ermakov–ray–reid reductions of variational approximations in nonlinear optics

Ermakov–Ray–Reid Reductions of VariationalApproximations in Nonlinear Optics

By Colin Rogers, Boris Malomed, and Hongli An

Integrable Hamiltonian systems of Ermakov-Ray-Reid type are shown to ariseout of variational approximation to certain modulated NLS models as well asin spiralling elliptic soliton systems and their generalization in a Bose-Einsteinsetting.

1. Introduction

The variational approximation (VA) method introduced by Anderson andBonnedal [1] has been extensively employed in nonlinear optics (see, e.g., thereview in [2]) . In particular, it has been applied to modulated versions ofthe nonlinear Schrodinger equations in the context of soliton management inperiodic nonlinear optics systems [3] . In recent work by Desyatnik et al. [4], thecontribution of orbital angular momentum to the suppression of the collapse ofspiralling elliptic solitons in Kerr media has been analysed by means of avariational approach. An analogous system modeled by a 2+1-dimensionalNLS equation incorporating an harmonic trap was subsequently derived byAbdullaev et al. [5] via a variational approximation in the context of ellipticcloud evolution in a Bose–Einstein condensate. The remarkable occurrence ofErmakov–Ray–Reid systems in a nonlinear optics setting through the paraxialapproximation is well documented [6–12]. In particular, such systems model

Address for correspondence: Colin Rogers, Australian Research Council Centre of Excellence forMathematics, and Statistics of Complex Systems, School of Mathematics, The University of New SouthWales, Sydney, NSW 2052, Australia; [email protected]

DOI: 10.1111/j.1467-9590.2012.00557.x 1STUDIES IN APPLIED MATHEMATICS 0:1–25C© 2012 by Massachusetts Institute of Technology

2 C. Rogers et al.

the evolution of the size and shape of the light spot and wave front in anelliptical Gaussian beam [7, 8].

Nonlinear coupled systems of Ermakov–Ray–Reid type have their origin inclassical work of Ermakov [13] and were originally introduced by Ray andReid in [14, 15]. These systems adopt the form

α + ω2(t)α = 1

α2βF(β/α),

β + ω2(t)β = 1

αβ2G(α/β)

(1)

and admit a distinctive integral of motion, namely, the Ray–Reid invariant

I = 1

2(αβ − βα)2 +

∫ β/α

F(z) dz +∫ α/β

G(w) dw (2)

and concomitant nonlinear superposition principles. Subsequently, 2+1-dimensional Ermakov–Ray–Reid systems were constructed by Rogers et al.in [16]. Extensions to arbitrary order and dimension which preserveRay–Reid invariants were presented in [17]. In [18], it was shown that theErmakov–Ray–Reid system admits interesting underlying linear structure.

In terms of applications, Ermakov–Ray–Reid systems have arisen mostnotably in nonlinear optics in the description of elliptic Gaussian beampropagation. In hydrodynamics, multicomponent Ermakov–Ray–Reid systemswere introduced in [19] where application was made to an N-layer fluidmodel. Novel algebraic structure underlying these multicomponent systemswas uncovered in [20].

In recent work in [21], a Hamiltonian Ermakov–Ray–Reid system has beenobtained in the context of 2+1-dimensional rotating shallow water system withunderlying a rigid circular paraboloidal bottom topography. Ermakov–Ray–Reidsystems have also been uncovered in magneto-gasdynamics [22, 23] andshown to admit pulsrodon-type solutions analogous to those originally derivedin an elliptic warm core eddy theory (Rogers [24], Holm [25], Rubinoand Brandt [26]). The magneto-gasdynamics pulsrodon describes a rotatingpulsating elliptic cylindrical plasma column bounded by a vacuum state [23].Ermakov–Ray–Reid structure has also recently been revealed in [27] in anisothermal cloud system originally investigated by Ovsiannikov [28] andDyson [29] and subsequently analysed by Gaffet (see [30] and work citedtherein). Connection with integrability has been made in the 2+1-dimensionalcase by construction of a Lax pair in [27]. Lyapunov stability aspects ofpulsrodons and their duals have been discussed by Holm [25]. Stability andperiodic properties of a Ermakov–Ray–Reid system arising out of a two-layerhydrodynamic model have been investigated by Athorne [31].

