analytic solutions for long internal wave models with improved nonlinearity

ANALYTIC SOLUTIONSANALYTIC SOLUTIONS FOR LONG INTERNAL WAVE MODELS FOR LONG INTERNAL WAVE MODELS

WITH IMPROVED NONLINEARITYWITH IMPROVED NONLINEARITY

Alexey Slunyaev Alexey Slunyaev Insitute of Applied Physics RASInsitute of Applied Physics RAS

Nizhny Novgorod, RussiaNizhny Novgorod, Russia

z

x

1 U1

g

0

H

U2 2

2-layer fluid rigid-lid boundary conditionBoussinesq approximation

0

Xp

XU

UT

U jjj

jj

0

jjj UH

XTH

HHH 21

02211 HUHU

z

x

1 U1

g

0

H

U2 2 012

w

xt

012

w

xtw

HHWU 2

1 2HHWU 1

2 2

21HHH

22HHH

HgC

21

21

CwW HH

tCHT HxX

12

0

xSV

tS

)1)(1(22 22 wwS

)1)(1(2 22 wwV

Representation inRepresentation inRiemann invariantsRiemann invariants

[Baines, 1995;Lyapidevsky & Teshukov 2000;

Slunyaev et al, 2003]

z

x

1 U1

g

0

H

U2 2


0

xV

t

212121

22121

22

231

hhhhhh

hhhhV

z

x

1 U1

g

0

H

U2 2

The fully nonlinear The fully nonlinear (but dispersiveless) (but dispersiveless)

modelmodel

The full nonlinear velocityThe full nonlinear velocity

[Slunyaev et al, 2003; Grue & Ostrovsky, 2003]

z

x

1 U1

g

0

H

U2 2

0

xV

t

212121

22121

22

231

hhhhhh

hhhhV


00 cos412cos

43 V

)sin(121 h )sin(1

21

01 h

1

1

2

2

u1

u1

u2u2

clinclin

V+

V+

Velocity profilesVelocity profiles

hh = 0.1 = 0.1 hh = 0.5 = 0.5

z

x

1 U1

g

0

H

U2 2

0

xV

t

212121

22121

22

231

hhhhhh

hhhhV


asymptotic expansions for asymptotic expansions for any-order nonlinear coefficientsany-order nonlinear coefficients

0

xV

t

543

32

211 OVV lin

21

21

23

HHHH

221

212

14

83

HHHHH

321

212

2 163

HHHHH

4

21

221

2

3 12815

HHHHH

etc…etc…


0

xV

t

543

32

211 OVV lin

Exact Exact relation relation

for for HH11 = = HH22


2

2

121H

VV lin

Corresponds to the Corresponds to the Gardner eqGardner eq


Exact fully nonlinear velocity for asymp eqsExact fully nonlinear velocity for asymp eqs

Exact velocity fields (hydraulic approx)Exact velocity fields (hydraulic approx)

Strongly nonlinear wave steepening (dispersionless approx)Strongly nonlinear wave steepening (dispersionless approx)

The GE is exact when the layers have equal depthsThe GE is exact when the layers have equal depths

z

x

(z)

U (z)

g

0

H

Rigorous way for Rigorous way for obtaining asymptotic eqsobtaining asymptotic eqs

stratified fluid free surface condition

z

x

(z)

U (z)

g

0

H

Rigorous way for Rigorous way for obtaining asymptotic eqsobtaining asymptotic eqs

stratified fluid free surface condition

02

512

33323

31

214

33

2

21

Ouuuuuuu

uuuuuuuu

uuuuuu

xxxxxxxx

xxxxxxxx

xxxxxt extGE

0 Ouuuu xxxxt

02121

21

Ouuuuuuu

uuuu

xxxxxxxxxxxx

xxxxt

xxxx

x

xxxx vvvxvdxvvvvu 432

21

0

02 Ovvvv xxxxt

Asymptotical integrability Asymptotical integrability (Marchant&Smyth, Fokas&Liu 1996)

