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Multi-component Ermakov Systems: Structure and Linearization
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
198, 194]220 Ž1996.
0076
Multi-component Ermakov Systems: Structure and Linearization C. Rogers and W. K. Schief School of Mathematics, The Uni¨ ersity of New South Wales, Sydney, New South Wales 2052, Australia Submitted by William F. Ames Received March 30, 1995
A symmetry reduction of a Ž2 q 1.-dimensional nonlinear N-layer fluid model is shown to lead to a prototype for an N-component extension of the classical Ermakov system. The general N-component Ermakov system introduced here has the attractive property that it may be iteratively reduced to a system of N y 2 linear equations augmented by a canonical 2-component Ermakov system. The recently established linearization procedure for the latter may then be used to solve the N-component system N ) 2 in generality. The procedure is illustrated, in detail, for a 3-component system. Sequences of classical 2-component Ermakov systems are shown to be linked via Darboux transformations. Q 1996 Academic Press, Inc.
1. INTRODUCTION The study of the coupled nonlinear ordinary equations now known as Ermakov systems originated in 1880 w1x. In the intervening years there has been an extensive literature devoted to their analysis w2]30x. In terms of applications, Ermakov systems arise, notably, in nonlinear optics w31]35x and nonlinear elasticity w36, 37x. The main theoretical interest in such systems originated in the fact that they admit a novel constant of motion, namely, the celebrated Lewis]Ray]Reid invariant. In a recent key development, it was shown by Athorne et al. w24x that the classical Ermakov system is, in fact, linearizable, that is, C-integrable in the terminology of Calogero and Eckhaus w38]40x. It turns out that the Lewis]Ray]Reid invariant plays a key role in that linearization. The classical Ermakov system consists of a coupled system of ordinary differential equations involving two dependent variables. It is natural to enquire as to the existence of multi-component Ermakov systems which share the recently established C-integrability of the classical two194 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
MULTI-COMPONENT ERMAKOV SYSTEMS
195
component Ermakov system. It turns out that a prototype for such a multi-component system resides in a symmetry reduction of a multi-layer fluid system. Thus, it was recently shown by Rogers et al. w41, 42x that a nonlinear Ž2 q 1.-dimensional shallow water system due to Stoker admits a symmetry reduction to a particular classical Ermakov system. Here, an analogous reduction for an N-layer fluid leads to what may be termed a multi-component Ermakov system. It is demonstrated that such multicomponent nonlinear systems are, like the classical Ermakov system, linearizable.
2. THE CLASSICAL ERMAKOV SYSTEM The classical Ermakov system, as extended by Ray and Reid in 1979, adopts the form w6x u ¨ q vŽ t. u s ¨¨ q v Ž t . ¨ s
f Ž ¨ ru . u2 ¨ g Ž ur¨ . ¨2u
Ž 1. ,
where f, g, and v are arbitrary functions of their indicated arguments and the overdot denotes the derivative drdt. Here, the linearization procedure for the system Ž1. as originally presented in Athorne et al. w24x is reviewed since it is used in our subsequent linearization of the multi-component Ermakov system. Thus, new dependent and independent variables a, b, and z are introduced into Ž1. according to w24x u s aŽ z . f ,
¨ s bŽ z . f ,
z s crf ,
Ž 2.
where c and f are linearly independent solutions with unit Wronskian fc˙ y cf˙ of the base equation
f¨ q v Ž t . f s 0.
Ž 3.
Under Ž2., the original Ermakov system Ž1. is reduced to the autonomous system aY s
f Ž bra. 2
a b
,
bY s
g Ž arb . b2a
.
Ž 4.
It is observed that the Wronskian W Ž u, ¨ . s u¨˙ y ¨ u ˙ is invariant under Ž2. so that W Ž u, ¨ . s w Ž a, b . s abX y baX .
196
ROGERS AND SCHIEF
On introduction of the new independent variable t via dt s by2 dz,
Ž 5.
it is seen that the system Ž4. is, in turn, equivalent to the coupled system d2
a
1
dt 2
ž / b
q
b
1
ž /
g Ž arb .
a
2
ž / ž /
dt 2 b d2
b
s
b
a
f Ž bra. y g Ž arb .
Ž 6.
s 0.
In terms of the new canonical group variables 2
x s Ž bra. ,
y s 1rb,
Ž 7.
the system Ž6. yields the first integral 1 dx x dt
s
1 dx xy 2 dz
s h,
Ž 8.
where h Ž x ; I . s 2 x 1r2 2 I q
x
H
1r2
sy3r2 g Ž sy1r2 . y sy1r2 f Ž s 1r2 . d s
Ž 9.
together with the linear equation d dx
dy
ž / xh
dx
q
g Ž xy1r2 . x 1r2 h
y s 0.
Ž 10 .
The relation Ž8. produces the Lewis]Ray]Reid invariant Is
Ž u¨˙ y ¨ u˙. 2
2
q
ur¨
H
g Ž l. d l q
¨ ru
H
f Ž m . dm
Ž 11 .
of the original Ermakov system Ž1.. The solution of the system Ž1. may now, in principle, be readily generated modulo the solution of the linear equation Ž6. 2 or, equivalently, Ž10.. Thus, if f and g are specified, hŽ x; I . is determined via Ž9.. If hŽ x; I . is then substituted into Ž10. to produce, on integration, y s y Ž x ; I, c1 , c 2 . s c1 y 1 Ž x ; I . q c 2 y 2 Ž x ; I .
Ž 12 .
then substitution of Ž12. into Ž8. and integration yields z s z Ž x ; I, c1 , c2 , c 3 . s c Ž t . rf Ž t . ,
Ž 13 .
MULTI-COMPONENT ERMAKOV SYSTEMS
197
where c 3 is an additional constant of integration. The relations Ž7., on the other hand, give as
1
'x
c1 y 1 Ž x ; I . q c 2 y 2 Ž x ; I .
,
bs
1 c1 y 1 Ž x ; I . q c 2 y 2 Ž x ; I .
.
Ž 14 . At this stage, we may proceed to the solution of the original Ermakov system in two ways. Thus, if Ž13. may be inverted to yield t s T Ž x ; I, c1 , c 2 , c 3 .
Ž 15 .
then the original Ermakov variables u, ¨ 4 are given parametrically in terms of x via Ž15. together with the relations us
f Ž T Ž x ; c1 , c 2 , c 3 . .
'x
c1 y 1 Ž x ; I . q c 2 y 2 Ž x ; I .
,
¨s
f Ž T Ž x ; c1 , c 2 , c 3 . . c1 y 1 Ž x ; I . q c 2 y 2 Ž x ; I .
.
Ž 16 . On the other hand, if Ž13. may be inverted to yield x s X Ž crf ; I, c1 , c 2 , c 3 .
Ž 17 .
then we obtain a nonlinear superposition principle for the original Ermakov variables u, ¨ 4 in terms of two linearly independent solutions f , c 4 of the linear base equation Ž3. via us
f
'x
c1 y 1 Ž x ; I . q c 2 y 2 Ž x ; I .
,
¨s
f c1 y 1 Ž x ; I . q c 2 y 2 Ž x ; I .
.
Ž 18 . This generalizes the well-known nonlinear superposition principle for Pinney’s equation w43x retrieved in the decoupled reduction f s ¨ ru, g s 0 of the Ermakov system Ž1.. In what follows, guided by the structure of a symmetry reduction of an N-layer fluid system, a multi-component Ermakov system is introduced which inherits linearization from the classical 2-component Ermakov system.
198
ROGERS AND SCHIEF
3. A SYMMETRY REDUCTION OF A MULTI-LAYER FLUID SYSTEM: AN N-COMPONENT PINNEY SYSTEM The following Ž2 q 1.-dimensional N-layer fluid system is considered:
ui t ¨i t t
Ž hiy1 y hi . q
x
q ui q ui
ui x ¨i x
q ¨i q ¨i
u i Ž hiy1 y hi . q
ui y ¨i
y
y
y f¨ i q q fu i q
pi x pi
s0
s 0, y i s 1, . . . , N
¨ i Ž hiy1 y hi . s 0.
Ž 19 . Here, the quantities u i , ¨ i 4 denote fluid current components and hiy1 s hiy1Ž x, y, t . denotes the surface of the upper boundary of the ith layer. In the hydrostatic approximation adopted here, the pressure terms pi are given by iy1
pi s
Ý g j hj ,
Ž 20 .
js0
where g j represents the buoyancy jump at the interface h s hj Ž x, y, t .. In what follows, the Coriolis parameter f / 0 is scaled to be unity under t ª ft, x ª fx, y ª fy. In w41, 42x, it was shown that an appropriate symmetry reduction of the two-layer system results in a particular classical Ermakov system of the type Ž1.. The linearization procedure for Ž1. may be used to construct analytic solutions which, in particular, provide valuable necessary checks on numerical schemes. Here, an analogous symmetry reduction is sought for the N-layer system Ž19. with a view to discovering a prototype of an N-component Ermakov system. The existence of such systems, until now, has remained an open question. Thus, on introduction of the ansatz u i s xu ˜i Ž t . q y¨˜i Ž t . ,
¨ i s yx¨˜i Ž t . q yu ˜i Ž t . ,
hi s Ž x q y 2 . H˜i Ž t . , 2
Ž 21 .
MULTI-COMPONENT ERMAKOV SYSTEMS
199
into the N-layer system Ž19., we obtain
˙i q 2 u˜i¨˜i y u˜i s 0 ¨˜ iy1
˙˜i y ¨˜i2 q u˜2i q ¨˜i q 2 u
Ý g j H˜j s 0
Ž 22 .
js0
˜iy1 y H˙ ˜i q 4 H˜iy1 y H˜i u˜i s 0 H˙
ž
/
˜N s 0 in the case of a rigid bottom topography. Here, we set with H˙ H˜N s 0. The solution of the nonlinear system Ž22. admits a parametrization in terms of an N-tuple of dependent variables V 1 , . . . , V N 4 governed by the coupled system of the form ¨iq V
1 4
Vi q
ž
ai j
N
Ý
V 4j
js1
/
V i s 0,
Ž 23 .
