Mathematical Models of the Evolution of Surface Waves on...

MathematicalModels of theEvolution of

Surface Waveson DeepWater

Crystal Lee

Derivation of1D NLS

NumericalSolutions of2D NLS

Solitary Waves

PerturbedSolitary Wave

Unperturbedvs PerturbedSolitary Wave

Mathematical Models of the Evolution ofSurface Waves on Deep Water

Crystal Lee

University of Washington

May 15, 2008



Crystal Lee

Derivation of1D NLS


Solitary Waves



1 Derivation of 1D NLS

2 Numerical Solutions of 2D NLS

3 Solitary Waves

4 Perturbed Solitary Wave

5 Unperturbed vs Perturbed Solitary Wave



Crystal Lee

Derivation of1D NLS


Solitary Waves



Derivation of 1D NLSGoverning Equations

z

x

Φzz+∆2Φxx = 0

z = 0

z = 1+¶Ζ

φzz + δ2φxx = 0 on 0 < z < 1 + εζ

φz = δ2(ζt + εφxζx) on z = 1 + εζ

φt + ζ+ε

2

( 1

δ2φ2

z + φ2x

)=

T ζxx

(1 + ζ2x )3/2

on z = 1 + εζ

φz = 0 on z = 0



Crystal Lee

Derivation of1D NLS


Solitary Waves



Derivation of 1D NLS

Introduce the transformation

ξ = x − cp(k)t η = ε(x − cg (k)t) τ = ε2t

k =2π

λ

Look for an asymptotic solution of the form

φ ∼∞∑

n=0

εnφn(ξ, η, τ, z) ζ ∼∞∑

n=0

εnζn(ξ, η, τ)



Crystal Lee

Derivation of1D NLS


Solitary Waves




Solution takes the form

φ0 = f0(η, τ) + F0(z , η, τ)e ikξ + F ∗0 (z , η, τ)e−ikξ

ζ0 = A0(η, τ)e ikξ + A∗0(η, τ)e−ikξ

and

φn =n+1∑

m=0

Fnm(z , η, τ)e ikmξ + F ∗nm(z , η, τ)e−ikmξ

ζn =n+1∑

m=0

Anm(η, τ)e ikmξ + A∗nm(η, τ)e−ikmξ.



Crystal Lee

Derivation of1D NLS


Solitary Waves




Finally,−2ikcpAτ + αAηη + β|A|2A = 0,

whereα = c2

g − (1− δktanh δk)sech2δk

and

β =k2

c2p

(1

2(1 + 9coth2δk − 13sech2δk − 2tanh4δk)

− (2cp + cg sech2δk)2(1− c2g )−1

)



Crystal Lee

Derivation of1D NLS


Solitary Waves



Numerical Solutions of 2D NLS

Determine the numerical solution to

iuτ + αuξξ + βuηη + γ|u|2u = 0

with ICu(ξ, η, 0) = f (ξ, η)

and periodic BCs in ξ and η on [0, L].



Crystal Lee

Derivation of1D NLS


Solitary Waves



Numerical Solutions of 2D NLSPDE

iuτ + αuξξ + βuηη + γ|u|2u = 0

u = u(ξ, η, τ) = uR(ξ, η, τ) + iuI (ξ, η, τ)

Re: − uIτ + αuRξξ+ βuRηη + γu3

R + γu2I uR = 0

Im: uRτ + αuIξξ+ βuIηη + γuIu

2R + γu3

I = 0



Crystal Lee

Derivation of1D NLS


Solitary Waves



Numerical Solutions of 2D NLSIC

u(ξ, η, 0) = f (ξ, η)

u(ξ, η, 0) = uR(ξ, η, 0) + iuI (ξ, η, 0)

Re: uR(ξ, η, 0) = Re(f (ξ, η))

Im: uI (ξ, η, 0) = Im(f (ξ, η))



Crystal Lee

Derivation of1D NLS


Solitary Waves



Numerical Solutions of 2D NLSBCs

u(0, η, τ) = u(L, η, τ)

u(ξ, 0, τ) = u(ξ, L, τ)

Re: uR(0, η, τ) = uR(L, η, τ)

uR(ξ, 0, τ) = uR(ξ, L, τ)

Im: uI (0, η, τ) = uI (L, η, τ)

uI (ξ, 0, τ) = uI (ξ, L, τ)



Crystal Lee

Derivation of1D NLS


Solitary Waves



Numerical Solutions of 2D NLSCOV

Letτ = ε2

√gkt

ξ = εk(x − g + 3k2T

2√

gk + k3Tt)

η =εky

b.

Then, the surface displacement is given by

ζ(x , y , t) =−2ε

√gk

k√

k(g + k2T )

(uR(ξ, η, τ)sin(kx −

√k(g + k2T )t)

+ uI (ξ, η, τ)cos(kx −√

k(g + k2T )t))



Crystal Lee

Derivation of1D NLS


Solitary Waves



Solitary Waves

A soliton solution of the 1D NLS equation is given by

u(ξ, τ) = ±√

2α

γsech(ξ)e iατ .

