Self-adaptive moving mesh discrete integrable systemsstaff.norbert/JNMP-Conference-2013/maruno.pdf · 2013. 6. 5. · Self-adaptive moving mesh discrete integrable systems Kenichi

. . . . . .

.

......

Self-adaptive moving mesh discrete integrablesystems

Kenichi Maruno

Department of Mathematics, The University of Texas - Pan American

Joint work with Bao-Feng Feng (UTPA), Kenji Kajiwara (Kyushu University), Yasuhiro Ohta(Kobe University)

Conference on Nonlinear Mathematical Physics, Sophus Lie Conference Center, Norway

June 4–14, 2013

K.Maruno (UTPA) Self-adaptive moving mesh discrete integrable systems June 4–14, 2013 1 / 34

. . . . . .

Good discretizations of nonlinear wave equations

It is possible to discretize integrable PDEs, such as the KdV equation, thesine-Gordon equation, the nonlinear Schrödinger equation, keeping theintegrability. (Hirota, Ablowitz-Ladik, Suris, etc.)

There are integrable PDEs having singularities (loop, cusp, etc.) in theirsolutions, e.g., the Camassa-Holm equation, the short pulse equation, theWKI elastic beam equation, the Dym equation). However, their integrablediscretizations are unknown. (It is very difficult to construct numericaldifference schemes for these equations because of singularities in theirsolutions.)

Goal:Develop methods to discretize these PDEs keeping the integrabilityImportant Keys → Gauge transformations, reciprocal transformations(hodograph transformations), discrete differential geometry.

⇓Find a method to discretize geometric nonlinear PDEs (including

nonintegrable systems.)


. . . . . .




Goal:Develop methods to discretize these PDEs keeping the integrability

Important Keys → Gauge transformations, reciprocal transformations(hodograph transformations), discrete differential geometry.




. . . . . .








. . . . . .






nonintegrable systems.)K.Maruno (UTPA) Self-adaptive moving mesh discrete integrable systems June 4–14, 2013 2 / 34

. . . . . .

Camassa-Holm equation

wt + 2κ2wx − wtxx + 3wwx = 2wxwxx + wwxxx ,or

(∂t + w∂x)(κ2 + w − wxx) = −2wx(κ2 + w − wxx) .History:Camassa & Holm (1993) :Derivation from shallow water waveCamassa & Holm (1993), Camassa, Holm & Hyman(1994): Peakon


. . . . . .

Parametric form of exact solutions of the CH equation.Theorem..

......

Exact soliton and cusped soliton solutions of the Camassa-Holmequation

wt + 2κ2wx − wtxx+ 3wwx = 2wxwxx+ wwxxx ,

is expressed by

w =(lng(x1, x−1)

h(x1, x−1)

)x−1

through the hodograph (reciprocal) transformation

x(x1, x−1) = 2cx1 +∫ x−1

−∞w(x1, x′−1)dx

′−1

= 2cx1 + lng

h,

t(x1, x−1) = x−1 ,K.Maruno (UTPA) Self-adaptive moving mesh discrete integrable systems June 4–14, 2013 4 / 34

. . . . . .

A conservation form of the Camassa-Holm equation

The Camassa-Holm equation can be written in the form of

ρx−1 = wx1 ,1

2(ln ρ)x1x−1 =

ρ

2c−

2c

ρ+wρ

2,

with the hodograph (reciprocal) transformation

x = 2cx1 +∫ x−1

−∞w(x1, x′−1)dx

′−1

t = x−1 .

Note that the second equation is the deformed sinh-Gordon equation.


. . . . . .

Bilinear equations of the Camassa-Holm equation

.Bilinear equations..

......

The Camassa-Holm equation is decomposed into the bilinear equations(1

2Dx1Dx−1 − 1

)f · f + gh = 0 ,

(Dx1 + 2c)g · h = 2cf2 ,(1

2Dx1Dx−1 + cDx−1 − 1

)g · h+ f2 = 0 ,

through the dependent variable transformation w =(ln g

h

)x−1

and thehodograph (reciprocal) transformation

x = 2cx1 +∫ x−1

−∞w(x1, x′−1)dx

′−1

= 2cx1 + lng

h,

t = x−1 .