Ermakov–Ray–Reid Reductions of Variational Approximations in Nonlinear Optics 3

Here, the remarkable occurrence of integrable Hamiltonian Ermakov–Ray–Reid structure in nonlinear optics is shown to extend to variational approximationsof certain modulated NLS models as well as to the spiralling elliptic solitonsystem of [4] and its generalization in the Bose–Einstein setting of [5]. A classof modulated 3+1-dimensional NLS equations is considered initially whichincorporates both de Broglie–Bohm and Bialynicki-Birula quantum potentialterms as well as a harmonic trap. It is noted that a de Broglie–Bohm-typepotential arises in nonlinear optics in the pioneering work of Wagner et al. [6],while a NLS equation involving a logarithmic potential was proposed by Snyderand Mitchell [32] in connection with the propagation of Gaussian beams ina saturable medium. The logarithmic Schrodinger equations was originallyintroduced by Bialynicki-Birula and Mycielski [33, 34] in a nonlinear wavemechanics connection and is noteworthy for its admittance of gausson-typesolutions [35]. The dynamics of partially coherent beams in logarithmicnonlinear media has subsequently been investigated by Krolikowski et al.[36]. Soliton interaction in logarithmically nonlinear saturable media has beenanalysed by Christodoulidis et al. [37] and; by Hansson et al. [38]. IntegrableErmakov–Ray–Reid reductions of 2+1-dimensional hydrodynamic Madelungsystems incorporating logarithmic and de Broglie–Bohm quantum potentialshave recently been obtained in [39].

In the present work, a variational approximation procedure is applied toobtain a reduction of the class of modulated 3+1-dimensional NLS equations toa coupled nonlinear system for beam widths. Integrable Hamiltonian reductionsto Ermakov–Ray–Reid systems are obtained. A nonlinear coupled systemobtained via variational approximation applied to a 2+1-dimensional model ofoscillating self-trapped beams in [4] is likewise shown to be reducible to anintegrable Ermakov–Ray–Reid system, as is the generalization obtained in aBose–Einstein condensate context in [5].

2. A 3+1-dimensional modulated NLS equation: The variationalapproximation

Here, we consider the class of modulated 3+1-dimensional nonlinear Schrodingerequations

i∂u

∂t+[

1

2

(∂2

∂x2+ ∂2

∂y2+ ζ (t)

∂2

∂z2

)− s

∇2|u||u| + δ(t) ln |u|

+ε(t)|u|2n − 1

2ω2(t)( x2 + y2 + z2)

]u = 0. (3)

wherein, in the present nonlinear optics context [2], t plays the role ofthe propagation distance, while z is the temporal variable (local time).

4 C. Rogers et al.

Accordingly, the term ∂2u/∂z2 represents the group-velocity dispersion, whilethe term (∂2/∂x2 + ∂2/∂y2) u ≡ ∇2

T u accounts for the paraxial diffraction in thetransverse plane. The NLS equation (3) incorporates a de-Broglie–Bohm-typequantum potential term, ∇2|u|/|u|, a Bialynicki-Birula logarithmic term and apower nonlinearity term |u|2n together with an harmonic trap. Further, (3) alsoincludes effects of the dispersion management through the coefficient ζ (t),nonlinear management, accounted for by δ(t) and ε(t), together with modulationof the transverse structure of the optical waveguide via the term involvingω2(t).

The object of the analysis is to apply the VA, of a kind originally introducedby Anderson and Bonnedal ([1], [2]) to isolate integrable Ermakov–Ray–Reidreductions of (3).

It is readily seen that the class of NLS equations (3) can be derived via theEuler–Lagrange equation

δL = ∂L∂u∗ − d

dt

∂L∂u∗

t− d

dx

∂L∂u∗

x

− d

dy

∂L∂u∗

y

− d

dz

∂L∂u∗

z

= 0, (4)

where

L =∫ ∫ ∫ ∫

L(u, u∗, ut , u∗t , ux , u∗

x , uy, u∗y, uz, u∗

z )dxdydzdt, (5)

and L is the Lagrangian density given by

L = i

2(u∗ut − uu∗

t ) − 1

2|ux |2 − 1

2|uy|2 − 1

2ζ (t)|uz|2 − 1

2δ(t) |u|2

+ δ(t) |u|2 ln |u| + ε(t)

n + 1|u|2n+2 + s

2(|ux |2 + |uy|2 + |uz|2)

+ s

4

u∗

u

(u2

x + u2y + u2

z

)+ s

4

u

u∗(u∗

x2 + u∗

y2 + u∗

z2)

− 1

2ω2(t)(x2 + y2 + z2)|u|2,

(n �= −1).(6)

The VA is based on the Gaussian wave ansatz

u(x, y, z, t) = A(t) exp

{iφ(t) − 1

2

[x2

W 2(t)+ y2

V 2(t)+ z2

T 2(t)

]

+ i

2[b(t)x2 + c(t)y2 + d(t)z2]

}, (7)


and substitution of the latter into (5) reduces the Lagrangian density to

L = 1

2

{i(A∗ A − AA∗) − 2|A|2φ − |A|2[ x2(b + b2) + y2(c + c2)

+ z2(d + ζd2) ] − δ |A|2 − ω2|A|2(x2 + y2 + z2) − |A|2(1 − 2s)