H˜iy1 y H˜i s c i Vy4 i ,
Ž 24 .
where a i j s const and
˙ irV i , u ˜i s V
¨˜i s 12 y Vy2 i ,
so that N
H˜j s
Ý
c k Vy4 k .
Ž 25 .
ksjq1
On introduction of the dependent variables a i and independent variable z according to V i s ai Ž z . f ,
z s crf ,
Ž 26 .
where f s '2 cosŽ tr2. and c s '2 sinŽ tr2. are solutions with unit Wronskian of the linear base equation
f¨ q 14 f s 0,
Ž 27 .
the system Ž23. reduces to the autonomous form aYi q
ž
N
Ý js1
ai j a4j
/
a i s 0,
Ž 28 .
X where [ drdz. The system Ž28. may be regarded as an N-component Pinney system with Pinney’s equation being retrieved in the case N s 1.
200
ROGERS AND SCHIEF
The basic idea in what follows involves linearization of the N-component system Ž28. via an iteration process. Thus, we observe that Ž28. implies a1 aYj y a j aY1 s
a 1 k y a jk
N
ž
Ý
/
a4k
ks1
j s 2, . . . , N,
a1 a j ,
Ž 29 .
so that d2 dt
2
aj
ž / a1
s Ž a 11 y a j1 .
aj
ž / a1
N
Ý Ž a 1 k y a jk .
q
ks2
a1
4
aj
ž /ž / ak
a1
, Ž 30 .
where dt s a1y2 dz.
Ž 31 .
Accordingly, if we set Uj s a jra1 ,
j s 2, . . . , N,
Ž 32 .
the system Ž30. reduces to d 2 Uj dt
2
N
q Ž a j1 y a 11 . Uj q
ž
Ý
a jk y a 1 k Uk4
ks2
/
Uj s 0,
Ž 33 .
together with the linear Schrodinger equation ¨ d2 dt
2
1
N
ž / ž a1
s a 11 q
Ý
a1 k
4 ks2 Uk
1
/ž / a1
Ž 34 .
in 1ra1. It is noted that the method of variation of parameters and the relation Ž31. yield z s a1raU1 ,
Ž 35 .
where 1ra and 1raU are linearly independent solutions of Ž34. with unit Wronskian. If it is now required that
a j1 y a 11 s const [ a ,
j s 2, . . . , N,
Ž 36 .
then on introduction of the new variables bj 4 and w via Uj s bj Ž w . x ,
w s jrx ,
Ž 37 .
MULTI-COMPONENT ERMAKOV SYSTEMS
201
where x and j are solutions with unit Wronskian of the linear base equation
xtt q a x s 0
Ž 38 .
the system Ž33. then reduces to the Ž N y 1.-component Pinney-type system d 2 bj dw
2
N
q
ž
Ý
a jk y a 1 k
/
bk4
ks2
bj s 0.
Ž 39 .
It is interesting to note that the conditions Ž36. hold for the N-layer model Žwith a s 1.. The system Ž39., in turn, yields d2 dt
N
bk
2
ž / b2
q Ž b k 2 y b 22 . q
Ý Ž bk l y b 2 l . ls3
b2
4
bk
ž /ž / bl
b2
s 0,
k s 3, . . . , N,
Ž 40 .
where
bk j s a k j y a 2 j
Ž 41 .
dt s by2 dw. 2
Ž 42 .
and
It is now further required that
b k 2 y b 22 s const [ b ,
k s 3, . . . , N,
Ž 43 .
whence
a 32 s ??? s a N 2 .
Ž 44 .
Again, remarkably, this condition holds for the N-layer fluid system. Therein a k 2 s 2Ž g 0 q g 1 . c 2 and b s 0. The residual equation in b 2 in Ž39. reduces to the linear Schrodinger ¨ equation d2 dt
2
1
ž / b2
N
s Ž a 12 y a 22 . q
Ý Ž a1 k y a2 k . ks3
b2
4
1
ž /ž / bk
b2
Ž 45 .
in 1rb 2 . Consequently, w s b 2rbU2 , where 1rbU2 is a second solution of Ž45. with unit Wronskian. Iteration of the above procedure shows that the original N-component Pinney-type system Ž23. may be sequentially reduced to N y 2 linear
202
ROGERS AND SCHIEF
Schrodinger equations supplemented by a residual classical two-compo¨ nent Ermakov system provided that the constraints
am n s aN n ,
m ) n,
Ž 46 .
hold. The conditions Ž46. are identically satisfied for the N-layer system. The N-component Pinney-type system Ž23. with entries of the matrix Ž a m n . constrained by the condition Ž46. may now, in principle, be solved via the linearization procedure for the classical Ermakov system Ž1.. This, in turn, is equivalent to performing another step in the above iteration process so that the N-component Pinney system is reducible to N y 1 linear Schrodinger equations and integration of the Lewis]Ray]Reid ¨ invariant relation. The procedure will be subsequently illustrated for a 3-component Pinney system. It is emphasized that the sole aim of the present work is to exploit the above symmetry reduction to an N-component Pinney-type system to point the way to the structure of general N-component Ermakov systems. The systematic application of general Lie group analysis to the N-layer model to derive Ball-type theorems w44, 45x and to search for physically significant model solutions remains. However, in terms of physical applications, it is noted that a parametrization analogous to Ž24. and applied to a model in elliptic warm-core eddy theory has recently led to the construction of so-called pulsrodon solutions with accompanying underlying Hamiltonian structure w46, 47x. Lie group theory was used in w48x to reveal the rich Lie-algebraic structure underlying single-layer rotating shallow water models with elliptic and circular paraboloidal bottom topographies. In particular, Ball-type theorems were group-generated and subalgebras of the symmetry algebra systematically investigated. Certain symmetry reductions for a circular paraboloidal basin were shown to lead to exact solutions of the shallow water system wherein the temporal evolution of the moving shoreline is driven by Pinney’s equation. Indeed, it was this observation that led to the present study wherein an analogous symmetry reduction of an N-layer system leads to the N-component Pinney system Ž23..
4. THE N-COMPONENT ERMAKOV SYSTEM Here, we start with the extension of the 3-component Pinney system to what we shall term the 3-component Ermakov system. This will motivate the form of the general N-component Ermakov system which is subsequently introduced.
MULTI-COMPONENT ERMAKOV SYSTEMS
203
Thus, we consider 3-component systems of the form
¨ 1 q v Ž t . V1 s V ¨ 2 q v Ž t . V2 s V ¨ 3 q v Ž t . V3 s V
F 1 Ž V 1rV 3 , V 2rV 3 . V 33 F 2 Ž V 1rV 3 , V 2rV 3 .
Ž 47 .
V 33 F 3 Ž V 1rV 3 , V 2rV 3 . V 33
,
where the Fi are dependent on their indicated arguments. Introduction of the change of dependent and independent variables Vi s ViŽ z. f,
z s crf ,
Ž 48 .
where f and c are solutions with unit Wronskian of the linear base equation
f¨ q v Ž t . f s 0,
Ž 49 .
reduces Ž47. to the autonomous form VYi s
Fi V 1rV 3 , V 2rV 3
ž
/
Ž 50 .
V 33
X
with [ drdz. The latter system, in turn, shows that V 3 VY1 y V i VY3 s V 3 VY2 y V 2 VY3 s
1 V 23 1 V 23
F1 y F2 y
V1 V 33 V2 V 33
F3
Ž 51 . F3 .
If we set u s V 1rV 3 ,
¨ s V 2rV 3 ,
Ž 52 .
then Ž51. becomes d2 u dt 2 d2 ¨ dt 2
s F 1 Ž u, ¨ . y uF 3 Ž u, ¨ .
Ž 53 . s F 2 Ž u, ¨ . y ¨ F 3 Ž u, ¨ . ,
204
ROGERS AND SCHIEF
where dt s Vy2 3 dz.
Ž 54 .
If the system Ž53. is now set in correspondence with the classical 2-component Ermakov system in the form d2 u dt
2
d2 ¨ dt 2
q v2 u s q v 2¨ s
u ¨4 ¨
u4
f 2 Ž ur¨ .
Ž 55 . g 2 Ž ur¨ .
with constant v 2 , then the original system Ž47. becomes
¨ 1 q v Ž t . V1 s V
V1
q
¨ 2 q v Ž t . V2 s V
V1 V 43
V2
V2 V 43
1 V 33
F 3 Ž V 1rV 3 , V 2rV 3 . y v 2
g 2 Ž V 1rV 2 .
V 41 q
¨ 3 q v Ž t . V3 s V
f 2 Ž V 1rV 2 .
V 42
Ž 56 .
F 3 Ž V 1rV 3 , V 2rV 3 . y v 2
F 3 Ž V 1rV 3 , V 2rV 3 . ,
where F 3 remains an arbitrary function of its indicated arguments. It is natural to term Ž56. the 3-component Ermakov system since, by construction, it inherits the C-integrability property of the 2-component Ermakov system. Thus, application of the sequence of transformations Ž48., Ž52., and Ž54. reduces Ž56. to the classical 2-component Ermakov system Ž55. augmented by Ž47. 3 . The latter reduces to the linear Schro¨ dinger equation d2 dt
2
1
ž / V3
q F 3 V 1rV 3 , V 2rV 3
ž
/
1
ž / V3
s0
Ž 57 .
in 1rV 3 . Thus, once the 2-component Ermakov system Ž55. has been solved for uŽ t . s V 1rV 3 and ¨ Ž t . s V 2rV 3 , the linear equation Ž57. determines V 3 s V 3 Ž t . and the original independent variable z according to z s V 3rVU3 ,
Ž 58 .
MULTI-COMPONENT ERMAKOV SYSTEMS
205
where 1rVU3 is a second solution of Ž57. with unit Wronskian. The relation Ž58. gives z s z Ž t . and hence, in principle, t s t Ž t . through Ž48. 2 and Ž49.. The solution of the original 3-component Ermakov system is then given parametrically in terms of t via the relations Vi s ViŽ t . f Ž t. ,
t s tŽ t. .
Ž 59 .