. . . . . .

τ -functions

.τ -functions..

......

The previous bilinear equations have a determinant solution

f = det(ψ(j−1)i )1≤i,j≤N , g = det(ψ(j)i )1≤i,j≤N ,

h = det(ψ(j−2)i )1≤i,j≤N

where

ψ(j)i = (p2i−1 − c)

jep2i−1x1+(p2i−1−c)−1x−1+θ2i−1,0

+(−p2i−1 − c)je−p2i−1x1−(p2i−1−c)−1x−1+θ2i,0 .


. . . . . .

Discretization of the Camassa-Holm equation

.Bilinear equations..

......

Consider the following τ -functions: fk = det(ψ(j−1)i (k))1≤i,j≤N , gk =

det(ψ(j)i (k))1≤i,j≤N , hk = det(ψ(j−2)i (k))1≤i,j≤N where

ψ(j)i (k) = (p2i−1 − c)

j(1 − ap2i−1)−kep2i−1x1+(p2i−1−c)−1x−1+θ2i−1,0

+(−p2i−1 − c)j(1 + ap2i−1)−ke−p2i−1x1−(p2i−1−c)−1x−1+θ2i,0 .

These τ -functions satisfy the bilinear equations(1

aDx−1 − 1

)fk+1 · fk +

1

2(gk+1hk + gkhk+1) = 0 ,(

(1 + ac)Dx−1 − a)gk+1 · hk −

((1 − ac)Dx−1 + a

)gk · hk+1

= −2afk+1fk ,1

a(gk+1hk − gkhk+1) + c(gk+1hk + gkhk+1) = 2cfk+1fk .


. . . . . .

Semi-discrete Camassa-Holm equation.Self-adaptive moving mesh semi-discrete integrable systems..

......

The system of differential-difference equations

∂x−1ρk =wk+1 − wk

a, δk = 2

(1 + ac)ea(ρk−2c) − (1 − ac)(1 + ac)ea(ρk−2c) + (1 − ac)

,

2

δk(wk+1 − wk) −

2

δk−1(wk − wk−1) =

δk

2(wk+1 + wk)

+δk

c

(1 −

4a2c2

δ2k

)+δk−1

2(wk + wk−1) +

δk−1

c

(1 −

4a2c2

δ2k−1

).

(ρk = (Xk+1 −Xk)/a) with the hodograph (reciprocal) transformation

Xk = 2cx1,k +∫ x−1

−∞wk(x′−1)dx

′−1 , t = x−1 ,

gives a discrete analogue of the Camassa-Holm equation. Xk ≡ X(k, x−1),wk ≡ w(k, x−1), and ρk ≡ ρ(k, x−1).


. . . . . .

Numerical evolution of cuspon-soliton interaction

0 10 20 30 40 50 60 70−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

X

w(X

,t)

0 10 20 30 40 50 60 70−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

X

w(X

,t)

0 10 20 30 40 50 60 70−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

X

w(X

,t)

0 10 20 30 40 50 60 70−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

X

w(X

,t)

0 10 20 30 40 50 60 70−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

X

w(X

,t)

0 10 20 30 40 50 60 70−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

X

w(X

,t)

0 10 20 30 40 50 60 70−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

X

w(X

,t)

Figure: Numerical solution for cuspon-soliton collision with p1 = 9.12,p2 = 10.98 and c = 10.0: t = 0.0; t = 12.0; t = 14.4; t = 14.6; t = 14.8;t = 17.0; t = 25.0.


. . . . . .

Short Pulse Equation

uxt = u+1

6(u3)xx

Schäfer & Wayne(2004): Derived from Maxwell equation on the settingof ultra-short optical pulse in silica optical fibers. The pulse spectrum isnot narrowly localized around the carrierVariety of short pulse solitons (Sakovich & Sakovich 2006)


. . . . . .