×(

x2

W 4+ y2

V 4+ ζ z2

T 4

)− δ|A|2

(x2

W 2+ y2

V 2+ z2

T 2− 2 ln |A|

)}

× exp

(− x2

W 2− y2

V 2− z2

T 2

)+ ε

n + 1|A|2n+2

× exp

[− (n + 1)

(x2

W 2+ y2

V 2+ z2

T 2

)]. (8)

The corresponding reduced Lagrangian adopts the form

〈L〉=∫ +∞

−∞

∫ +∞

−∞

∫ +∞

−∞L dxdydz = π

32 W V T

{i

2(A∗ A − AA∗)

− |A|2φ + δ |A|2 ln |A| − 5

4δ |A|2 − 1

4|A|2[ (b + b2 + ω2)W 2

+ (c + c2 + ω2)V 2 + (d + ζd2 + ω2)T 2 ] + s

2|A|2

(1

W 2+ 1

V 2+ 1

T 2

)

− 1

4|A|2

(1

W 2+ 1

V 2+ ζ

T 2

)+ ε

(n + 1)5/2|A|2n+2

}. (9)

The Euler–Lagrange equations

∂〈L〉∂p

− d

dt

∂〈L〉∂ p

= 0, p = {A,W, V, T, b, c, β} (10)

demonstrate the existence of a dynamical invariant

|A|2W V T = const ≡ N . (11)

Further, the chirps in the ansatz (7) are expressed through derivatives of therespective widths via the relations

b(t) = W/W, c(t) = V /V, d(t) = ζ−1(T /T ), (12)

while the following nonlinear coupled system for the evolution of the widths{W, V, T } is obtained:

6 C. Rogers et al.

lW + ω2(t)W= 1 − 2s

W 3− δ(t)

W− 2nε(t)|A|2n

W (n + 1)52

= 1 − 2s

W 3− δ(t)

W− 2n

(n + 1)52

ε(t)N n

W n+1V nT n,

V + ω2(t)V = 1 − 2s

V 3− δ(t)

V− 2nε(t)|A|2n

V (n + 1)52

= 1 − 2s

V 3− δ(t)

V− 2n

(n + 1)52

ε(t)N n

W nV n+1T n,

d

dt

(T

ζ (t)

)+ ω2(t)T = ζ (t) − 2s

T 3− δ(t)

T− 2nε(t)|A|2n

T (n + 1)52

= ζ (t) − 2s

T 3− δ(t)

T− 2n

(n + 1)52

ε(t)N n

W nV nT n+1.

(· ≡ d/dt)

(13)

In general, the above system is analytically intractable. However, for certainclasses of modulations {δ(t), ε(t), ζ (t)} it can be reduced to considerationof an integrable Ermakov–Ray–Reid system. Such systems have extensiveapplication in nonlinear optics [6–11] and, in particular, have recently beenshown to arise in the variational treatment of 2+1-dimensional nonlinear opticalpulse propagation [12].

3. Ermakov–Ray–Reid reductions

In I—III below three distinct reductions of the nonlinear coupled system (13)to one of integrable Ermakov–Ray–Reid type are exhibited.

I

If

δ(t) = 1

W V�

(W

V

), ε(t) = 1

W V |A|2n�

(W

V

), (14)

then (13) reduces to a Ermakov–Ray–Reid-type system

W + ω2(t)W = 1

W 2V

[(1 − 2s)

V

W+

(W

V

)],

V + ω2(t)V = 1

W V 2

[(1 − 2s)

W

V+

(W

V

)] (15)


augmented by the driven equation

d

dt

(T

ζ (t)

)+ ω2(t)T = ζ (t) − 2s

T 3+ 1

W V T

(W

V

), (16)

where

(W

V

)= −�

(W

V

)− 2n

(n + 1)5/2�

(W

V

). (17)

If ω = 0 and in the absence of a de-Broglie–Bohm potential, introduction ofthe variable t∗ according to

dt∗ = ζ (t)dt (18)

when the modulation function ζ (t) adopts the form

ζ (t) = k T −2, (19)

reduces (16) to a classical Steen–Ermakov-type equation

Tt∗t∗ − 1

kW V

(W

V

)T = 1

T 3. (20)

The latter admits general solution the nonlinear superposition

T =√λ T 2

1 + 2μ T1T2 + ν T 22 (21)

where T1, T2 are linearly independent solutions of

Tt∗t∗ − 1

kW V

(W

V

)T = 0 (22)

with unit Wronskian and

λ ν − μ2 = 1. (23)