The procedure will be illustrated, in detail, in the next section for a particular 3-component Pinney-type system. It is important to remark that the choice of the new independent variable in the reduction of the system Ž47. may be taken to be any of dt s Vy2 dt, i g 1, 2, 34 . Indeed, the choice may be crucial in the determii nation of whether a specified system of the type Ž47. represents a 3-component Ermakov system or not. This point is well-illustrated by the 3-layer fluid model
¨1 q V ¨2q V ¨3q V
ž ž ž
1 4 1 4 1 4
q q q
a1
q
V 43
a2
q
V 41
a3
q
V 42
b1
/ / /
V 42
b2 V 43
b3 V 41
V1 s V2 s V3 s
g1 V 31
g2
Ž 60 .
V 32
g3 V 33
,
where
a 1s2 c 3 g 0 ,
a 2s2 c1 g 0 ,
b 1s2 c 2 g 0 ,
b 2s2 c3 Ž g 0 q g 1 . ,
g 1s1y2 c1 g 0 ,
a 3s2 c 2 Ž g 0 q g 1 . b 3s2 c1 g 0
g 2s1y2 c 2 Ž g 0 q g 1 . ,
Ž 61 .
g 3s1y2 c3 Ž g 0 q g 1 q g 3 . . Thus, with the choice dt s Vy2 3 dt, identification of the 3-layer system with Ž56. yields V2
f2 s Ž g 1 q b3 .
q a 3 y b1
ž / ž / ž / ž / V1
g2 s Ž g 2 q a3 . F3 s g 3 y a3
4
V1
V2
V3 V2
4
y a 2 q b3
4
y b3
V3 V1
4
Ž 62 .
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ROGERS AND SCHIEF
together with the necessary restrictions
a1 s v 2 y g 3 s b2 .
Ž 63 .
Inspection of the relations Ž61. shows that the restriction Ž63. is not satisfied by the 3-layer system Ž60. ] Ž61.. However, on cyclic interchange in Ž56., corresponding to reduction via dt s Vy2 dt, the 3-component 1 Ermakov system
¨ 1 q v V1 s V ¨ 2 q v V2 s V ¨ 3 q v V3 s V
1 V 31 V2 V 43 V3 V 42
F 1 Ž V 2rV 1 , V 3rV 1 . V2
f 2 Ž V 2rV 3 . q
F 1 Ž V 2rV 1 , V 3rV 1 . y v 2
V 41
g 2 Ž V 2rV 3 . q
V3
Ž 64 .
F 1 Ž V 2rV 1 , V 3rV 1 . y v 2
V 41
is obtained. Identification of the 3-layer system Ž60. with Ž64. results in
g1 V 31
g2 V 32
g3 V 33
y a1 y a2 y a3
V1 V 43 V2 V 41 V3 V 42
y b1 y b2 y b3
V1 V 42 V2 V 43 V3 V 41
s s s
1 V 31 V2 V 43 V3 V 42
F1 f2 q g2 q
V2 V 41 V3 V 41
Ž F1 y v 2 .
Ž 65 .
Ž F1 y v 2 . ,
whence we obtain the restrictions
a 2 s v 2 y g 1 s b3 .
Ž 66 .
It is observed that the condition a 2 s b 3 is indeed satisfied by the 3-layer system Ž60. ] Ž61.. Moreover, the choice v 2 s g 1 q a 2 s 1 allows the restrictions Ž66. to be satisfied. This emphasizes the importance of the presence of the v 2-term in the 3-component Ermakov system. In its absence the condition Ž66. would have required that a 2 s yg 1 and this is not the case for the above 3-layer fluid model. It is noted that the remaining cyclic interchange produces the restrictions a 3 s v 2 y g 2 s b 1. The condition a 3 s b 1 is not satisfied by the 3-layer system. Thus, the choice dt s Vy2 dt turns out to be crucial in the reduction of the 3-layer 1 fluid model to a Ermakov system.
MULTI-COMPONENT ERMAKOV SYSTEMS
207
If we now proceed to the 4-component system Fi Ž V 1rV 4 , V 2rV 4 , V 3rV 4 .
¨ i q v Ž t. Vi s V
,
V 34
i s 1, . . . , 4, Ž 67 .
then the requirement that it be reducible to a 3-component Ermakov system of the type Ž64. leads to the 4-component Ermakov system which, in extenso, is V1
¨ 1 q v Ž t . V1 s V
V 42 q
V1 V 44
V2
¨ 2 q v Ž t . V2 s V
V 41 q
V 44
V 33 q
¨ 4 q v Ž t . V4 s V
V 44
V 34
V 43
h 3 Ž V 1rV 3 , V 2rV 3 . y v 2
V2 V 43
h 3 Ž V 1rV 3 , V 2rV 3 . y v 2
F4 Ž V 1rV 4 , V 2rV 4 , V 3rV 4 . y v 3
Ž 68 .
h 3 Ž V 1rV 3 , V 2rV 3 .
V3
1
V1
F4 Ž V 1rV 4 , V 2rV 4 , V 3rV 4 . y v 3
g 2 Ž V 1rV 2 . q
V2
1
¨ 3 q v Ž t . V3 s V
f 2 Ž V 1rV 2 . q
F4 Ž V 1rV 4 , V 2rV 4 , V 3rV 4 . y v 3
F4 Ž V 1rV 4 , V 2rV 4 , V 3rV 4 . ,
where h 3 is an arbitrary function of its indicated arguments. The form of the N-component Ermakov system is now clear, namely
¨ i q v Ž t. Vi s V
FiŽ N . Ž V 1rV N , . . . , V Ny1rV N . V 3N
,
i s 1, . . . , N, Ž 69 .
where F kŽ n. V 3n F 1Ž1.
s s
F kŽ ny1. V 3ny1 F 1Ž1.
q
Vk V 4n
Ž V 1rV 2 . ,
FnŽ n. Ž V 1rV n , . . . , V ny1rV n . y v ny1
Ž 70 .
n s 2, . . . , N ; k s 1, . . . , n y 1.
It is noted that FnŽ n., n s 1, . . . , N are arbitrary functions of their indicated arguments. The cases N s 1, 2 correspond to the classical 1-component
208
ROGERS AND SCHIEF
Ermakov ŽPinney. equation and 2-component classical Ermakov system, respectively. The N-component Ermakov system determined by Ž69., Ž70. may be sequentially reduced to a classical 2-component Ermakov system augmented by N y 2 linear equations. Thus, the N-component system yields Vk
d2 dt
2
ž / VN
s F kŽ N . y
Vk VN
FNŽ N . ,
k s 1, . . . , N y 1,
Ž 71 .
together with the linear Schrodinger equation ¨ d2 dt
1
2
ž / VN
q FNŽ N .
1
ž /
s 0,
VN
Ž 72 .
where V N and t are introduced via Vi s ViŽ z. f,
dt s Vy2 N dz,
z s crf ,
Ž 73 .
and f and c are solutions with unit Wronskian of Ž49.. Thus, the N-component Ermakov system reduces to d2 dt 2 d2 dt 2
Vk
ž / ž / VN 1
VN
q v Ny 1 q FNŽ N .
Vk
ž / ž / VN 1
VN
s
F kŽ Ny1. V 3Ny1
Ž 74 .
s 0.
The system Ž74.1 is an Ž N y 1.-component Ermakov system for the quantities V krV N with F kŽ Ny1. Ž V lrV Ny1 . s F kŽ Ny1. Ž w V lrV N x r w V Ny1rV N x . .
Ž 75 .
Iteration of the procedure reduces the N-component Ermakov system to a classical 2-component Ermakov system augmented by N y 2 linear Schrodinger equations. Accordingly, the N-component Ermakov system is ¨ linearizable. The solution procedure is illustrated for a 3-component Ermakov system in the next section.
5. ILLUSTRATION: A 3-COMPONENT ERMAKOV SYSTEM Here, we consider a 3-component Ermakov system of the type
¨ i q v2Vi q V
ž
3
Ý js1
ai j V 4j
/
V i s 0,
i s 1, 2, 3,
Ž 76 .
MULTI-COMPONENT ERMAKOV SYSTEMS
209
where v is a constant. On introduction of the transformation V i s ai Ž z .
cos Ž v t .
'v
z s tan Ž v t . ,
,
Ž 77 .
the system Ž76. reduces to the autonomous form aYi q
ž
3
Ý js1
ai j a4j
/
ai s 0
Ž 78 .
X
with [ drdz. In terms of the new independent variables Uj s a jra1 ,
j s 2, 3,
Ž 79 .
the system Ž78., in turn, reduces to the 2-component classical Ermakov system dU2 dt
2
dU3 dt 2
q Ž a 21 y a 11 . U2 q q Ž a 31 y a 11 . U3 q
ž ž
a 22 y a 12 U24
a 32 y a 12 U24
a 23 y a 13
q
U34
a 33 y a 13
q
U34
/ /
U2 s 0
Ž 80 . U3 s 0
augmented by the linear equation d2 dt
1
ž /
2
a1
s a 11 q
a 12 U24
q
a 13 U34
1
ž / a1
Ž 81 .
in 1ra1. In the above, dt s a1y2 dz and it is required that
a 21 y a 11 s a 31 y a 11 \ ya 2
Ž 82 .
as indicated in the previous section. The 2-component Ermakov system Ž80. as derived in w42x has been recently investigated for a 22 / a 32 n a 23 / a 33 by Athorne w29x. It is interesting to note that a particular subcase was earlier derived by Wagner et al. in w31x in connection with the self-trapping of optical beams. Therein, a numerical treatment of the Ermakov system was presented. Here, by contrast, we focus on the general case in order to illustrate the solution procedure of the 3-component Ermakov system. Thus, new dependent and independent variables a, b, and s are introduced according to U2 s a Ž s .
cosh Ž at .
'a
,
U3 s b Ž s .
cosh Ž at .
'a
,
s s tanh Ž at . , Ž 83 .
210
ROGERS AND SCHIEF
so that the system Ž80. reduces to the autonomous form as s q
f Ž bra. a
3
s 0,
bs s q
g Ž bra. a3
s 0,
Ž 84 .
where f Ž q . s c1 q
c2 q
4
,
g Ž q . s c3 q q
c4 q
3
qs
,
b a
,
Ž 85 .
with the identifications c1 s a 22 y a 12 , c 2 s a 23 y a 13 , c 3 s a 32 y a 12 , c 4 s a 33 y a 13 . Ž 86 . The solution procedure presented in Section 2 may now be applied to the canonical system Ž84. leading to the linear second order equation Ž10.. This equation may subsequently be transformed into a Schrodinger equa¨ tion by changing the independent variable appropriately. This process is here performed directly by introducing a new independent variable via d s s ay2 ds
Ž 87 .
to produce the decoupled system
fss s f Ž q . f
Ž 88 .
qss s qf Ž q . y g Ž q . , where
f s 1ra,
q s bra.