Short Pulse Equation

Sakovich & Sakovich (2005): A Lax pair of WKI type, relationship withsine-Gordon equation; Sakovich & Sakovich (2006): Exact solutionsMatsuno (2007): Systematic construction of multisoliton solutions through ahodograph (reciprocal) transformation.Parametric form of the soliton solution..

......

The soliton solution u(x, t):

u(x1, x−1) =∂

∂x−1

(2i ln

F ∗(x1, x−1)

F (x1, x−1)

)=∂θ(x1, x−1)

∂x−1

through the hodograph transformation

x(x1, x−1) = x1 − 2 (lnF ∗F )x−1 , t(x1, x−1) = x−1

A function θ = 2i ln F∗

Fsatisfies the sine-Gordon equation θx1x−1 = sin θ.


. . . . . .

A conservation form of the short pulse equation

The short pulse equation uxt = 4u+ 23(u3)xx can be written in the

form of

ρx−1 = −(2u2)x1 ,

ux−1x1 = 4uρ ,

with the hodograph (reciprocal) transformation

x = x1 +∫ x1

−∞ρ(x′1, x−1)dx

′1

t = x−1 .


. . . . . .

Semi-discrete short pulse equation

The system of differential-difference equations

∂x−1ρk =−2u2k+1 + 2u2k

a,

∂x−1(uk+1 − uk) = 2(Xk+1 −Xk)(uk+1 + uk) .

(ρk = (Xk+1 −Xk)/a) with the hodograph (reciprocal) transformation

Xk = x0 +k−1∑k=0

aρk

t = x−1 ,

gives a discrete analogue of the short pulse equation. Here

Xk ≡ X(k, x−1), uk ≡ u(k, x−1), and ρk ≡ ρ(k, x−1) are functionsdepending on a discrete variable k and a continuous variable x−1. A set of

(Xk, uk) gives a solution of the semi-discrete analogue of the short pulse

equation.K.Maruno (UTPA) Self-adaptive moving mesh discrete integrable systems June 4–14, 2013 14 / 34

. . . . . .

Problems

Find good discretizations of the following PDEs

WKI (Wadati-Konno-Ichikawa) elastic beam equation(Wadati-Konno-Ichikawa 1979)

vt = −(

vxx

(1 + v2x)32

)x

This PDE has N -loop soliton and N -breather solutions.This PDE is obtained as compatibility conditions ofWKI(Wadati-Konno-Ichikawa) type linear eigenvalue problemCurve shortening equation

vt =vxx

1 + v2x.

This equation is a geometric PDE which describes the mortion of a plane

curve, but nonintegrable.→ We cannot use τ -functions to discretize thieequation!


. . . . . .

Problems

Find good discretizations of the following PDEsWKI (Wadati-Konno-Ichikawa) elastic beam equation(Wadati-Konno-Ichikawa 1979)

vt = −(

vxx

(1 + v2x)32

)x


vt =vxx

1 + v2x.




. . . . . .

Problems


vt = −(

vxx

(1 + v2x)32

)x

This PDE has N -loop soliton and N -breather solutions.This PDE is obtained as compatibility conditions ofWKI(Wadati-Konno-Ichikawa) type linear eigenvalue problem

Curve shortening equationvt =

vxx

1 + v2x.




. . . . . .

Problems


vt = −(

vxx

(1 + v2x)32

)x


vt =vxx

1 + v2x.




. . . . . .

A geometric approach: motions of a plane curve

Goldstein & Petrich 1991, Nakayama-Segur-Wadati 1992, Doliwa-Santini1994A curve on γ(s): Euclidean plane R2,s: arc length parameter.

Tangent vector T = ∂γ∂s

=

[cos θsin θ

], |T| = 1.

Normal vector N =

[0 −11 0

]T =

[− sin θcos θ

].

θ = θ(s): an angle function (an angle measured from x-axiscounterclockwise)


. . . . . .