Thus, the modulation function ζ (t) should be chosen as

ζ = k

λ T 21 + 2μ T1T2 + ν T 2

2

, (24)

where

t = 1

k

∫ [λ T 2

1 + 2μ T1T2 + ν T 22

]dt∗. (25)

and, ζ (t) is then given parametrically in terms of t∗ via (24) and (25). LikewiseT (t) is given parametrically in terms of t∗ through the relations (21) and (25).In particular, the Ermakov–Ray–Reid system (15) with ω = 0 adopts theHamiltonian form

W = − ∂�

∂W, V = −∂�

∂V(26)

8 C. Rogers et al.

iff = const = C, in which case

� = 1

2

[(1 − 2s)

(1

W 2+ 1

V 2

)+ C

W V

]

and the system (26) becomes

W = (1 − 2s)

W 3+ C

W 2V,

V = (1 − 2s)

V 3+ C

V 2W. (27)

The latter system arises in a variety of nonlinear optics contexts via theparaxial approximation. Its admittance of two independent integrals of motion,namely, the Ray–Reid invariant and Hamiltonian readily allows its completeintegration [12].

In the simplest case, when C = 0, together with ω = 0 and s = 0, the widthsreduce to a triad of autonomous Steen–Ermakov equations

W = 1

W 3, V = 1

V 3,

Tt∗t∗ = 1

T 3.

Thus,

W =[

1

c( c t + d )2 + 1

]1/2

, V =[

1

c( c t + d )2 + 1

]1/2

,

T =[

1

a( a t∗ + b )2 + 1

]1/2

,

where, on integration of (18),

t = 1

k

[1

3a(at∗ + b)3 + t∗

]+ l.

In the above, a, b, c, d, c, d, and l are arbitrary constants of integration.The modulation ζ (t) is given by (19) and the modulations δ(t) and ε(t) by (14)with � and � related by (17) with = 0.

II

If ζ = 1 together with T = V then, for the class of modulations (14), thenonlinear system (13) reduces to consideration of the Ermakov–Ray–Reidsystem (15) alone. On introduction of new dependent and independent variables


W ∗, V ∗, and τ according to

W ∗ = W/φ, V ∗ = V/φ

τ = ψ/φ (28)

where φ,ψ are linearly independent solutions with unit Wronskian of

�+ ω2(t)� = 0 (29)

it is seen that (15) is reduced to the autonomous Ermakov–Ray–Reid system

W ∗ττ = 1

W ∗2V ∗

[(1 − 2s)

V ∗

W ∗ +

(W ∗

V ∗

) ],

V ∗ττ = 1

W ∗V ∗2

[(1 − 2s)

W ∗

V ∗ +

(W ∗

V ∗

) ](30)

An integrable reduction

If s = 1/2 and

= CW ∗2V ∗2

(W ∗2 + V ∗2)2= CW 2V 2

(W 2 + V 2)2

(31)

then the system (30) adopts the form

W ∗ττ = −∂�

∗

∂V ∗ , V ∗ττ = − ∂�∗

∂W ∗ (32)

where

�∗ = C

2(W ∗2 + V ∗2). (33)

This system belongs to the more general class

α +�′(αβ)α = 1

α2βF(β/α),

β +�′(αβ)β = 1

αβ2G(α/β) (34)

where

α = ∂�

∂β, β = ∂�

∂α. (35)

These Ermakov–Ray–Reid systems are readily shown to be integrable.Thus, (35) implies that

∂

∂α

[1

α2βF(β/α)

]= ∂

∂β

[1

αβ2G(α/β)

]

10 C. Rogers et al.

whence

2F(z) + z d F(z)/dz = 2w2G(w) + w3d G(w)/dw (36)

where z = β/α, w = α/β. Accordingly,

w2d(F/w2)/dw = −d(w2G)/dw

so that, on integration

w2G = 2∫

F

wdw − F. (37)

If we now set F ≡ w�′(w) then the system (34) adopts the form

α +�′(αβ) α = 1

α2β

[−αβ�′(α/β)

],

β +�′(αβ) β = 1

αβ2

[�′(α/β)

α/β− 2

(α/β)2�(α/β)

](38)

with associated Ray–Reid invariant

I=1

2(αβ − αβ)2 +

∫−1

z�′(

1

z

)dz +

∫ [�′(w)

w− 2

w2�(w)

]dw

=1

2(αβ − αβ)2 + 2� (α/β)

β

α.