2nd order KdV

KdV

021 Ouuuuuu xxxxxt

02

512

33323

31

214

33

2

21

Ouuuuuuu

uuuuuuuu

uuuuuu

xxxxxxxx

xxxxxxxx

xxxxxt

xxxxx

x

xx

x

xxxx vvvvvxdxvvvdxvvvvvu 2

162

5433

22

1

00

0243

32

21 Ovvvvvvv xxxxt

Almost asymptotical integrabilityAlmost asymptotical integrability

GE

extGE

0243

32

21 Ovvvvvvv xxxxt

Sv

v

dVxx0

2/163524132

0 151063β

Solitary waves

2-order GE theory as perturbations of the GE solutions2-order GE theory as perturbations of the GE solutions

Qualitative closeness of the GE and its extensionsQualitative closeness of the GE and its extensions

066 2 xxxxxt uuuuuuGE

-20 0 20

0.0

0.2

0.4

0.6

0.8

1.0

-4 0 4

-6

-4

-2

0

2

4


2

1

x

x

uu

1

Initial ProblemAKNS approach


x

x

uu

1

AKNS approach

mKdV )(qQ AKNS approach

06 2 xxxxt qqqq

x

x

qq

Q


mKdV 06 2 xxxxt qqqq

066 2 xxxxxxt uuuuucuu

26ac a2

atxutxq ),(),(

GE



)(uU

x

x

auu

U2

AKNS approach

22)(

222 aaUQ u

22)(

2)( aqu

GE



26ac a2

a – is an arbitrary number

GE 066 2 xxxxxxt uuuuucuu

26ac a2

Passing through a turning point? t

Tasks:


Passing through a turning point? t

Tasks:

A solitary-like wave over a long-scale wave

22)(

2)( aqu


A solitary-like wave over a long-scale wave

22)(

2)( aqu

222)(

2)( aauu

GE+mKdV+ 06 2 xxxxt qqqq


22)(

2)( aqu

a soliton cannot pass a soliton cannot pass through a too high through a too high

wave being a solitonwave being a soliton

discrete eigenvalues discrete eigenvalues may become may become continuouscontinuous

a

GE+mKdV+ 06 2 xxxxt qqqq


soliton amplitude soliton amplitude ((ss denotes polarity) denotes polarity)

asA qsolu 22 )(

)()(

22)(

)()( 24 aC q

solu

soliton velocitysoliton velocity )(2

2)(

2)()(

cosh411

4

uu

usol

s

u

tctx uuu2

)()()( 42

Solitons

GE-mKdV- 06 2 xxxxt qqqq


22)(

2)( aqu 02

)( q

22)( au

at the turning point at the turning point all spectrum becomes all spectrum becomes

continuouscontinuous

GE-mKdV- 06 2 xxxxt qqqq


soliton amplitudesoliton amplitudeaA q

solu 22 2

)()(

)(

22)(

)()( 24 aC q

solu

soliton velocitysoliton velocity

This approach was applied to the NLS eq

peri

odic

al b

ound

ary

cond

ition

s

peri

odic

al b

ound

ary

cond

ition

s

an e

nvel

ope

solit

onplane wave

plane wave

The initial conditions: an envelope soliton and a plane wave background

Spatio-temporal evolution NLS “breather”

envelope soliton

This approach was applied to the NLS eq

Solitary wave dynamics on pedestals may be interpretedSolitary wave dynamics on pedestals may be interpreted

Strong change of waves may be predicted (“turning” points)Strong change of waves may be predicted (“turning” points)

Thank you for attention!Thank you for attention!

Gavrilyuk S.

Grimshaw R.

Pelinovsky E.

Pelinovsky D.

Polukhina O.

Talipova T.

Co-authorsCo-authors

analytic solutions for long internal wave models with improved nonlinearity

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