The Frenet frame

F = (T,N), T =∂γ

∂s,

The Frenet equation∂

∂sF = F

[0 −κκ 0

],

κ = ∂θ∂s

: a curvatureConsider the time evolution of a plane curve:

∂

∂tγ(s, t) = g(s, t)T(s, t) + f(s, t)N(s, t) .

A non-stretching condition gs = fκ →

∂

∂tF = F

[0 −fs − gκ

fs + gκ 0

].


. . . . . .


The compatibility condition gives

Ut − Vs − [U, V ] = 0.

U =

[0 −κκ 0

], V =

[0 −fs − gκ

fs + gκ 0

].

Then we obtainκt = (fs + gκ)s .

Thus the equation describing the motion of a plane curve is.The equation of motion of a plane curve..

......

κt = (fs + gκ)s ,

gs = fκ .

Setting f = −κs gives g = −κ2

2, then we obtain modified KdV equation

κt + 32κ2κs + κsss = 0.


. . . . . .

A geometric approach: An Eulerian description of themotion of a plane curve

Describe the motion of a plane curve in the rectangular coordinates (x, v)(an Eulerian description of the motion of a plane curve):


. . . . . .

A geometric approach: An Eulerian description of themotion of a plane curve

x and v are described by an angle function θ:

γ(s, t) =

[x(s, t)v(s, t)

]=∫ s0

[cos θ(s′, t)sin θ(s′, t)

]ds′ +

[x0v0

].

Thus consider the transformation from (s, t) to (x, t′):

(x, t′) =(∫ s

0

cos θ(s′, t) ds′ + x0, t).

Find an equation in (x, t′). (we use t for t′.)Remark 1:This transformation is often called the hodograph transformation orthe reciprocal transformation.Remark 2:cos θ is a conserved density of the mKdV equation.


. . . . . .

A geometric approach: the WKI elastic beam equation

Write down geometric quantities in terms of x,v,t:

s(x, t) =∫ √

1 + v2x dx, κ(x, t) =vxx

(1 + v2x)32

,

N =1√

1 + v2x

[−vx1

], T =

1√1 + v2x

[1vx

].

By ∂∂tγ = −κsN − 12κ

2T, we obtain −κs = γt · N = vt√1+v2x

. By usingdsdx

=√

1 + v2x, we obtain

vt = −κs√

1 + v2x= −κx .

By using the formula of κ, we obtain the WKI elastic beam equation

vt = −(

vxx

(1 + v2x)32

)x

,

(or ut = −

(ux

(1 + u2)32

)xx

, u = vx

).


. . . . . .

A geometric approach: the curve shortening equation

κt = (fs + gκ)s ,

gs = fκ ,

Setting f = κ, we obtain g =∫ s0κ2ds′. Thus

κt = κss + κsss + κs

∫ s0

κ2ds′.

Write down geometric quantities in terms of x, v, t. By∂∂tγ = κN +

(∫ s0κ2ds′

)T, we obtain

vt = κ√

1 + v2x .

By the formula of κ, we obtain the curve shortening equation

vt =vxx

1 + v2x.

This equation appears in pattern formations such as viscous fingering

(Nakayama-Iizuka-Wadati 1994). Nonintegrable.K.Maruno (UTPA) Self-adaptive moving mesh discrete integrable systems June 4–14, 2013 22 / 34

. . . . . .

A geometric approach: the motion of a discrete curve

Hisakado-Nakayama-Wadati 1995, Doliwa-Santini 1995, Nishinari 1998,Inoguchi-Kajiwara-Matsuura-Ohta 2011Tangent vector

Tl =γl+1 − γl

al,

∣∣∣∣γl+1 − γlal∣∣∣∣ = 1.

sl =l−1∑k=0

ak, ψl and κl are given by

γl+1 − γlal

=

[cosψlsinψl

], Tl+1 · Tl = cosκl ,


. . . . . .