Ž 89 .
The solution of the autonomous equation Ž88. 2 can now be given implicitly in terms of two quadratures, namely 2s Ž q. s
1
H 'Hqf Ž q . y g Ž q . dq
dq,
Ž 90 .
while Ž88.1 represents a linear Schrodinger equation with potential f Ž q Ž s ... ¨ In the present case, it is possible to invert the relation Ž90. although certain subcases need to be considered. Thus, the case c1 / c 3 leads to the explicit formula q2 s
)
c52 y c 42 c13
cosh D q
c5
'c
13
,
D s 2 c13 Ž s q c6 . , Ž 91 .
'
MULTI-COMPONENT ERMAKOV SYSTEMS
211
where c5 Žproportional to the Lewis]Ray]Reid invariant I . and c6 are two constants of integration and c13 s c1 y c 3 , c 42 s c 4 y c 2 . Hence, the Schrodinger equation Ž88.1 assumes the explicit form ¨
fss s c1 q
fss s c1 q
fss s c1 q
fss s c1 q
c 2 c13
ž'
c52
y c 42 cosh D q c5 c2 c13
Ž
1 2
exp D q c5 .
2
2
2
c1 / c 3 , c 42 / c52
c1 / c 3 , c 42 s c52
Ž 92 .
c52 Ž s q c6 . y c 42
4 c 42 Ž s q c6 .
f,
/
f,
c 2 c52
c2
2
f,
2
f,
c1 s c 3 , c 5 / 0
c1 s c 3 , c5 s 0.
For c13 s yc 2 the ‘‘generic’’ case Ž92.1 turns out to be the Ince equation. The problems of stability and periodicity of the solutions of the 2-component Ermakov system Ž80. reduce to those for this linear equation w29x. A particular case is given by c5 s 0 which can be solved explicitly in terms of hypergeometric functions. Interestingly, these hypergeometric functions reduce to rationals if 4 c 2rŽ c 42 y c52 . s lŽ l q 1., where l is an integer. In this case, the solutions of the Schrodinger equation Ž92.1 may also be ¨ derived via a composite Darboux transformation of the order l. This observation provides a point of contact with Darboux transformations. The generation of sequences of classical Ermakov systems by means of Darboux transformations will be the subject of the following section. Another important subcase is represented by c 2 s y3c 3 , c 4 s y3c1 , which makes the 2-component Ermakov system Hamiltonian w49x. Hamiltonian Ermakov systems have been recently derived in connection with the propagation of elliptic Gaussian beams in nonlinear optics w35x. It can be shown w50x that Ermakov systems which are based on Hamiltonians that are quadratic in the conjugate momenta are solvable in terms of quadratures. Finally, Ž92. 2 is a degenerate case of the generic case, whereas the remaining equations Ž92. 3, c1s0 and Ž92.4 may be solved explicitly in terms of trigonometricrhyperbolic and Bessel functions respectively w51x. To illustrate the remaining steps in the solution procedure, we choose the simplest nontrivial case c 2 s 0, c1 / c 3 . This produces the decoupled
212
ROGERS AND SCHIEF
Pinney system dU2
y a 2 U2 y
dt 2 dU3
y a U3 q 2
dt 2
ž
c3
q
U24
c4 U34
c 02
s0
U23
/
Ž 93 .
U3 s 0,
where c1 s yc02 . It is emphasized that this does not mean that the original 3-component Ermakov system Ž76. decouples. The Schrodinger ¨ equation Ž92.1 may then be solved to produce as
a0 cos Ž c 0 s q c 7 .
,
Ž 94 .
whence, on integration of Ž87., ss
a20 c0
tan Ž c 0 s q c 7 . y c10 .
Ž 95 .
If we invert this relation and insert s into a, q as given by Ž94., Ž91. we finally obtain explicit formulae for the original variables U2 , U3 by means of the transformation Ž83., viz. U2 s
U3 s
) (
a20
a
q
c 02
a a02
c8 cosh
2 c0
2 Ž s q c10 . cosh Ž at .
'
c13 arctan
ž
c0 a20
Ž 96 . Ž s q c10 . q c6 q c9 U2
/
s s tanh Ž at . , where c8 s
)
c52 y c 4 c13
,
c9 s
c5
'c
.
Ž 97 .
13
It is readily seen that this class of solutions possesses the correct number of arbitrary constants, namely a0 , c6 , c9 , and c10 . The remaining constants are given in terms of the coefficients c1 , c 3 , c 4 contained in the 2-component system Ž93.. There remains the problem of solving the linear equation Ž81.. It is evident that, in general, the potential in this Schrodinger equation is ¨
MULTI-COMPONENT ERMAKOV SYSTEMS
213
complicated. Accordingly, we here restrict our attention to a subcase which allows the solution procedure to be completed explicitly. Thus, we set c 0 s a02 , c6 s ipr2, c 3 s c5 s c10 s 0, thereby decoupling the 2-component system into two Pinney equations. The original 3-component Ermakov system Ž78. is then symmetric in the dependent variables a i due to the special structure of the matrix
a 11 Ž a i j . s a 21 a 21
a 12 a 22 a 12
a 13 a 13 . a 33
0
Ž 98 .
The expressions for U2 and U3 reduce to U2 s a0 cosh Ž 2 at . ,
'
U3 s a0 c8 sinh Ž 2 at . ,
'
a0 s a0r'a , Ž 99 .
and the linear equation Ž81. assumes the form of the second Poschl]Teller ¨ equation w52x 1
ž / a1
q 4a 2 tt
m Ž m y 1. cosh Ž 2 at . 2
y
n Ž n y 1. sinh Ž 2 at . 2
y p2
1
ž / a1
s 0 Ž 100 .
on appropriate absorption of constants into m, n, and p. It is interesting to note that the trigonometric version of this equation was discussed by Darboux w53x. It may be solved via the change of variables 1ra1 s h Ž j . cosh m Ž 2 at . sinh n Ž 2 at . ,
j s ysinh 2 Ž 2 at . , Ž 101 .
leading to the hypergeometric equation
j Ž j y 1 . hY q Ž A q B q 1 . j y C hX q ABh s 0
Ž 102.
with A, B s 12 Ž m q n . " 12 p,
C s n q 12 .
Ž 103.
Hence, the general solution of the second Poschl]Teller equation is given ¨ by 1ra1 s g 1cosh m Ž 2 at . sinh n Ž 2 at . F A, B, C, ysinh 2 Ž 2 at . q g 2 cosh m Ž 2 at . sinhŽ1yn. Ž 2 at . =F A y C q 1, B y C q 1, 2 y C, ysinh 2 Ž 2 at . , Ž 104 . where g i are arbitrary constants and F w x are the usual hypergeometric functions.
214
ROGERS AND SCHIEF
The independent variable z is now given in terms of 1ra1 and a second solution 1raU1 with unit Wronskian, viz. z s aU1 ra1 ,
Ž 105.
whence we obtain the solution of the canonical 3-component system Ž78. parametrically via a1 s a1 Ž t . a2 s a1 Ž t . a0 cosh Ž 2 at . a3
' s a Ž t . a 'c sinh Ž 2 at . 1
0
Ž 106.
8
z s a1 Ž t . raU1 Ž t . . Finally, the solution of the original 3-component Ermakov system Ž76. is obtained by means of the transformation Ž77..
6. SEQUENCES OF CLASSICAL ERMAKOV SYSTEMS LINKED VIA DARBOUX TRANSFORMATIONS It has been established that the linearization procedure for N-component Ermakov system leads to linear Schrodinger equations. In general, ¨ the solutions of the latter cannot be given explicitly. It is therefore natural to enquire as to the form of N-component systems which reduce to ‘‘solvable’’ Schrodinger equations. Here, we consider sequences of classical ¨ Ermakov systems which are linked via Darboux transformations. The Darboux theorem w54x provides both potentials and solutions of Schro¨ dinger equations so that the solution of the corresponding Ermakov systems may, in principle, be given in terms of the solution of a ‘‘seed’’ Ermakov system. The first step in this process is to determine the equivalence class of Ermakov systems that is defined by a given linear Schro¨ dinger equation. 6.1. The In¨ erse Procedure Let w be a solution of the Schrodinger equation ¨
ws s s U Ž s . w ,
Ž 107.
where UŽ s . is a given ‘‘potential.’’ If we set s s SŽ q . and introduce f Ž q . and g Ž q . according to f Ž q . s UŽ SŽ q . . ,
g Ž q . s qf Ž q . y
1 2
1
ž / S q2
, q
Ž 108 .
MULTI-COMPONENT ERMAKOV SYSTEMS
215
then the equations Ž107. and Ž108. 2 assume the form of the decoupled system Ž88., namely
ws s s f Ž q . w
Ž 109.
q s s s qf Ž q . y g Ž q . . The change of independent variable dz s wy2 ds
Ž 110.
then takes the system Ž109. to the canonical Ermakov system aY q
f Ž bra. a
3
s 0,
bY q
g Ž bra. a3
s 0,
Ž 111.
where a s 1rw ,
b s qrw .
Ž 112.
It is emphasized that the variable z may be expressed explicitly in terms of w and a second solution with unit Wronskian w U of the Schrodinger ¨ equation Ž107., viz. z s w Urw .
Ž 113.
It is concluded that the Schrodinger equation Ž107. defines canonical ¨ Ermakov systems modulo the arbitrary function S. This implies that Ermakov systems possess a hidden gauge freedom which acts within the equivalence classes given by the underlying Schrodinger equation Ž107.. ¨ Classical Ermakov systems are now generated via the change of variables u Ž t . s aŽ z . f Ž t . ,
¨ Ž t . s bŽ z . f Ž t . ,
z s c Ž t . rf Ž t . , Ž 114 .
where c and f are solutions with unit Wronskian fc˙ y cf˙ of the linear base equation
f¨ q v Ž t . f s 0,
Ž 115.
which takes the canonical system Ž111. to u ¨ q vŽ t. u q ¨¨ q v Ž t . ¨ q
f Ž ¨ ru . u3 g Ž ¨ ru . u3
s0
Ž 116. s 0.
