dispersionlessveselov-novikovhierarchy in ...misgam.sissa.it/rhpia05/talks/moro.pdfparaxial laser...

Dispersionless Veselov-Novikov hierarchyin nonlinear geometrical optics and its hydrodynamic-type reductions

Antonio Moro

(joint work with Boris Konopelchenko )

Dipartimento di Fisica dell’ Università di Lecce

and Istituto Nazionale di Fisica Nucleare - Sezione di Lecce

Conference on Riemann-Hilbert problems, Integrability and asymptotics

Sissa - Trieste, September 20-25, 2005

2

Introduction

• Celebrated Nonlinear Schrödinger equation (NLSE) is an useful model to describe paraxial

laser beams in nonlinear (Kerr) media.

• 3(or 2+1)D-NLSE is not integrable (unlike 1+1D-NLSE solitons).

• Recently, it was shown both theoretically (Snyder&Mitchell, Science, 276, 1538, 1997) and

experimentally (Conti, Peccianti and Assanto, PRL, 92, 1139021, 2004) that high

nonlocal effects play a crucial role in the propagation of stable laser beams (spatial solitons).

In that limit NLS equation is reduced to the “quantum harmonic oscillator” equation.

In this sense the high nonlocality regime is integrable.

• In the present talk we would like to investigate possible existence of new integrable regimes

considering both nonlinear and nonlocal effects.



3

The Model

Paraxial Laser beam

z

x

y•• NonlinearityNonlinearity

• Spatial Nonlocality

( ) 02

22 =⋅∇∇−

∂

∂−∇ E

t

DE

rr

rNonlinear

Maxwell equations

Wave equation

weak



4

( ) ( ) 1 ~ 2

0 <<+=+= αεαεεαε EFExD NLoc

rrrrr

+

∂∂

∂+

∂

∂== ∑∑∫

==

...3

1,

23

1 mn mn

nm

n n

nNLocxx

Ec

x

EcdVEF

rrrr

αρ

Displacement vector

Nonlocal contribution

Stationary wave equation in paraxial approximation gives

( ) 0~2 22

0 =++∇+∂

∂⊥ NLocFEEE

z

Ei

rrrrr

εωε

Nonlocal NLS type equation (NNLSE)

nonlinearity

Max. order of

derivative = nonlocality

degree

2

~~ Er

ε 0≡NLocFr

⇒ (3D)-NLS equation(Kerr)

Particular case

Linear superposition of Dirac delta-function derivatives

5

Slow dependence on the variable (asymptotic expansion w.r.t. )

High frequency limit for a Cole-Cole nonlocal medium

( )νω 2−= OFNLoc

r

zz νω 2→

( )

+++=

νωτ

εεαεε

2

0

0

2

01

~~

i 2

10 <<ν

nonlinearity

Phenomenological

parameter

Let us consider a medium satisfying the Cole-Cole dispersion law (Cole&Cole, J.Chem.Phys, 9, 341, 1941)(Liquid crystals, several liquid and solid polar media, biological tissues).

weak nonlocal perturbations

SieE ωϕrr

= geometric optics limit

νω 2−



6

( )zyxuSS yx ,,422 =+

Phase equations

( ) ( )zyxuuzyx ,, ,, =Φ=Φ

( ) ( ) yx SzyxSzyx ,,,, 21 αα +=Φ ( ) ( )yxz

xyyx

uuu 21

2121

αα

αααα

+=

−==

( ) ( )

yxyxyx

yxz

uVVuVV

uVuVu

3 3 2121

21

=+−=−

+=yxyx SVSVSS 21

33

4

3

3

1++−=Φ

zero deg. polynomial – Local case

1st deg. polynomial – 1st deg. NonLocality

3rd degree polynomial – 3rd degree NonLocality

Function is a differential polynomial. Refractive index must satisfy so-called dispersionless Veselov-Novikov

hierarchy (dVN).

u

−− νω 2 ( )yxz SSzyxS ,,,,Φ=

leading order)

order)

Φ

…and so on

dVN equation

Even degree polynomials are not compatible with phase inversion symmetry of the eikonal equation.

2D EIKONAL EQT2D EIKONAL EQT



7

NonlocalityNonlocality

degreesdegreesEquationsEquations of of

dVNdVN hierarchyhierarchy

Correspondence

is realised!!!



8

( ) ( )( )tSWtS ;,,,;, λλλλλλ λλ=

( ) ( )nn tt SS

λµ

λ ∂

∂=

∂

∂Beltrami’s equation (BE)

( ) ( ) CffC NN ∈∀=⇒=∈∃ λψψψψψψλ 0,...,,0,...,,: 2121

Nψψψψ ,.....,, 321 a set of solutions of BE , an arbitrary combination f

Vekua’Vekua’s s theoremtheorem

Quasiclassical Dbar-dressing method

QC Dbar-problem

Let us be

),,( ξλλξ

µ∂

∂=

W

where

bounded on the complex plane and satisfies suitable mild conditions .