The Frenet frame

Fl = (Tl,Nl), Tl =γl+1 − γl

al,

The discrete Frenet equation

Fl+1 = Fl

[cosκl − sinκlsinκl cosκl

],

The time evolution of a discrete plane curve:

∂

∂tγl = glTl + flNl .

A non-streching condition gl+1 cosκl − gl = fl+1 sinκl gives

∂

∂tF = F

[0 gl+1 sinκl+fl+1 cosκl−fl

al

−gl+1 sinκl+fl+1 cosκl−flal

0

].


. . . . . .


The compatibility condition givesd

dtUl + VlUl − UlVl+1 = 0,

U =[

cos κl − sin κlsin κl cos κl

], V =

[0 Al

−Al 0

],

Al = (gl+1 sinκl + fl+1 cosκl − fl)/al. Then we obtaindκl

dt=Al+1

al+1−Al

al.

The equation of motion of a discrete plane curve isdκl

dt=Al+1

al+1−Al

al,

gl+1 cosκl − gl = fl+1 sinκl .

Setting fl = −ul−1 = − tan κn2 , gl = 1, al = a, we obtain thesemi-discrete mKdV equation:

dul

dt=

1

2a(1 + u2l )(ul+1 − ul−1).


. . . . . .

A geometric approach: the discrete WKI elastic beamequation

Describe the motion of a discrete plane curve in the rectangular coordinates(Xl, vl) (An Eulerian description of the motion of a discrete plane curve):


. . . . . .


Consider the following coordinate transformation:

γl(t) =[Xl(t)

vl(t)

]=

l−1∑j=0

[� cos

`

ψj´

� sin`

ψj´

]+[X0v0

].

Write down geometric quantities in terms of Xl,vl,t:

sl =

l−1X

k=0

q

(Xk+1 − Xk)2 + (vk+1 − vk)2,

Nl =

2

4

−vl+1−vl

aXl+1−Xl

a

3

5 , Tl =

2

4

Xl+1−Xla

vl+1−vla

3

5 .

Taking inner products of ∂∂tγl = (− tan

κl−12

)Nl + Tl with Tl, Nl, we obtain

Xl+1 −Xla

dXl

dt+vl+1 − vl

a

dvl

dt= 1 ,

−vl+1 − vl

a

dXl

dt+Xl+1 −Xl

a

dvl

dt= − tan

κl−1

2.


. . . . . .


After some calculations, we obtain

tanκl−1

2=

vl+1 − 2vl + vl−1Xl+1 − 2Xl +Xl−1

.

Solving a linear system in the previous slide, we obtain.The semi-discrete WKI elastic beam equation..

......

dXl

dt=Xl+1 −Xl

a+vl+1 − vl

a

vl+1 − 2vl + vl−1Xl+1 − 2Xl +Xl−1

,

dvl

dt=vl+1 − vl

a+Xl+1 −Xl

a

vl+1 − 2vl + vl−1Xl+1 − 2Xl +Xl−1

.

This equation adjusts mesh points Xl automatically when the height of the

curve vl is changed. Near singularities this scheme generates many mesh

points. This is a self-adaptive moving mesh scheme!K.Maruno (UTPA) Self-adaptive moving mesh discrete integrable systems June 4–14, 2013 28 / 34

. . . . . .


There is another form:.The semi-discrete WKI elastic beam equation..

......

d

dtδl = −

vl+1 − vla

(Gl+1 +Gl) ,

d

dt(vl+1 − vl) =

δl

a(Gl+1 +Gl) ,

δl ≡ Xl+1 −Xl, Gl ≡vl+1 − 2vl + vl−1

δl − δl−1, δl = aρl.

δl: lattice spaces

Similar schemes were obtained for the Camassa-Holm, short pulse,

Hunter-Saxton, Dym equations. These self-adaptive moving mesh schemes

have exact soliton solutions (τ -functions).K.Maruno (UTPA) Self-adaptive moving mesh discrete integrable systems June 4–14, 2013 29 / 34

. . . . . .