216
ROGERS AND SCHIEF
This inverse procedure ŽSchrodinger ª Ermakov. will now be exploited ¨ to derive sequences of classical Ermakov systems related via Darboux transformations. 6.2. The Darboux Theorem The application of the classical Darboux theorem and its extensions to construct multi-soliton solutions and induce auto-Backlund transforma¨ tions for integrable systems is well-established w55]57x. Here, by contrast, Darboux’s original result may be used to relate classical Ermakov systems. The Darboux theorem reads as follows: DARBOUX THEOREM. The linear Schrodinger equation ¨
lQ s Q x x q w Ž x . Q
Ž 117.
is form-in¨ ariant under the transformation
˜ s Qx y QªQ
Q0 x Q0
Q
Ž 118.
wªw ˜ s w q 2 Ž ln Q 0 . x x , where Q 0 is a solution of the Schrodinger equation Ž117. with parameter l0 . ¨ Crum w58x has shown that the above Darboux transformation can be iterated in a purely algebraic manner so that one has to solve only the Schrodinger equation Ž117. with the seed potential w in order to obtain ¨ ˜ The sequences of new potentials w ˜ and corresponding eigenfunctions Q. explicit compound Darboux transformation is given in: CRUM’S ITERATION FORMULA. n-fold application of the Darboux transformation Ž118. produces the in¨ ariance
˜s QªQ
W Ž Q, Q1 , . . . , Qn . W Ž Q 1 , . . . , Qn .
Ž 119.
wªw ˜ s w q 2 ln W Ž Q1 , . . . , Qn .
xx,
where W Ž . is the usual Wronskian of eigenfunctions Q and Qi associated with the parameters l i . In the present context, the Darboux theorem may be applied in two important and distinct ways. Firstly, if we set x s t,
w s v,
Q s f,
l s 0,
Ž 120.
MULTI-COMPONENT ERMAKOV SYSTEMS
217
the Darboux theorem produces an invariance of the linear base equation Ž115.. Consequently, the Darboux transformation maps the Ermakov system Ž116. to Ermakov systems of the form u ¨ q v˜ Ž t . u q ¨¨ q v ˜Ž t.¨ q
f Ž ¨ ru . u3 g Ž ¨ ru . u3
s0
Ž 121. s 0,
where the functions f and g remain unchanged since the canonical Ermakov system Ž111. is not affected by this transformation. Secondly, the identifications x s s,
Q s w,
w s yU,
l s 0,
Ž 122.
allow an application to the Schrodinger equation Ž107.. Hence, the ¨ Darboux transformation maps the Ermakov system Ž116. to associated Ermakov systems u ¨ q vŽ t. u q ¨¨ q v Ž t . ¨ q
f˜Ž ¨ ru . u3
˜g Ž ¨ ru . u3
s0
Ž 123. s 0.
In this case, only the canonical Ermakov system is transformed so that the base equation Ž115. remains the same. As far as praticability is concerned, an important feature of the Darboux theorem is that its application to the constant coefficient Schrodinger ¨ equation leads to a nontrivial sequence of potentials and corresponding eigenfunctions. Thus, we illustrate the second application of the Darboux theorem by starting with the seed solution U Ž s . s n s const.
Ž 124.
The inverse procedure as given in the previous section then generates the corresponding decoupled classical Ermakov system u ¨ q vŽ t. u q ¨¨ q v Ž t . ¨ q
n u3
s0
g Ž ¨ ru . u3
Ž 125. s 0,
where g is an arbitrary function of its argument. By construction, this Pinney-type system is gauge-equivalent Žin the sense of subsection 6.1. to
218
ROGERS AND SCHIEF
the case g s 0 which is exactly the classical system originally introduced by Ermakov w1x. In order to generate a sequence of related Ermakov systems it is necessary to solve the Schrodinger equations ¨
l i w s ws s y nw
Ž 126.
according to Crum’s iteration formula. Here, we indicate the procedure only for the single Darboux transformation as given in the Darboux theorem. Thus, on inserting the general solution of the above equation for i s 0 into the transformation formula Ž118. 2 we obtain the new potential U˜ s n y
2 m2 cosh 2 Ž m s .
,
m2 s l0 q n ,
Ž 127.
where we have suppressed an irrelevant integration constant. The new functions f˜ and ˜ g in the transformed Ermakov system Ž123. are given by the tilde version of Ž108., namely f˜Ž q . s n y
2 m2 cosh 2 Ž m S Ž q . .
˜g Ž q . s n q y
2 m2 q cosh 2 Ž m S Ž q . .
y
1 2
Ž 128.
1
ž / S q2
. q
The Darboux transformation therefore maps the functions f s n and g to the functions f˜ and ˜ g as given above. Different choices of the arbitrary function S correspond to different ‘‘gauges’’ of the same Ermakov system. It is natural to ask whether the above class of Ermakov system contains a known system for a particular gauge. In this connection, we recall the formula Ž91. associated with the 2-component canonical Ermakov system Ž84., Ž85.. This suggests the ansatz q 2 s r cosh Ž m S Ž q . .
Ž 129.
and in this particular gauge it is seen that f˜Ž q . s n y
2 m2r 2 q4
,
˜g Ž q . s Ž n y m2r4 . q y
9m2r 2r4 q3
, Ž 130 .
MULTI-COMPONENT ERMAKOV SYSTEMS
219
corresponding to the Ermakov system u ¨ q vŽ t. u q ¨¨ q v Ž t . ¨ q
ž
ž
n y m2r4 u4
n u4 y
y
2 m 2r 2 ¨4
9m2r 2r4 ¨4
/ /
us0
Ž 131. ¨ s 0.
We therefore note that the Darboux transformation links the equivalence class of Ermakov systems represented by the original Ermakov system Ž125. gs 0 to an equivalence class of Ermakov systems represented by the three-parameter class of Pinney-type equations Ž131.. Interestingly, for constant v the latter is exactly of the form Ž80. derived in connection with a three-layer fluid system.
ACKNOWLEDGMENTS One of the authors ŽC.R.. acknowledges with gratitude support under an ARC grant.
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0076
Multi-component Ermakov Systems: Structure and Linearization C. Rogers and W. K. Schief School of Mathematics, The Uni¨ ersity of New South Wales, Sydney, New South Wales 2052, Australia Submitted by William F. Ames Received March 30, 1995
A symmetry reduction of a Ž2 q 1.-dimensional nonlinear N-layer fluid model is shown to lead to a prototype for an N-component extension of the classical Ermakov system. The general N-component Ermakov system introduced here has the attractive property that it may be iteratively reduced to a system of N y 2 linear equations augmented by a canonical 2-component Ermakov system. The recently established linearization procedure for the latter may then be used to solve the N-component system N ) 2 in generality. The procedure is illustrated, in detail, for a 3-component system. Sequences of classical 2-component Ermakov systems are shown to be linked via Darboux transformations. Q 1996 Academic Press, Inc.
1. INTRODUCTION The study of the coupled nonlinear ordinary equations now known as Ermakov systems originated in 1880 w1x. In the intervening years there has been an extensive literature devoted to their analysis w2]30x. In terms of applications, Ermakov systems arise, notably, in nonlinear optics w31]35x and nonlinear elasticity w36, 37x. The main theoretical interest in such systems originated in the fact that they admit a novel constant of motion, namely, the celebrated Lewis]Ray]Reid invariant. In a recent key development, it was shown by Athorne et al. w24x that the classical Ermakov system is, in fact, linearizable, that is, C-integrable in the terminology of Calogero and Eckhaus w38]40x. It turns out that the Lewis]Ray]Reid invariant plays a key role in that linearization. The classical Ermakov system consists of a coupled system of ordinary differential equations involving two dependent variables. It is natural to enquire as to the existence of multi-component Ermakov systems which share the recently established C-integrability of the classical two194 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
MULTI-COMPONENT ERMAKOV SYSTEMS
195
component Ermakov system. It turns out that a prototype for such a multi-component system resides in a symmetry reduction of a multi-layer fluid system. Thus, it was recently shown by Rogers et al. w41, 42x that a nonlinear Ž2 q 1.-dimensional shallow water system due to Stoker admits a symmetry reduction to a particular classical Ermakov system. Here, an analogous reduction for an N-layer fluid leads to what may be termed a multi-component Ermakov system. It is demonstrated that such multicomponent nonlinear systems are, like the classical Ermakov system, linearizable.
2. THE CLASSICAL ERMAKOV SYSTEM The classical Ermakov system, as extended by Ray and Reid in 1979, adopts the form w6x u ¨ q vŽ t. u s ¨¨ q v Ž t . ¨ s
f Ž ¨ ru . u2 ¨ g Ž ur¨ . ¨2u
Ž 1. ,
where f, g, and v are arbitrary functions of their indicated arguments and the overdot denotes the derivative drdt. Here, the linearization procedure for the system Ž1. as originally presented in Athorne et al. w24x is reviewed since it is used in our subsequent linearization of the multi-component Ermakov system. Thus, new dependent and independent variables a, b, and z are introduced into Ž1. according to w24x u s aŽ z . f ,
¨ s bŽ z . f ,
z s crf ,
Ž 2.
where c and f are linearly independent solutions with unit Wronskian fc˙ y cf˙ of the base equation
f¨ q v Ž t . f s 0.
Ž 3.
Under Ž2., the original Ermakov system Ž1. is reduced to the autonomous system aY s
f Ž bra. 2
a b
,
bY s
g Ž arb . b2a
.
Ž 4.
It is observed that the Wronskian W Ž u, ¨ . s u¨˙ y ¨ u ˙ is invariant under Ž2. so that W Ž u, ¨ . s w Ž a, b . s abX y baX .
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ROGERS AND SCHIEF
On introduction of the new independent variable t via dt s by2 dz,
Ž 5.
it is seen that the system Ž4. is, in turn, equivalent to the coupled system d2
a
1
dt 2
ž / b
q
b
1
ž /
g Ž arb .
a
2
ž / ž /
dt 2 b d2
b
s
b
a
f Ž bra. y g Ž arb .