(39)

Now, the system,

α +�′(αβ)α = ∂�

∂β,

β +�′(αβ)β = ∂�

∂α, (40)

generically admits the invariant J where

J = αβ +�(αβ) −�. (41)

Thus, in particular, system (38), in addition to the Ray–Reid invariant (39)admits the constant of motion

J = αβ +�(αβ) − 1

α2�

(α

β

). (42)

Further, a combination of (39) and (42) shows that

I = 1

2(αβ + αβ)2 + 2αβ [�(αβ) − J ] . (43)

Hence, setting Z = αβ we obtained

(d Z/dt)2 = 2 [I − 2z(�(Z ) − J )]


so that

± 1√2

∫d Z√

I − 2Z [�(Z ) − J ]= t + t0, (44)

where t0 is a constant of integration. In particular, if �(Z ) = Z , it is noted that

Z = J ± 1

2

√2I + J 2 sin 2(t + t0). (45)

On introduction of the new variable t according to

Z dt = dt (46)

into the Ray–Reid invariant (39), it is seen that

1

w2

(dw

dt

)2

= 2

[I − 2�(w)

w

]

whence

± 1√2

∫dw√

I w2 −�(w)w= t + t0. (47)

In particular, the procedure is readily applied to the present system in which is given by (31) wherein � = − Cw2

2(1+w2) .It is noted that the general procedure given above for the Ermakov–

Ray–Reid system (38) complements that presented in [12] for HamiltonianErmakov–Ray–Reid systems of the form

α +�′(α2 + β)2α = 1

α2βF

(β

α

)= −∂V

∂α,

β +�′(α2 + β2)β = 1

αβ2G

(α

β

)= −∂V

∂β.

(48)

III. An Ovsiannikov–Dyson reduction

If ζ = 1 together with s = 1/2, ω2(t) = 0 and

δ(t) = λ

(W V T )γ−1, ε(t) = μ

|A|2n(W V T )γ−1(49)

12 C. Rogers et al.

then, on appropriate scaling, the system (13) reduces to the form

W = 1

(W V T )γ−1W,

V = 1

(W V T )γ−1V,

T = 1

(W V T )γ−1T .

(50)

The latter system has its origin in work of Ovsiannikov [28] and Dyson[29] on spinning anisentropic gas clouds. It has subsequently been thesubject of an extensive series of papers (see Gaffet [30] and referencescited therein). Importantly, in the case of an ideal monatomic gas withadiabatic index γ = 5/3 as appropriate in certain astrophysical contexts, ithas been established by Gaffet via a Painleve procedure that the systemis integrable. It is of interest to remark that, in this case, the quantityln |u| with {W, V, T } given by the nonlinear system (51) admits anovel ellipsoidal flip over phenomenon. This is exhibited in Figure 1 andFigure 2 corresponding to initial data (W (0), V (0), T (0)) = (0.1, 0.25, 0.5)and (W (0), V (0), T (0)) = (0.5, 0.25, 0.1), respectively, and is obtained bynumerical integration of the system (51). It is seen that ln |u| changes itsshape from vertically elongated (cigar-like shape) to horizontally elongated(pancake-like shape) as it evolves (Figure 1) and vice-versa (Figure 2). Sucha flip-over effect has been observed experimentally in plasma ellipsoids byGornushkin et al. [40] in a model that describes asymmetric expansion of alaser-induced plasma into a vacuum.

Dyson [29] considered the large-time asymptotics of the special case of(50) when T = V so that the system (50) reduces to

W = W V

(W 2V )γ,

V = W 2

(W 2V )γ.

(51)

Here, we observe that the above system adopts Ermakov–Ray–Reid form iffγ = 5/3 in which case it becomes

W = 1

W 2V

(V

W

)1/3

,

V = 1

W V 2

(V

W

)1/3(52)


Figure 1. The time evolution of ellipsoids ln |u| = const with initial asymmetry ratios0.1:0.25:0.5.

with Ray–Reid invariant

I = 1

2(W V − W V )2 + 3

4

[ (V

W

)4/3

+ 2

(W

V

)2/3]. (53)

If we now set W = W ∗, V = √2 V ∗ then the system (52) becomes

W ∗ = V ∗−2/3 W ∗−7/3 2−1/3,

V ∗ = W ∗−4/3 V ∗−5/3 2−4/3. (54)

14 C. Rogers et al.

Figure 2. The time evolution of ellipsoids ln |u| = const with initial asymmetry ratios0.5:0.25:0.1.

The latter is Hamiltonian with

W ∗ = − ∂�∗

∂W ∗ , V ∗ = −∂�∗

∂V ∗ , (55)

where

�∗ = 3

21/34W ∗−4/3V ∗−2/3. (56)

The corresponding Hamiltonian is given by

H∗ = 1

2

[W ∗2 + V ∗2

]+ 3

21/34W ∗−4/3V ∗−2/3, (57)


while the Ray–Reid invariant (53) becomes

I ∗ ≡ I

2= 1

2(W ∗V ∗ − W ∗V ∗)2 + 3

21/34

[(V ∗

W ∗

)4/3

+(

W ∗

V ∗

)2/3].