It provides us with a procedure to construct several dispersionless integrable hierarchies.



Konopelchenko, Martinez Alonso, Ragnisco, Medina 2000-2003

µ ( )1<≤ Kµ

9

Looking for suitable combinations of infinitesimal symmetries

bounded on the complex plane, one can construct an infinite set of equations of the form

( ) 0 ;, =Ω λntnm St

hierarchy of dispersionless integrable equations.

whose compatibility condition is equivalent to a

( ) ,ntnm StΩ

,...2,1=m



• BE describes quasi-conformal mappings of the plane.

Integrable deformations of quasiconformal mappings and dispersionless hierarchies.

(several papers by Konopelchenko, Martinez Alonso, Kodama, 2000-2004).

• Present study highlights a connection among quasiconformal mappings of the plane and laser beam profile

evolution under a weakly nonlocal response of a nonlinear medium.

10

Example: dKP

∑∞

=

=1

0

k

k

k tS λ 0≡W ∞

SSS~

0 += holomorphic part

0≠W

0S

( )122

1113

~ Res2

4

3

2

3 ttttttt Suuuuu

∞=−=+=

λ

singular part

21113

12

4

3

2

33

2

ttttt

tt

uUUSuSS

uSS

=++=

+=



λ

11

∑ ∑∞

=

∞

=

+−− +=1 1

1212

0

n n

n

n

n

n ttS λλ

0≡W

∞( )

=

−=−−

λλλλ

λλλλ

1,

1,

),(),(

SS

SS

dVN hierarchy

0S0≠W



Discrete symmetries on the complex plane

λ

12

SetSet zttiyxt ==+= 221

Simplest “times”Simplest “times”--equations areequations are dVNdVN equationequation

1

~ Res1),,( 11 tSzttu

∞=+=

λ

Integrability of dVN

The solution can be expressed in terms of the function The solution can be expressed in terms of the function

( )zyxuSS yx ,,422 =+

yxyxz SVSVSSS 21

33

4

3

3

1++−=

( ) ( )

yxy

xyx

yxz

uVV

uVV

uVuVu

3

3

21

21

21

=+

−=−

+=



S

B. Konopelchenko, L. Martinez Alonso Stud. Appl. Math. (2002)

B. Konopelchenko, A. M. , Stud. Appl. Math. (2004)

13

Intensity equations

*

*

S

SIP∇

∇=r

...1

2

0 ++= − III νω

0=⋅∇ Pr

0

0

00

2

11

2

00

=∂

∂+∇+∇⋅∇

=∇+∇⋅∇

⊥⊥⊥

⊥⊥⊥

z

ISIIS

SIIS

ε

Poynting vector

Due to paraxial approximation, function S

describes deviations from the plane

wavefront. The `total’ phase is

( )zyxSzS νωε 2

0

* ,, −+=

Expansion

Conservation law const* =S

Pr

optical

wavefront

Light ray

direction.

gives



14

0

0

00

2

11

2

00

=∂

∂+∇+∇⋅∇

=∇+∇⋅∇

⊥⊥⊥

⊥⊥⊥

z

ISIIS

SIIS

ε

( )

( )yxz

yx

SSzyxS

zyxuSS

,,,,

,,422

Φ=

=+

Response of several media is

described by “Intensity Law” ( )uII =

Intensity Law and compatibility

( )uII 00 =

Transverse

equations



15

Transverse condition is interesting in itself since it describesTransverse condition is interesting in itself since it describes the phase structurethe phase structure

of a paraxial beam for nonlinear Schrof a paraxial beam for nonlinear Schröödingerdinger--type equation.type equation.

02 =++ xyyyxx CSBSAS

2' 2 += xSA χ 2' 2 += ySB χ yxSSC 'χ=

( )1'242 +=−=∆ uCAB χ

( )[ ]uILog 0=χ

0

0

0

<∆

=∆

>∆ elliptic

parabolic

hyperbolic

Several cases of physical

Interest (Kerr-type and

logarithmic saturable media)

satisfy it uniformly.

Tansverse condition

Compatibility of transverse equations is equivalent to



16

Minimal surfaces

Proposition

0=+ yyxx SS

02 =++ xyyyxx CSBSAS ( ) ( ) 02 1 1 22 =++++ xyyxyyxxxy SSSSSSS

Let us be

⇔(minimal surfaces equation)

for any intensity law .( )uII 00 =

Harmonic minimal surfaces satisfy transverse condition!Harmonic minimal surfaces satisfy transverse condition!

