

Strain of spring between particle k and k + 1:

εk+1/2(t) =Xk+1(t) − Xk(t)

∆x− 1

= 0 if unstretched

Restoring force (= stress): σ(εk+1/2)

or σk+1/2(εk+1/2) if heterogeneous

Velocity: Uk(t) = X ′

k(t)

Lattice model for vibration of solids

With mk = ∆x ρk, the equations of motion are

ε′k+1/2(t) =Uk+1(t) − Uk(t)

∆x

ρkU ′

k(t) =σ(εk+1/2(t)) − σ(εk−1/2(t))

∆x

These can be viewed as a discretization of the continuum equations

εt = ux

ρut = σx

Linear springs: Hooke’s law, σ(ε) = Kε.

Toda Lattice

Layered Medium

PSfrag replacements

AAAAA BBBB

“particle layers:” ρA = O(1/wA), KA � 1

“spring layers:” ρB � 1, KB = O(1),

σB(ε) = exp(KBε) − 1

Take cA, cB � c̄ so that states equilibrate quickly within layers.

Comparison of layered medium with Toda lattice

200 210 220 230 240 250 260

0

0.05

0.1

0.15

0.2

0.25

0.3

t = 250

940 960 980 1000 1020 1040

0

0.05

0.1

0.15

0.2

0.25

0.3

t = 1000

Homogenization theory

for long wavelengths (relative to layer width)

εt − ux = 0

ρ(x)ut − σx = 0

Rewrite as equations for σ and u since these are continuous.

σt = K(x)εt

Obtain

1

K(x)σt − ux = 0

ρ(x)ut − σx = 0

Homogenization theoryAverage over distance long relative to variation in K, ρ,but short relative to variation in u, σ:

⟨

1

K(x)

⟩

σt − ux = 0 + O(δ2)

〈ρ(x)〉ut − σx = 0 + O(δ2)

or

σt − K̂ux = 0 + O(δ2)

ρ̄ut − σx = 0 + O(δ2)

where ρ̄ = 〈ρ〉, K̂ =⟨

K−1⟩

−1= harmonic average of K.

Wave speeds ±c̄ = ±√

K̂/ρ̄.

Linear periodic layered medium

PSfrag replacementsρAρA

σA(ε)σA(ε)

ρBρB

σB(ε)σB(ε)

σA = KAε, σB = KBε.

Averaged parameters: ρ̄ = 〈ρ〉 = wAρA + wBρB

K̂ =

⟨

1

K

⟩

−1

=

(

wA

KA

+wB

KB

)

−1

Effective wave speed: c̄ =

√

K̂/ρ̄.

Ex: (Santosa & Symes SIAP 51 (1991) p. 984)

ρA = KA = 4, ρB = KB = 1, wA = wB = 1/2.

Then c ≡ 1 everywhere but c̄ = 0.8.

Linear periodic layered medium

5 10 15 20 25 300

2

4

6

8

10

12

14

16

18

20

Also dispersive, with dispersion relation

ω(ξ) = c̄ξ + dξ3 + · · ·

Ref: Santosa & Symes SIAP 51 (1991) p. 984

Nonlinear homogenization theory

Work of Darryl Yong, based on his thesis with J. Kevorkian.

For the nonlinear stress strain relation

σ(ε, x) = K(x)ε + βK2(x)ε2

and layers of equal width δ/2, small relative to the wavelength.

ρ̄ut − σx = δ2

(

(ρA − ρB)(Z2A − Z2

B)

24(KA + KB)(ρA + ρB)2

)

σxxx + O(δ3)

σt − K̂ux = 2K̂(βσ − β2σ2)ux

+ δ2

(

K̂(KA − KB)(Z2A − Z2

B)

24(KA + KB)2(ρA + ρB)

)

uxxx + O(δ3).

Z =√

Kρ = impedance

• β = 0 =⇒ linear result of Santosa-Symes,• ZA = ZB =⇒ dispersive terms vanish.

Homogenized equations for exponential stress-strain

σ(ε, x) = eεK(x) − 1

ρA = KA = 4, ρB = KB = 1, wA = wB = 1/2.

ut =2σx

5+

3δ2σxxx

500+ δ4

(

3σxxxσ2x

15625(σ + 1)2−

72u2xxσx

15625(σ + 1)−

12σxxxxσx

15625(σ + 1)

−96uxxuxxx

15625−

12σxxσxxx

15625(σ + 1)−

357σxxxxx

1000000

)

+ O(δ6)

σt =8(σ + 1)ux

5+ δ2

(

3(σ + 1)uxxx

125+

3uxxσx

50

)

+ δ4

(

48uxu2xx

15625−

48σxσxxuxx

15625(σ + 1)−

4761σxxxuxx

500000−

72uxxxσ2x

15625(σ + 1)

−357(σ + 1)uxxxxx

250000−

3543uxxxxσx

500000−

3891uxxxσxx

500000

)

+ O(δ6).

Comparison of homogenized solution to DNS

-30 0 30

0

0.1

0.2

0.3

0.4 velocity at t = 0

-30 0 30

0

0.2

0.4

0.6

0.8

1 stress at t = 0

-30 0 30

0

0.1

0.2

0.3

0.4 velocity at t = 20

-30 0 30

0

0.2

0.4

0.6

0.8

1 stress at t = 20

-110 -100 -90 -80 -70 -60

0

0.1

0.2

0.3

0.4 velocity at t = 100

-110 -100 -90 -80 -70 -60

0

0.2

0.4

0.6

0.8

1 stress at t = 100

-450 -440 -430 -420 -410 -400

0

0.1

0.2

0.3

0.4 velocity at t = 500

-450 -440 -430 -420 -410 -400

0

0.2

0.4

0.6

0.8

1 stress at t = 500

Summary

• Elastic wave equations in one dimension• Hyperbolic in homogeneous medium• Layered medium with impedance mismatch leads to

dispersion• Dispersion plus nonlinearity leads to solitary waves• Relation to discrete solitons in Toda lattice• Homogenized equations can be derived

Many open questions, e.g.,• Are there exact “solitons” for the layered media equations for

particular choice of σ(ε, x)?• Behavior with other layer parameters, smooth periodic,

random media, ...• Multidimensional

Summary

• Elastic wave equations in one dimension• Hyperbolic in homogeneous medium• Layered medium with impedance mismatch leads to

dispersion• Dispersion plus nonlinearity leads to solitary waves• Relation to discrete solitons in Toda lattice• Homogenized equations can be derived

Many open questions, e.g.,• Are there exact “solitons” for the layered media equations for

particular choice of σ(ε, x)?• Behavior with other layer parameters, smooth periodic,

random media, ...• Multidimensional