(58)

Thus,

(W ∗2 + V ∗2) H∗ − I ∗ = 1

2(W ∗W ∗ + V ∗V ∗)2 (59)

and, if we set

�∗ = W ∗2 + V ∗2 (60)

it is seen that

1

8�∗2 − H∗�∗ + I ∗ = 0 (61)

so that

�∗ = 2H∗(t − t∗0 )2 + I ∗/H∗, (62)

where t∗0 is a constant of integration.

On introduction of θ via the parametrization

W ∗ = �∗1/2 cos θ, V ∗ = �∗1/2 sin θ (63)

(57) shows that

H∗ = 1

8

�∗2

�∗ + 1

2�∗θ2 + D

�∗ (cos−4/3 θ sin−2/3 θ ),

where D = 321/34 . Thus, on use of (61)

I ∗ = 1

2θ2

t∗ + D cos−4/3 θ sin−2/3 θ, (64)

where

t∗ = 1√2I ∗ tan−1

(H∗√

2

I ∗ ( t − t∗0 )

), I ∗ > 0. (65)

Hence, the Ray–Reid invariant I ∗ plays the role of the total energy in adynamical system with potential energy

�∗ = D cos−4/3 sin−2/3 θ. (66)

Accordingly, the Ermakov–Ray–Reid system (54) has been reduced to a singlequadrature for θ via (64). The Ermakov variables W ∗, V ∗ are then given bythe relations (62) and (63). Alternatively, an elliptic integral reduction maybe obtained via the procedure indicated for Hamiltonian Ermakov–Ray–Reid

16 C. Rogers et al.

systems in [12]. It is noted that an integral of motion of the type (64) hasrecently been derived in another manner in the context of expansion of laserinduced plasma into a vacuum alluded to earlier (Gornushkin et al. [40]).

The above procedure is readily extended to the Ermakov–Ray–Reid reduction

W + ω2(t)W = 1 − 2s

W 3+ C

W 2V

(V

W

)1/3

,

V + ω2(t)V = 1 − 2s

V 3+ C

W V 2

(V

W

)1/3

(67)

with ζ = 1 and T = V of the system (13) following an autonomization of thetype (28).

It is natural to enquire as to whether 3+1-dimensional modulated NLSequations of the type (3) admit exact reduction to integrable Ermakov–Ray–Reidsystems without recourse to variational approximation. In this connection,it is readily shown, in particular, that the 3+1-dimensional modulatedBialynicki-Birula-type NLS equation

i∂

∂t+ ∇2 − ξ (t) ln | | − s

( ∇2| || |

) − ω2(t)

4(x2 + y2 + z2) = 0

(68)

admits exact Gaussian solutions of the form

=(

cI

αβδ

)1/2

exp

[ −(x − q)2

2cI Iα2− (y − p)2

2cI I Iβ2− (z − r )2

2cI V δ2

+ i

2

(α

α(x − q)2 + q(x − q) + β

β(y − p)2 + p(y − p)

+ δ

δ(z − r )2 + r (z − r )

) ], (69)

where the modulation ξ (t) is given by

ξ (t) =(αβδ

C

)1−γ(70)

and α, β, δ are governed by the coupled nonlinear dynamical system

α + ω2(t)α = 2 Cγ−1

c2I Iα(αβδ)γ−1

+ 4(1 − s)

c4I Iα

3,

β + ω2(t)β = 2 Cγ−1

c2I I Iβ(αβδ)γ−1

+ 4(1 − s)

c4I I Iβ

3,

δ + ω2(t)δ = 2 Cγ−1

c2I V δ(αβδ)

γ−1+ 4(1 − s)

c4I V δ

3, (71)


while p, q, r obey by the harmonic oscillator equations.

p + ω2(t) p = 0, q + ω2(t) q = 0, r + ω2(t) r = 0. (72)

In particular, if γ = 5/3, s = 1/2, andω = 0 the integrable Ovsiannikov–Dysonsystem is retrieved. On the other hand, if α = β and again γ = 5/3 anintegrable two-component Ermakov–Ray–Reid system is obtained.

4. A spiralling elliptic soliton system. Ermakov–Ray–Reid reduction

Desyatnikov et al. in [4] recently established that, importantly, orbitalmomentum may act as a mechanism to suppress collapse of spiralling solitonsin nonlinear Kerr media. Therein, a variational procedure was applied to the2+1-dimensional NLS equation with Lagrangian

L = 1

2

∫[ E∗Ez − E E∗

z ] dr − H (73)

with Hamiltonian

H =∫ {

|∇E |2 − 1

2|E |4

}dr. (74)

In this nonlinear optics context, again the role of the time variable in theunderlying model NLS equation is taken by the spatial variable z. A trialfunction with Gaussian envelope was introduced according to

E(x, y, z) = A(z) exp

[−1

2

(X2

b2(z)+ Y 2

c2(z)

) ]exp (i ) (75)

with phase

= B(z)X2 +�(z)XY + C(z)Y 2 + φ(z). (76)

In the above, X = x cos θ + y sin θ , Y = −x sin θ + y cos θ where θ (z) is theangle of a rotating frame.