A geometric approach: the discrete curve shorteningequation

dκl

dt=Al+1

al+1−Al

al,

gl+1 cosκl − gl = fl+1 sinκl ,

Al =gl+1 sinκl + fl+1 cosκl − fl

al.

Setting fl = sinκl−1, we obtain

dκl

dt=Bl+1

al+1−Bl

al,

gl+1 =sin2 κl + gl

cosκl

Bl =gl+1 sinκl + sinκl cosκl − fl

al.


. . . . . .


Consider the following coordinate transformation:

γl(t) =

[Xl(t)vl(t)

]=

l−1∑j=0

� cos (ψj)� sin (ψj)

+ [ X0v0

].

Write down geometric quantities in terms of Xl,vl,t:

Nl =

2

4

−vl+1−vl

aXl+1−Xl

a

3

5 , Tl =

2

4

Xl+1−Xla

vl+1−vla

3

5 .

Taking inner products of ∂∂tγl = (sinκl−1)Nl + glTl with Tl, Nl

Xl+1 −Xla

dXl

dt+vl+1 − vl

a

dvl

dt= gl ,

−vl+1 − vl

a

dXl

dt+Xl+1 −Xl

a

dvl

dt= sinκl−1 .


. . . . . .


After some calculations, we obtainsin κl−1 = ((vl+1 − vl)(Xl − Xl−1) − (Xl+1 − Xl)(vl − vl−1))/a

2,

cos κl−1 = ((Xl+1 − Xl)(Xl − Xl−1) + (vl+1 − vl)(vl − vl−1))/a2.

Solving the linear system in the previous slide, we obtain.The semi-discrete curve shortening equation..

......

dXl

dt=Xl+1 −Xl

agl −

vl+1 − vla

Gl ,

dvl

dt=Xl+1 −Xl

aGl +

vl+1 − vla

gl ,

gl+1 =G2l+1 + glHl+1

,

Gl ≡ ((vl+1 − vl)(Xl −Xl−1) − (Xl+1 −Xl)(vl − vl−1))/a2,Hl ≡ ((Xl+1 −Xl)(Xl −Xl−1) + (vl+1 − vl)(vl − vl−1))/a2.


. . . . . .


There is another form:.The semi-discrete curve shortening equation..

......

d

dtδl = −

vl+1 − vla

[(gl+1 +Hl+1)Gl+1 −Gl] ,

d

dt(vl+1 − vl) =

δl

a[(gl+1 +Hl+1)Gl+1 −Gl] ,

gl+1 =G2l+1 + glHl+1

,

Gl ≡ ((vl+1 − vl)(Xl −Xl−1) − (Xl+1 −Xl)(vl − vl−1))/a2,Hl ≡ ((Xl+1 −Xl)(Xl −Xl−1) + (vl+1 − vl)(vl − vl−1))/a2.

δl ≡ Xl+1 −Xl, ρl =δl

a

δl: lattice spacesK.Maruno (UTPA) Self-adaptive moving mesh discrete integrable systems June 4–14, 2013 33 / 34

. . . . . .

Summary

A geometric construction of self-adaptive moving mesh schemes

Integrable self-adaptive moving mesh schemes have geometricmeaning.

The geometric method can be applied for nonintegrable geometric PDEssuch as the curve shortening equation.

We obtained self-adaptive moving mesh discrete schemes forCamassa-Holm, Hunter-Saxton, short pulse, WKI elastic beam equation,Harry Dym equation, coupled short pulse equation, modified magmaequation, curve shortening equation.

Self-adaptive moving mesh schemes can be used for non-integrablegeometric PDEs.

For some equations (short pulse, WKI elastic beam, Hunter-Saxton,Harry Dym, modified magma), we also obtained fully discreteself-adaptive moving mesh schemes. But the fully discretization of theCamassa-Holm equation is an open problem.


Self-adaptive moving mesh discrete integrable systemsstaff.norbert/JNMP-Conference-2013/maruno.pdf · 2013. 6. 5. · Self-adaptive moving mesh discrete integrable systems Kenichi

Documents