(transverse condition)



The only harmonic nonThe only harmonic non--parametric minimal surface is the parametric minimal surface is the helicoidhelicoid..

17

cost

arctan* =

++≡

x

yzKzS

β

a) one-start right screw; b) two-start right screw; c) two-start left screw.

πK2

“pitch”

The helicoid (Optical vortex)

Mixed edge-screw dislocation

Pure screw dislocation0=β



18Conference on Riemann-Hilbert problems, Integrability and asymptotics


OPTICAL VORTEX SOLITONS OBSERVED IN KERR OPTICAL VORTEX SOLITONS OBSERVED IN KERR

NONLINEAR MEDIANONLINEAR MEDIA

G.A. Swartzlander, Jr. and C.T. Law, Phys. Rev. Lett. (1992)

19

( )( ) ( )22

21

211

22

yxzxy

yxyx x

+Φ=+−

+=+

αα

ααα

constarctan1 2

* =

++≡

− x

yKzS

γω ν

1st degree nonlocal perturbations of the helicoid

0≡Φ Local caseLocal case

pitch compression

pitch stretching

0>γ

0<γ

( ) ( )( ) ( )ycxz

yxcz

02

01

+Φ=

−Φ=

α

α

Nolocal Data

0const 0 ==≡Φ cγ

Wavefronts

33rdrd degree degree nonlocalnonlocal perturbations of the perturbations of the helicoidhelicoid

It can be verified directly that the helicoid is not preserved for the present class of 3rd degree

nonlocal perturbations: dVN equation breaks the optical vortices!



20

Beltrami equation

Notations: 21 ippw −=

iyx +=λ

1 xSp = 2 ySp =

( )21, ppw −=

Vector field = it describes light-rays

distribution on the plane−w

0 =+++ λλλλ wbwawbwaTansverse condition

becomes

0=

Reciprocal coordinates( )( )λλ

λλ

,

,

ww

ww

=

= ( )( )ww

ww

,

,

λλ

λλ

=

=( ) ww ww λνλ ,= 1 <=

b

aν

Ellipticity condition

Linear!!!Linear!!!

)I(



21

Theorem.

There exists a homeomorphic mapping (BE solution)

of onto exterior of

mapping onto .

R ε

RD εD

Γ'Γ

iyx +=λ w

Let us consider, for instance, a solution having a simple pole at 0=w

εD

'Γ ΓRD

Physical consequence:Physical consequence:

Inverse mapping could be used to describe a laser beam selfguided around --axis.

Example: physical consequences of the Beltrami equation properties.

z



Using Vekua’s theorem one can show that nontrivial solutions must be singular

“somewhere” on the complex plane.

22

Solutions of Beltrami equation

Kerr-type intensity lawγuI =0

www

w

w

wλ

γ

γλ

1

2

2

11

22

−

−

+−−=

−

−= w

i

w

w

iF Log

2

1arctan 21

2

γγγ

λ

( ) ( )[ ]( ) ( )[ ]yxGyxp

yxGyxp

,Exp,sin

,Exp,cos

2

1

θ

θ

−=

=

we recall that

iyx

ippw

+=

−=

λ21

This relation can be inverted



2

22

2

1

145

12

γ

γγγ

γγ

++=

+=

23

( ) ( )( ) ( ) xxy LGG

yHx

θθθ

θθ

=+

=Φ++

tan

0

where and satisfy1G 2G



( ) ( ) ( ) ( )[ ]( ) ( )

( ) ( ) ( ) 1tan

122cos2

122cos2cot

11

2

2

11

2

2

−=

++−+

−−+−=

θθθ

γγγθ

γγγθθθ

HL

H

π

1=γ ( ) βθα +=Φ HExample

( )yx,θ

• Study of wave-fronts structure at these singularities and their nonlocal deformations is in progress!

• This study is interesting since Beltrami equation could lead to new kind of phase dislocations.

0 π

vortex-type singularity

24

Exponential intensity law

Saturable nonlinear media

[ ] [ ]00 log1

exp IuuIγ

γ =⇔=

( ) wwwwww

wλ

γ

γλ

82

222

2

++−=

General integral is given explicitly in terms of Kummer’s functions (solutions of degenerate

hypergeometric equation) .



25

0 2

00 =∇+∇⋅∇ ⊥⊥⊥ SIIS

yxyxz

yx

SVSVSSS

zyxuSS

21

33

22

4

3

3

1

),,(4

++−=

=+

Third degree nonlocal perturbations and hydrodynamic-typereductions

( )uII 00 =Intensity law appears to be a quite restrictive condition

making compatibility for higher degrees a rather

non-trivial problem.



In the following we show that suitable intensity laws there exist, such that the system is

compatible. Our approach consists in a reduction method based on the so-called symmetry constraints.