Application of the standard variational procedure resulted in the integrals [4]

P = π A2 bc, P = 0

M = 1

2P �(b2 − c2), M = 0,

(77)

where

B = b/4b,C = c/4c (78)

18 C. Rogers et al.

together with

θ = 2 �(b2 + c2)

(b2 − c2). (79)

and, in the above · ≡ d/dz .On introduction of the ellipticity parameter ε(z) and new spatial variable

a(z) according to

b =√

2 a cos ε, c =√

2 a sin ε, (80)

it was recorded in [4] that b(z) and c(z) are determined through a(z) and ε(z)by a Hamiltonian system

H = P

2

(a2

2+ μ

a2

), ˙H = 0, (81)

μ = a4 ε2

2+ 4 − p sin 2ε

2 sin2 2ε+ 2σ 2

cos2 2ε, μ = 0, (82)

with p ≡ P/π = A2bc, σ ≡ M/P = a2� cos 2ε.In [4], numerical results were presented for the above system. Here, it is shown

that the nonlinear coupled system (81)−(82) is, in fact, of Ermakov–Ray–Reidtype and that its exact solution may be rendered in terms of a standard ellipticintegral by means of the procedure described in [12, 21] as applied in theprevious section to the Ovsiannikov–Dyson system. Thus, it was shown in[12, 21] that Ermakov–Ray–Reid systems which are Hamiltonian may beparametrized according to

α +�′(α2 + β2)α = 2

α3J (β/α) + β

α4d J (β/α)/dz,

β +�′(α2 + β2)β = − 1

α3d J (β/α)/dz

(83)

where z = β/α. Herein, the Hamiltonian adopts the form

H = 1

2[α2 + β2 +�(α2 + β2)] + 1

α2J (β/α) (84)

while the Ray–Reid invariant is given by

I = 1

2(αβ − αβ)2 +

(α2 + β2

α2

)J (β/α). (85)


In the present nonlinear optics context, on use of the relations (80), theintegral of motion (82) is seen to correspond to the Ray-Reid invariant

1

2(αβ − αβ)2 +

[2(α2 + β2)

α2β2− p

αβ

](α2 + β2) + 8σ 2

(α2 + β2

α2 − β2

)2

= 4μ = I

(p (α2 + β2)

αβ+ I > 0

)(86)

whence, alignment with (85) shows that

J (β/α) =[

2

(α

β

)2

+ 2 − p

(α

β

) ]+

8σ 2

[1 +

(β

α

)2]

1 − 2(β

α

)2+(β

α

)4, (87)

Now, on use of the identity

(α2 + β2)(α2 + β2) − (αα + ββ)2 ≡ (αβ − αβ)2, (88)

the invariants (88) and (89) combine to show that

a2

2+ μ

a2= 1

4(2H −�), (89)

and alignment with the Hamiltonian relation (81) is obtained by setting � = 0together with H = 4H/P . Thus, the spiralling soliton system(81)−(82) of [4]is equivalent to the Hamiltonian Ermakov–Ray–Reid system

α = 2

α3J (β/α) + β

α4d J (β/α)/dz,

β = − 1

α3d J (β/α)/dz

(90)

where J (β/α) is given by (87). Accordingly, the system may be solved via theprocedure described in detail in [12, 21]. In general, the Hamiltonian andRay–Reid invariants (84) and (85) combine to give

(α2 + β2) H − I = 1

2(αα + ββ)2 + 1

2(α2 + β2) � (α2 + β2) (91)

whence, if we set � = α2 + β2 then

1

8�2 + 1

2��(�) − H� + I = 0. (92)

If � is now introduced according to

� = 2αβ

α2 + β2(93)

20 C. Rogers et al.

then, it is seen that

� = 2(α2 − β2)(αβ − αβ)

(α2 + β2)2(94)

and the Ray–Reid invariant becomes

I = 1

8�2 (α2 + β2)4

(α2 − β2)2+(α2 + β2

α2

)J (β/α). (95)

In terms of the new spatial measure z∗ given by

dz∗ = (1/�)dz, (96)

(95) shows that

(d�/dt∗)2 = 8

(α2 − β2

α2 + β2

)2 [I −

(α2 + β2

α2

)J (β/α)

](97)

whence reduction is obtained to the quadrature

± 1

2√

2

∫ √�

(1 −�2)(�I − 2L(�))d� = z∗ (98)

where L(�) = (β/α) J (β/α) and β/α is given in terms of � via

β/α = (1 ±√

1 −�2)/�. (99)