Ž 6.
s 0.
In terms of the new canonical group variables 2
x s Ž bra. ,
y s 1rb,
Ž 7.
the system Ž6. yields the first integral 1 dx x dt
s
1 dx xy 2 dz
s h,
Ž 8.
where h Ž x ; I . s 2 x 1r2 2 I q
x
H
1r2
sy3r2 g Ž sy1r2 . y sy1r2 f Ž s 1r2 . d s
Ž 9.
together with the linear equation d dx
dy
ž / xh
dx
q
g Ž xy1r2 . x 1r2 h
y s 0.
Ž 10 .
The relation Ž8. produces the Lewis]Ray]Reid invariant Is
Ž u¨˙ y ¨ u˙. 2
2
q
ur¨
H
g Ž l. d l q
¨ ru
H
f Ž m . dm
Ž 11 .
of the original Ermakov system Ž1.. The solution of the system Ž1. may now, in principle, be readily generated modulo the solution of the linear equation Ž6. 2 or, equivalently, Ž10.. Thus, if f and g are specified, hŽ x; I . is determined via Ž9.. If hŽ x; I . is then substituted into Ž10. to produce, on integration, y s y Ž x ; I, c1 , c 2 . s c1 y 1 Ž x ; I . q c 2 y 2 Ž x ; I .
Ž 12 .
then substitution of Ž12. into Ž8. and integration yields z s z Ž x ; I, c1 , c2 , c 3 . s c Ž t . rf Ž t . ,
Ž 13 .
MULTI-COMPONENT ERMAKOV SYSTEMS
197
where c 3 is an additional constant of integration. The relations Ž7., on the other hand, give as
1
'x
c1 y 1 Ž x ; I . q c 2 y 2 Ž x ; I .
,
bs
1 c1 y 1 Ž x ; I . q c 2 y 2 Ž x ; I .
.
Ž 14 . At this stage, we may proceed to the solution of the original Ermakov system in two ways. Thus, if Ž13. may be inverted to yield t s T Ž x ; I, c1 , c 2 , c 3 .
Ž 15 .
then the original Ermakov variables u, ¨ 4 are given parametrically in terms of x via Ž15. together with the relations us
f Ž T Ž x ; c1 , c 2 , c 3 . .
'x
c1 y 1 Ž x ; I . q c 2 y 2 Ž x ; I .
,
¨s
f Ž T Ž x ; c1 , c 2 , c 3 . . c1 y 1 Ž x ; I . q c 2 y 2 Ž x ; I .
.
Ž 16 . On the other hand, if Ž13. may be inverted to yield x s X Ž crf ; I, c1 , c 2 , c 3 .
Ž 17 .
then we obtain a nonlinear superposition principle for the original Ermakov variables u, ¨ 4 in terms of two linearly independent solutions f , c 4 of the linear base equation Ž3. via us
f
'x
c1 y 1 Ž x ; I . q c 2 y 2 Ž x ; I .
,
¨s
f c1 y 1 Ž x ; I . q c 2 y 2 Ž x ; I .
.
Ž 18 . This generalizes the well-known nonlinear superposition principle for Pinney’s equation w43x retrieved in the decoupled reduction f s ¨ ru, g s 0 of the Ermakov system Ž1.. In what follows, guided by the structure of a symmetry reduction of an N-layer fluid system, a multi-component Ermakov system is introduced which inherits linearization from the classical 2-component Ermakov system.
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3. A SYMMETRY REDUCTION OF A MULTI-LAYER FLUID SYSTEM: AN N-COMPONENT PINNEY SYSTEM The following Ž2 q 1.-dimensional N-layer fluid system is considered:
ui t ¨i t t
Ž hiy1 y hi . q
x
q ui q ui
ui x ¨i x
q ¨i q ¨i
u i Ž hiy1 y hi . q
ui y ¨i
y
y
y f¨ i q q fu i q
pi x pi
s0
s 0, y i s 1, . . . , N
¨ i Ž hiy1 y hi . s 0.
Ž 19 . Here, the quantities u i , ¨ i 4 denote fluid current components and hiy1 s hiy1Ž x, y, t . denotes the surface of the upper boundary of the ith layer. In the hydrostatic approximation adopted here, the pressure terms pi are given by iy1
pi s
Ý g j hj ,
Ž 20 .
js0
where g j represents the buoyancy jump at the interface h s hj Ž x, y, t .. In what follows, the Coriolis parameter f / 0 is scaled to be unity under t ª ft, x ª fx, y ª fy. In w41, 42x, it was shown that an appropriate symmetry reduction of the two-layer system results in a particular classical Ermakov system of the type Ž1.. The linearization procedure for Ž1. may be used to construct analytic solutions which, in particular, provide valuable necessary checks on numerical schemes. Here, an analogous symmetry reduction is sought for the N-layer system Ž19. with a view to discovering a prototype of an N-component Ermakov system. The existence of such systems, until now, has remained an open question. Thus, on introduction of the ansatz u i s xu ˜i Ž t . q y¨˜i Ž t . ,
¨ i s yx¨˜i Ž t . q yu ˜i Ž t . ,
hi s Ž x q y 2 . H˜i Ž t . , 2
Ž 21 .
MULTI-COMPONENT ERMAKOV SYSTEMS
199
into the N-layer system Ž19., we obtain
˙i q 2 u˜i¨˜i y u˜i s 0 ¨˜ iy1
˙˜i y ¨˜i2 q u˜2i q ¨˜i q 2 u
Ý g j H˜j s 0
Ž 22 .
js0
˜iy1 y H˙ ˜i q 4 H˜iy1 y H˜i u˜i s 0 H˙
ž
/
˜N s 0 in the case of a rigid bottom topography. Here, we set with H˙ H˜N s 0. The solution of the nonlinear system Ž22. admits a parametrization in terms of an N-tuple of dependent variables V 1 , . . . , V N 4 governed by the coupled system of the form ¨iq V
1 4
Vi q
ž
ai j
N
Ý
V 4j
js1
/
V i s 0,
Ž 23 .
H˜iy1 y H˜i s c i Vy4 i ,
Ž 24 .
where a i j s const and
˙ irV i , u ˜i s V
¨˜i s 12 y Vy2 i ,
so that N
H˜j s
Ý
c k Vy4 k .
Ž 25 .
ksjq1
On introduction of the dependent variables a i and independent variable z according to V i s ai Ž z . f ,
z s crf ,
Ž 26 .
where f s '2 cosŽ tr2. and c s '2 sinŽ tr2. are solutions with unit Wronskian of the linear base equation
f¨ q 14 f s 0,
Ž 27 .
the system Ž23. reduces to the autonomous form aYi q
ž
N
Ý js1
ai j a4j
/
a i s 0,
Ž 28 .
X where [ drdz. The system Ž28. may be regarded as an N-component Pinney system with Pinney’s equation being retrieved in the case N s 1.
200
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The basic idea in what follows involves linearization of the N-component system Ž28. via an iteration process. Thus, we observe that Ž28. implies a1 aYj y a j aY1 s
a 1 k y a jk
N
ž
Ý
/
a4k
ks1
j s 2, . . . , N,
a1 a j ,
Ž 29 .
so that d2 dt
2
aj
ž / a1
s Ž a 11 y a j1 .
aj
ž / a1
N
Ý Ž a 1 k y a jk .
q
ks2
a1
4
aj
ž /ž / ak
a1
, Ž 30 .
where dt s a1y2 dz.
Ž 31 .
Accordingly, if we set Uj s a jra1 ,
j s 2, . . . , N,
Ž 32 .
the system Ž30. reduces to d 2 Uj dt
2
N
q Ž a j1 y a 11 . Uj q
ž
Ý
a jk y a 1 k Uk4
ks2
/
Uj s 0,
Ž 33 .
together with the linear Schrodinger equation ¨ d2 dt
2
1
N
ž / ž a1
s a 11 q
Ý
a1 k
4 ks2 Uk
1
/ž / a1
Ž 34 .
in 1ra1. It is noted that the method of variation of parameters and the relation Ž31. yield z s a1raU1 ,
Ž 35 .
where 1ra and 1raU are linearly independent solutions of Ž34. with unit Wronskian. If it is now required that
a j1 y a 11 s const [ a ,
j s 2, . . . , N,
Ž 36 .
then on introduction of the new variables bj 4 and w via Uj s bj Ž w . x ,
w s jrx ,
Ž 37 .
MULTI-COMPONENT ERMAKOV SYSTEMS
201
where x and j are solutions with unit Wronskian of the linear base equation
xtt q a x s 0
Ž 38 .
the system Ž33. then reduces to the Ž N y 1.-component Pinney-type system d 2 bj dw
2
N
q
ž
Ý
a jk y a 1 k
/
bk4
ks2
bj s 0.
Ž 39 .
It is interesting to note that the conditions Ž36. hold for the N-layer model Žwith a s 1.. The system Ž39., in turn, yields d2 dt
N
bk
2
ž / b2
q Ž b k 2 y b 22 . q
Ý Ž bk l y b 2 l . ls3
b2
4
bk
ž /ž / bl
b2
s 0,
k s 3, . . . , N,
Ž 40 .
where
bk j s a k j y a 2 j
Ž 41 .
dt s by2 dw. 2
Ž 42 .
and
It is now further required that
b k 2 y b 22 s const [ b ,
k s 3, . . . , N,
Ž 43 .
whence
a 32 s ??? s a N 2 .
Ž 44 .