(Bogdanov, Konopelchenko 2003)



Symmetry constraints

In terms of the complex variable dVN equation looks likeiyx +=λ

( ) ( ) 3 λλ

λλ

uV

uVVuu z

−=

+=

21 iVVV +=

where

Infinitesimal symmetries are provided us by

( ) ( )( ) ( )λλλλ

λλ

δδ

δδδδδ

uVuV

uVuVVuuVu z

-3 3 =−=

+++=

uδ

TheoremTheorem

λλλλ

λλ

VSVSSSS

zyxuSS

z +++=

=

33

),,(( )1 ( )2

Suppose and are solutions of the equations (1). Then, the quantityiS iS

~

( )∑=

−=N

i

iii SScu1

~λλ

δ

where are arbitrary constants, is a symmetry of dVN equation.ic

L. Bogdanov, B. Konopelchenko and A.M. Fund. Prikl. Mat. (2004)

( )3



In particular, one can choose and( )ii SS λλ == ( )iii SS µλλ +==~

For and one gets the classe of symmetries0→iµiii cc µ/~=

( ) ( )ii

N

i

ii

Scu λλ

λφφδ

λλ=

∂

∂== ∑

=

~

1

Observing that (for instance) is an infinitesimal symmetry of dVN, an typical symmetry constraint isxu

uu x δ=

Symmetry constraints are compatible with the systems above and lead to hydrodynamic-type reductions

of dVN equation.

λλSu x =

( ) λλSSu x +=

λλφ=xu

S real( )I

( )II

( )III

Symmetry constraint

We analysed explicitly the following cases

28

Sux

2

⊥∇=

Case ( )I

In Cartesian coordinates

( )( ) ( ) x

H

p

p

ppppp

ppp

p

p

x

H

p

p

ppp

p

xz

xy

∂

∂=

+−−

−=

∂

∂=

−=

2

2

1

2

21112

211

2

1

1

2

1

212

1

613123

313

212

10

yx SpSp == 21

Setting



29

( )122 ppp =



Example

Look for solutions such that

( )

12

1

1

2

1

1

2

22

1

2

2

2

2

−+±==

+

++−+=

pppdp

dp

LogCp

ξ

ξ

ξξξ

( ) ( ) ( ) 0,, 1211211 =Φ+−− pzppHyppGx

Combining symmetry constraint and intensity equation one gets the following nice relation

20

1p

CeI−

=

Intensity law associated with the present symmetry constraint is

( )

−+

=

C

Ip

C

IIu 02

2

00 Log2

4

1Log

2

30

• dVN hierarchy can be used to describe the laser beams propagation in a nonlinear and nonlocal medium.

• dVN hierarchy and then an entire class of nonlocal responses are amenable by the QC Dbar-dressing method.

• Transverse condition can be solved explicitly singular phases - dislocations.

• Using the properties of Beltrami equation we can state about the existence of solutions describing

beams selfguided around one direction.

• Even if intensity law appears to be a quite restrictive condition for higher nonlocality degrees, symmetry

constraints allow us to construct hydrodynamic-type reductions of dVN and a class of compatible intensity laws.

Conclusions





A disadvantage:

High frequency limit allows us to detect singular phase solutions and study their structure,

but just at singularities wave corrections become relevant and geometric optics approximation fails.

An advantage:

We can study wavefronts structure for a very general class of nonlinear and nonlocal responses.

This analysis can lead to new non-trivial ansatz’s in the study of wave equation

with variational methods.

32

• A.M., “On the phase dislocations in nonlinear media”, in progress.

• B. Konopelchenko and A.M., “Paraxial light in a Cole-Cole nonlocal medium: integrable regimes and

singularities”, to appear on SPIE Proc. International Congress on Optics and Optoelectronics, Warsaw 2005.

• B. Konopelchenko and A.M. “Light propagation in a Cole-Cole nonlinear medium via Burgers-Hopf equation”,

Theor. Math. Phys. 144 (1), p. 968-974, 2005.

• Boris Konopelchenko and A.M. , “Integrable equations in nonlinear geometrical optics”,

Stud. Appl. Math., 113, 325-352, (2004).

• L. Bogdanov, B. Konopelchenko and A.M., “Symmetry constraints for real dispersionless Veselov-Novikov equation”,

Fund. Prikl. Mat., 10, 5-15, (2004).

• Boris Konopelchenko and A.M. “Geometrical optics in nonlinear media and integrable equations”,

J. Phys. A: Math. Gen, 37, L105-L111, (2004).

Bibliography

Acknowledgments:Acknowledgments:

Supported in part by COFIN Supported in part by COFIN -- PRIN (Project of National Interest)PRIN (Project of National Interest)

SintesiSintesi (Singularities, (Singularities, IntegrabilityIntegrability and Symmetries) 2004.and Symmetries) 2004.



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