The original Ermakov variables α, β are given in terms of � and � by

α = [√�(1 +�) +

√�(1 −�) ]/2, β = [

√�(1 +�) −

√�(1 −�) ]/2

(100)

if the positive sign is taken in (98).In the present case with J given by (87), it is seen that

L(�) = 4

�− p + 4σ 2�

1 −�2, (101)

and (97) produces to the elliptic integral relation

± 1

2√

2

∫�d�√

−I�4 − 2p�3 +�2(I + 8 − 8�2) + 2p�− 8= z∗, (102)

while (92) shows that

� = I

H+ 2H (z + C)2, (103)


with C an arbitrary constant of integration. Substitution of (102) into (96) andintegration produces, up to a constant,

z∗ = 1√2I

tan−1

[H

√2

I(z + C)

], I > 0. (104)

5. Incorporation of a harmonic trap: A Bose–Einstein condensate system

Abdullaev et al. in [5] derived an analogue of the system (81), (82) in thecontext of the collapse of elliptic clouds of Bose–Einstein condensates. Thesystem was derived by application of the Ritz variational procedure to a2+1-dimensional Gross–Pitaevskii model

i∂

∂t+� + | |2 − w−4(x2 + y2) = 0

(� = ∂2x + ∂2

y ) (105)

incorporating a radial symmetric trapping potential with constant depth w. Thevariational procedure with trial function of the type (76) with z replaced byt led to the system consisting again of (82) but now with (81) generalized toaccount for the harmonic trapping potential to

h = a2

2+ K

a2+ 2a2

w4. (106)

Alignment of (106) and (89) shows that

� = 8a2/w4 (107)

and

H = 2h, K = I/4. (108)

Thus, the variational approximation of the Bose–Einstein condensate systemin [5] reduces to the Hamiltonian Ermakov–Ray–Reid system

α + 4

w4α = 2

α3J (β/α) + β

α4d J (β/α)/dz,

β + 4

w4β = − 1

α3d J (β/α)/dz, (109)

where J is again given by (87). Thus, reduction may again be made to theelliptic integral (101) but now with z∗ replaced by t∗ with

dt∗ = 1

α2 + β2dt = 2

a2dt, (110)

22 C. Rogers et al.

where

a + 4a

w4=(

I

2

)1

a3. (111)

A classical result states that the Steen–Ermakov equation (111) has generalsolution given by the nonlinear superposition

a =√λa2

1 + 2μa1a2 + νa22, (112)

where λ, μ, ν are constants related by

λν − μ2 = I

2W 2(113)

and W is the Wronskian a1a2 − a1a2 of two linearly independent solutions ofthe harmonic oscillator equation

a + 4a

w4= 0. (114)

If a1, a2 are the linearly independent solutions

a1 = w0 cos

(2

w2t

)+ v0

2w2 sin

(2

w2t

),

a2 = sin

(2

w2t

)(115)

of (114) then the solution

a =[

a21 + I

2W 2a2

2

]1/2

(116)

obeys the general initial conditions

a(0) = w0, a(0) = v0,

(w0 �= 0).(117)

The relation (116) shows that, if the Ray–Reid invariant I > 0 then there isno collapse of the Bose–Einstein elliptic cloud and it exhibits an oscillatoryrotating mode. However, if I < 0 then collapse occurs at time.

tcollapse = w2

2tan−1

[(−w0v0 +

√−I

2

)/w2

(I

4w20

+ v20w

2

2

)]. (118)


To conclude, it is seen that (116) shows that

a2 = w20

2+ 1

8

(v2

0 + I

2w20

)w4 +

[w2

0

2− 1

8

(v2

0 + I

2w20

)w4

]cos

(4t

w2

)

+ v0w0w2

2sin

(4t

w2

)(119)

whence the relation (110) is readily integrated to determine t∗ in terms of t .

6. Conclusion

The occurrence of integrable Hamiltonian systems of Ermakov–Ray–Reidtype in areas of physical importance and notably as shown here in nonlinearoptics is well documented. In the modern soliton theory of integrable systems,solitonic phenomena with their associated remarkable characteristic nonlinearinteraction properties are typically associated with a delicate but stablebalance between nonlinearity and dispersion. Ermakov–Ray–Reid systemslikewise admit nonlinear superposition principles albeit of another kind.It would be of considerable interest to seek to characterise the universalphysical and mathematical properties that attend the admittance of integrableErmakov–Ray–Reid systems, not only in nonlinear optics, but in other nonlinearphysical contexts.

Acknowledgement

One of the authors acknowledges with gratitude support under the NationalNatural Science Foundation of China (Grant No. 41176005).

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THE UNIVERSITY OF NEW SOUTH WALES

TEL AVIV UNIVERSITY

THE HONG KONG POLYTECHNIC UNIVERSITY

(Received December 5, 2012)

ermakov–ray–reid reductions of variational approximations in nonlinear optics

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