Again, remarkably, this condition holds for the N-layer fluid system. Therein a k 2 s 2Ž g 0 q g 1 . c 2 and b s 0. The residual equation in b 2 in Ž39. reduces to the linear Schrodinger ¨ equation d2 dt
2
1
ž / b2
N
s Ž a 12 y a 22 . q
Ý Ž a1 k y a2 k . ks3
b2
4
1
ž /ž / bk
b2
Ž 45 .
in 1rb 2 . Consequently, w s b 2rbU2 , where 1rbU2 is a second solution of Ž45. with unit Wronskian. Iteration of the above procedure shows that the original N-component Pinney-type system Ž23. may be sequentially reduced to N y 2 linear
202
ROGERS AND SCHIEF
Schrodinger equations supplemented by a residual classical two-compo¨ nent Ermakov system provided that the constraints
am n s aN n ,
m ) n,
Ž 46 .
hold. The conditions Ž46. are identically satisfied for the N-layer system. The N-component Pinney-type system Ž23. with entries of the matrix Ž a m n . constrained by the condition Ž46. may now, in principle, be solved via the linearization procedure for the classical Ermakov system Ž1.. This, in turn, is equivalent to performing another step in the above iteration process so that the N-component Pinney system is reducible to N y 1 linear Schrodinger equations and integration of the Lewis]Ray]Reid ¨ invariant relation. The procedure will be subsequently illustrated for a 3-component Pinney system. It is emphasized that the sole aim of the present work is to exploit the above symmetry reduction to an N-component Pinney-type system to point the way to the structure of general N-component Ermakov systems. The systematic application of general Lie group analysis to the N-layer model to derive Ball-type theorems w44, 45x and to search for physically significant model solutions remains. However, in terms of physical applications, it is noted that a parametrization analogous to Ž24. and applied to a model in elliptic warm-core eddy theory has recently led to the construction of so-called pulsrodon solutions with accompanying underlying Hamiltonian structure w46, 47x. Lie group theory was used in w48x to reveal the rich Lie-algebraic structure underlying single-layer rotating shallow water models with elliptic and circular paraboloidal bottom topographies. In particular, Ball-type theorems were group-generated and subalgebras of the symmetry algebra systematically investigated. Certain symmetry reductions for a circular paraboloidal basin were shown to lead to exact solutions of the shallow water system wherein the temporal evolution of the moving shoreline is driven by Pinney’s equation. Indeed, it was this observation that led to the present study wherein an analogous symmetry reduction of an N-layer system leads to the N-component Pinney system Ž23..
4. THE N-COMPONENT ERMAKOV SYSTEM Here, we start with the extension of the 3-component Pinney system to what we shall term the 3-component Ermakov system. This will motivate the form of the general N-component Ermakov system which is subsequently introduced.
MULTI-COMPONENT ERMAKOV SYSTEMS
203
Thus, we consider 3-component systems of the form
¨ 1 q v Ž t . V1 s V ¨ 2 q v Ž t . V2 s V ¨ 3 q v Ž t . V3 s V
F 1 Ž V 1rV 3 , V 2rV 3 . V 33 F 2 Ž V 1rV 3 , V 2rV 3 .
Ž 47 .
V 33 F 3 Ž V 1rV 3 , V 2rV 3 . V 33
,
where the Fi are dependent on their indicated arguments. Introduction of the change of dependent and independent variables Vi s ViŽ z. f,
z s crf ,
Ž 48 .
where f and c are solutions with unit Wronskian of the linear base equation
f¨ q v Ž t . f s 0,
Ž 49 .
reduces Ž47. to the autonomous form VYi s
Fi V 1rV 3 , V 2rV 3
ž
/
Ž 50 .
V 33
X
with [ drdz. The latter system, in turn, shows that V 3 VY1 y V i VY3 s V 3 VY2 y V 2 VY3 s
1 V 23 1 V 23
F1 y F2 y
V1 V 33 V2 V 33
F3
Ž 51 . F3 .
If we set u s V 1rV 3 ,
¨ s V 2rV 3 ,
Ž 52 .
then Ž51. becomes d2 u dt 2 d2 ¨ dt 2
s F 1 Ž u, ¨ . y uF 3 Ž u, ¨ .
Ž 53 . s F 2 Ž u, ¨ . y ¨ F 3 Ž u, ¨ . ,
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ROGERS AND SCHIEF
where dt s Vy2 3 dz.
Ž 54 .
If the system Ž53. is now set in correspondence with the classical 2-component Ermakov system in the form d2 u dt
2
d2 ¨ dt 2
q v2 u s q v 2¨ s
u ¨4 ¨
u4
f 2 Ž ur¨ .
Ž 55 . g 2 Ž ur¨ .
with constant v 2 , then the original system Ž47. becomes
¨ 1 q v Ž t . V1 s V
V1
q
¨ 2 q v Ž t . V2 s V
V1 V 43
V2
V2 V 43
1 V 33
F 3 Ž V 1rV 3 , V 2rV 3 . y v 2
g 2 Ž V 1rV 2 .
V 41 q
¨ 3 q v Ž t . V3 s V
f 2 Ž V 1rV 2 .
V 42
Ž 56 .
F 3 Ž V 1rV 3 , V 2rV 3 . y v 2
F 3 Ž V 1rV 3 , V 2rV 3 . ,
where F 3 remains an arbitrary function of its indicated arguments. It is natural to term Ž56. the 3-component Ermakov system since, by construction, it inherits the C-integrability property of the 2-component Ermakov system. Thus, application of the sequence of transformations Ž48., Ž52., and Ž54. reduces Ž56. to the classical 2-component Ermakov system Ž55. augmented by Ž47. 3 . The latter reduces to the linear Schro¨ dinger equation d2 dt
2
1
ž / V3
q F 3 V 1rV 3 , V 2rV 3
ž
/
1
ž / V3
s0
Ž 57 .
in 1rV 3 . Thus, once the 2-component Ermakov system Ž55. has been solved for uŽ t . s V 1rV 3 and ¨ Ž t . s V 2rV 3 , the linear equation Ž57. determines V 3 s V 3 Ž t . and the original independent variable z according to z s V 3rVU3 ,
Ž 58 .
MULTI-COMPONENT ERMAKOV SYSTEMS
205
where 1rVU3 is a second solution of Ž57. with unit Wronskian. The relation Ž58. gives z s z Ž t . and hence, in principle, t s t Ž t . through Ž48. 2 and Ž49.. The solution of the original 3-component Ermakov system is then given parametrically in terms of t via the relations Vi s ViŽ t . f Ž t. ,
t s tŽ t. .
Ž 59 .
The procedure will be illustrated, in detail, in the next section for a particular 3-component Pinney-type system. It is important to remark that the choice of the new independent variable in the reduction of the system Ž47. may be taken to be any of dt s Vy2 dt, i g 1, 2, 34 . Indeed, the choice may be crucial in the determii nation of whether a specified system of the type Ž47. represents a 3-component Ermakov system or not. This point is well-illustrated by the 3-layer fluid model
¨1 q V ¨2q V ¨3q V
ž ž ž
1 4 1 4 1 4
q q q
a1
q
V 43
a2
q
V 41
a3
q
V 42
b1
/ / /
V 42
b2 V 43
b3 V 41
V1 s V2 s V3 s
g1 V 31
g2
Ž 60 .
V 32
g3 V 33
,
where
a 1s2 c 3 g 0 ,
a 2s2 c1 g 0 ,
b 1s2 c 2 g 0 ,
b 2s2 c3 Ž g 0 q g 1 . ,
g 1s1y2 c1 g 0 ,
a 3s2 c 2 Ž g 0 q g 1 . b 3s2 c1 g 0
g 2s1y2 c 2 Ž g 0 q g 1 . ,
Ž 61 .
g 3s1y2 c3 Ž g 0 q g 1 q g 3 . . Thus, with the choice dt s Vy2 3 dt, identification of the 3-layer system with Ž56. yields V2
f2 s Ž g 1 q b3 .
q a 3 y b1
ž / ž / ž / ž / V1
g2 s Ž g 2 q a3 . F3 s g 3 y a3
4
V1
V2
V3 V2
4
y a 2 q b3
4
y b3
V3 V1
4
Ž 62 .
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ROGERS AND SCHIEF
together with the necessary restrictions
a1 s v 2 y g 3 s b2 .
Ž 63 .
Inspection of the relations Ž61. shows that the restriction Ž63. is not satisfied by the 3-layer system Ž60. ] Ž61.. However, on cyclic interchange in Ž56., corresponding to reduction via dt s Vy2 dt, the 3-component 1 Ermakov system
¨ 1 q v V1 s V ¨ 2 q v V2 s V ¨ 3 q v V3 s V
1 V 31 V2 V 43 V3 V 42
F 1 Ž V 2rV 1 , V 3rV 1 . V2
f 2 Ž V 2rV 3 . q
F 1 Ž V 2rV 1 , V 3rV 1 . y v 2
V 41
g 2 Ž V 2rV 3 . q
V3
Ž 64 .
F 1 Ž V 2rV 1 , V 3rV 1 . y v 2
V 41
is obtained. Identification of the 3-layer system Ž60. with Ž64. results in
g1 V 31
g2 V 32
g3 V 33
y a1 y a2 y a3
V1 V 43 V2 V 41 V3 V 42
y b1 y b2 y b3
V1 V 42 V2 V 43 V3 V 41
s s s
1 V 31 V2 V 43 V3 V 42
F1 f2 q g2 q
V2 V 41 V3 V 41
Ž F1 y v 2 .
Ž 65 .
Ž F1 y v 2 . ,
whence we obtain the restrictions
a 2 s v 2 y g 1 s b3 .
Ž 66 .
It is observed that the condition a 2 s b 3 is indeed satisfied by the 3-layer system Ž60. ] Ž61.. Moreover, the choice v 2 s g 1 q a 2 s 1 allows the restrictions Ž66. to be satisfied. This emphasizes the importance of the presence of the v 2-term in the 3-component Ermakov system. In its absence the condition Ž66. would have required that a 2 s yg 1 and this is not the case for the above 3-layer fluid model. It is noted that the remaining cyclic interchange produces the restrictions a 3 s v 2 y g 2 s b 1. The condition a 3 s b 1 is not satisfied by the 3-layer system. Thus, the choice dt s Vy2 dt turns out to be crucial in the reduction of the 3-layer 1 fluid model to a Ermakov system.
MULTI-COMPONENT ERMAKOV SYSTEMS
207
If we now proceed to the 4-component system Fi Ž V 1rV 4 , V 2rV 4 , V 3rV 4 .