-10 -5 5 10

0.2

0.4

0.6

0.8

1sechHΞL

-10 -5 5 10Ξ

-1

-0.5

0.5

1



Crystal Lee

Derivation of1D NLS


Solitary Waves



Solitary WavesSolitary Wave IC



Crystal Lee

Derivation of1D NLS


Solitary Waves




0 50 100 150 200 250 3000

25

50

75

100

125

150

175

x cm

y cm t=5sec



Crystal Lee

Derivation of1D NLS


Solitary Waves




-3 -2 -1 1 2 3t sec

-0.2

-0.1

0.1

0.2

0.3

x=0 cm y=22.86 cmΖHx,y,tL

2 4 6 8 10t sec

-0.2

-0.1

0.1

0.2

ΖHx,y,tL x=150 cm y=22.86 cm

11 12 13 14 15 16 17t sec

-0.2

-0.1

0.1

0.2




Crystal Lee

Derivation of1D NLS


Solitary Waves




-3 -2 -1 1 2 3t sec

-0.2

-0.1

0.1

0.2

0.3


2 4 6 8 10t sec

-0.2

-0.1

0.1

0.2


11 12 13 14 15 16 17t sec

-0.2

-0.1

0.1

0.2




Crystal Lee

Derivation of1D NLS


Solitary Waves



Perturbed Solitary Wave

-10 -5 5 10Ξ

-3-2-1

123

Re@UHΞLD,Im@UHΞLD-10 -5 5 10

Ξ

-4

-2

2

4

Re@VHΞLD,Im@VHΞLD



Crystal Lee

Derivation of1D NLS


Solitary Waves




Consider the following IC.

pertsoln(ξ, η, 0, eps) =

r2α

γsech ξ

+ eps((<[U(ξ)] + i=[U(ξ)])e iρ(η−π2

) + (<[U(ξ)]− i=[U(ξ)])e−iρ(η−π2

))

+ ieps((<[V (ξ)] + i=[V (ξ)])e iρ(η−π2

) + (<[V (ξ)]− i=[V (ξ)])e−iρ(η−π2

))



Crystal Lee

Derivation of1D NLS


Solitary Waves




Re@pertsolnD

-10-5

05

10

x

-10

-5

0

5

10

y

-5

0

5

-10-5

05

10

x

Im@pertsolnD

-10-5

05

10

x

-10

-5

0

5

10

y

-50

5

-10-5

05

10

x



Crystal Lee

Derivation of1D NLS


Solitary Waves






Crystal Lee

Derivation of1D NLS


Solitary Waves




0 50 100 150 200 250 3000

25

50

75

100

125

150

175

x cm

y cm t=4.6sec



Crystal Lee

Derivation of1D NLS


Solitary Waves




-3 -2 -1 1 2 3t sec

-0.2

-0.1

0.1

0.2

0.3


2 4 6 8 10t sec

-0.2

-0.1

0.1

0.2


11 12 13 14 15 16 17t sec

-0.2

-0.1

0.1

0.2




Crystal Lee

Derivation of1D NLS


Solitary Waves



Unperturbed vs Perturbed Solitary Wave

At x = 0 cm, y = 22.86 cm,

-3 -2 -1 1 2 3t sec

-0.02

-0.01

0.01

0.02

0.03

x=0 cm y=22.86 cmΖ-Ζ0



Crystal Lee

Derivation of1D NLS


Solitary Waves




At x = 300 cm, y = 22.86 cm,

11 12 13 14 15 16 17t sec

-0.02

-0.01

0.01

0.02

Ζ-Ζ0 x=300 cm y=22.86 cm



Crystal Lee

Derivation of1D NLS


Solitary Waves




At x = 0 cm, y = 34.29 cm,

-3 -2 -1 1 2 3t sec

-0.00002

0.00002

0.00004

x=0 cm y=34.29 cmΖ-Ζ0Ζ-Ζ0



Crystal Lee

Derivation of1D NLS


Solitary Waves




At x = 300 cm, y = 34.29 cm,

11 12 13 14 15 16 17t sec

-0.02

-0.01

0.01

0.02

Ζ-Ζ0 x=300 cm y=34.29 cm



Crystal Lee

Derivation of1D NLS


Solitary Waves



Results

The soliton solution is not stable with respect to theperturbation.



Crystal Lee

Derivation of1D NLS


Solitary Waves



References

Ablowitz, Mark and Segur, Harvey. Solitons and theInverse Scattering Transform. Philadelphia: SIAMPublishing, 1981.

Byrd, P.F. and Friedman, M.D. Handbook of EllipticIntegrals for Engineers and Scientists, Second Edition.New York: Springer-Verlag, 1971

Dr. John Carter, Department of Mathematics, SeattleUniversity.

Johnson, R.S. A Modern Introduction to theMathematical Theory of Water Waves. Cambridge:Cambridge University Press, 1997.

Mathematical Models of the Evolution of Surface Waves on...

Documents

Transcript of Mathematical Models of the Evolution of Surface Waves on...