¨ i q v Ž t. Vi s V
,
V 34
i s 1, . . . , 4, Ž 67 .
then the requirement that it be reducible to a 3-component Ermakov system of the type Ž64. leads to the 4-component Ermakov system which, in extenso, is V1
¨ 1 q v Ž t . V1 s V
V 42 q
V1 V 44
V2
¨ 2 q v Ž t . V2 s V
V 41 q
V 44
V 33 q
¨ 4 q v Ž t . V4 s V
V 44
V 34
V 43
h 3 Ž V 1rV 3 , V 2rV 3 . y v 2
V2 V 43
h 3 Ž V 1rV 3 , V 2rV 3 . y v 2
F4 Ž V 1rV 4 , V 2rV 4 , V 3rV 4 . y v 3
Ž 68 .
h 3 Ž V 1rV 3 , V 2rV 3 .
V3
1
V1
F4 Ž V 1rV 4 , V 2rV 4 , V 3rV 4 . y v 3
g 2 Ž V 1rV 2 . q
V2
1
¨ 3 q v Ž t . V3 s V
f 2 Ž V 1rV 2 . q
F4 Ž V 1rV 4 , V 2rV 4 , V 3rV 4 . y v 3
F4 Ž V 1rV 4 , V 2rV 4 , V 3rV 4 . ,
where h 3 is an arbitrary function of its indicated arguments. The form of the N-component Ermakov system is now clear, namely
¨ i q v Ž t. Vi s V
FiŽ N . Ž V 1rV N , . . . , V Ny1rV N . V 3N
,
i s 1, . . . , N, Ž 69 .
where F kŽ n. V 3n F 1Ž1.
s s
F kŽ ny1. V 3ny1 F 1Ž1.
q
Vk V 4n
Ž V 1rV 2 . ,
FnŽ n. Ž V 1rV n , . . . , V ny1rV n . y v ny1
Ž 70 .
n s 2, . . . , N ; k s 1, . . . , n y 1.
It is noted that FnŽ n., n s 1, . . . , N are arbitrary functions of their indicated arguments. The cases N s 1, 2 correspond to the classical 1-component
208
ROGERS AND SCHIEF
Ermakov ŽPinney. equation and 2-component classical Ermakov system, respectively. The N-component Ermakov system determined by Ž69., Ž70. may be sequentially reduced to a classical 2-component Ermakov system augmented by N y 2 linear equations. Thus, the N-component system yields Vk
d2 dt
2
ž / VN
s F kŽ N . y
Vk VN
FNŽ N . ,
k s 1, . . . , N y 1,
Ž 71 .
together with the linear Schrodinger equation ¨ d2 dt
1
2
ž / VN
q FNŽ N .
1
ž /
s 0,
VN
Ž 72 .
where V N and t are introduced via Vi s ViŽ z. f,
dt s Vy2 N dz,
z s crf ,
Ž 73 .
and f and c are solutions with unit Wronskian of Ž49.. Thus, the N-component Ermakov system reduces to d2 dt 2 d2 dt 2
Vk
ž / ž / VN 1
VN
q v Ny 1 q FNŽ N .
Vk
ž / ž / VN 1
VN
s
F kŽ Ny1. V 3Ny1
Ž 74 .
s 0.
The system Ž74.1 is an Ž N y 1.-component Ermakov system for the quantities V krV N with F kŽ Ny1. Ž V lrV Ny1 . s F kŽ Ny1. Ž w V lrV N x r w V Ny1rV N x . .
Ž 75 .
Iteration of the procedure reduces the N-component Ermakov system to a classical 2-component Ermakov system augmented by N y 2 linear Schrodinger equations. Accordingly, the N-component Ermakov system is ¨ linearizable. The solution procedure is illustrated for a 3-component Ermakov system in the next section.
5. ILLUSTRATION: A 3-COMPONENT ERMAKOV SYSTEM Here, we consider a 3-component Ermakov system of the type
¨ i q v2Vi q V
ž
3
Ý js1
ai j V 4j
/
V i s 0,
i s 1, 2, 3,
Ž 76 .
MULTI-COMPONENT ERMAKOV SYSTEMS
209
where v is a constant. On introduction of the transformation V i s ai Ž z .
cos Ž v t .
'v
z s tan Ž v t . ,
,
Ž 77 .
the system Ž76. reduces to the autonomous form aYi q
ž
3
Ý js1
ai j a4j
/
ai s 0
Ž 78 .
X
with [ drdz. In terms of the new independent variables Uj s a jra1 ,
j s 2, 3,
Ž 79 .
the system Ž78., in turn, reduces to the 2-component classical Ermakov system dU2 dt
2
dU3 dt 2
q Ž a 21 y a 11 . U2 q q Ž a 31 y a 11 . U3 q
ž ž
a 22 y a 12 U24
a 32 y a 12 U24
a 23 y a 13
q
U34
a 33 y a 13
q
U34
/ /
U2 s 0
Ž 80 . U3 s 0
augmented by the linear equation d2 dt
1
ž /
2
a1
s a 11 q
a 12 U24
q
a 13 U34
1
ž / a1
Ž 81 .
in 1ra1. In the above, dt s a1y2 dz and it is required that
a 21 y a 11 s a 31 y a 11 \ ya 2
Ž 82 .
as indicated in the previous section. The 2-component Ermakov system Ž80. as derived in w42x has been recently investigated for a 22 / a 32 n a 23 / a 33 by Athorne w29x. It is interesting to note that a particular subcase was earlier derived by Wagner et al. in w31x in connection with the self-trapping of optical beams. Therein, a numerical treatment of the Ermakov system was presented. Here, by contrast, we focus on the general case in order to illustrate the solution procedure of the 3-component Ermakov system. Thus, new dependent and independent variables a, b, and s are introduced according to U2 s a Ž s .
cosh Ž at .
'a
,
U3 s b Ž s .
cosh Ž at .
'a
,
s s tanh Ž at . , Ž 83 .
210
ROGERS AND SCHIEF
so that the system Ž80. reduces to the autonomous form as s q
f Ž bra. a
3
s 0,
bs s q
g Ž bra. a3
s 0,
Ž 84 .
where f Ž q . s c1 q
c2 q
4
,
g Ž q . s c3 q q
c4 q
3
qs
,
b a
,
Ž 85 .
with the identifications c1 s a 22 y a 12 , c 2 s a 23 y a 13 , c 3 s a 32 y a 12 , c 4 s a 33 y a 13 . Ž 86 . The solution procedure presented in Section 2 may now be applied to the canonical system Ž84. leading to the linear second order equation Ž10.. This equation may subsequently be transformed into a Schrodinger equa¨ tion by changing the independent variable appropriately. This process is here performed directly by introducing a new independent variable via d s s ay2 ds
Ž 87 .
to produce the decoupled system
fss s f Ž q . f
Ž 88 .
qss s qf Ž q . y g Ž q . , where
f s 1ra,
q s bra.
Ž 89 .
The solution of the autonomous equation Ž88. 2 can now be given implicitly in terms of two quadratures, namely 2s Ž q. s
1
H 'Hqf Ž q . y g Ž q . dq
dq,
Ž 90 .
while Ž88.1 represents a linear Schrodinger equation with potential f Ž q Ž s ... ¨ In the present case, it is possible to invert the relation Ž90. although certain subcases need to be considered. Thus, the case c1 / c 3 leads to the explicit formula q2 s
)
c52 y c 42 c13
cosh D q
c5
'c
13
,
D s 2 c13 Ž s q c6 . , Ž 91 .
'
MULTI-COMPONENT ERMAKOV SYSTEMS
211
where c5 Žproportional to the Lewis]Ray]Reid invariant I . and c6 are two constants of integration and c13 s c1 y c 3 , c 42 s c 4 y c 2 . Hence, the Schrodinger equation Ž88.1 assumes the explicit form ¨
fss s c1 q
fss s c1 q
fss s c1 q
fss s c1 q
c 2 c13
ž'
c52
y c 42 cosh D q c5 c2 c13
Ž
1 2
exp D q c5 .
2
2
2
c1 / c 3 , c 42 / c52
c1 / c 3 , c 42 s c52
Ž 92 .
c52 Ž s q c6 . y c 42
4 c 42 Ž s q c6 .
f,
/
f,
c 2 c52
c2
2
f,
2
f,
c1 s c 3 , c 5 / 0
c1 s c 3 , c5 s 0.
For c13 s yc 2 the ‘‘generic’’ case Ž92.1 turns out to be the Ince equation. The problems of stability and periodicity of the solutions of the 2-component Ermakov system Ž80. reduce to those for this linear equation w29x. A particular case is given by c5 s 0 which can be solved explicitly in terms of hypergeometric functions. Interestingly, these hypergeometric functions reduce to rationals if 4 c 2rŽ c 42 y c52 . s lŽ l q 1., where l is an integer. In this case, the solutions of the Schrodinger equation Ž92.1 may also be ¨ derived via a composite Darboux transformation of the order l. This observation provides a point of contact with Darboux transformations. The generation of sequences of classical Ermakov systems by means of Darboux transformations will be the subject of the following section. Another important subcase is represented by c 2 s y3c 3 , c 4 s y3c1 , which makes the 2-component Ermakov system Hamiltonian w49x. Hamiltonian Ermakov systems have been recently derived in connection with the propagation of elliptic Gaussian beams in nonlinear optics w35x. It can be shown w50x that Ermakov systems which are based on Hamiltonians that are quadratic in the conjugate momenta are solvable in terms of quadratures. Finally, Ž92. 2 is a degenerate case of the generic case, whereas the remaining equations Ž92. 3, c1s0 and Ž92.4 may be solved explicitly in terms of trigonometricrhyperbolic and Bessel functions respectively w51x. To illustrate the remaining steps in the solution procedure, we choose the simplest nontrivial case c 2 s 0, c1 / c 3 . This produces the decoupled
212
ROGERS AND SCHIEF
Pinney system dU2
y a 2 U2 y
dt 2 dU3
y a U3 q 2
dt 2
ž
c3
q
U24
c4 U34
c 02
s0
U23
/
Ž 93 .
U3 s 0,
where c1 s yc02 . It is emphasized that this does not mean that the original 3-component Ermakov system Ž76. decouples. The Schrodinger ¨ equation Ž92.1 may then be solved to produce as
a0 cos Ž c 0 s q c 7 .
,
Ž 94 .
whence, on integration of Ž87., ss
a20 c0
tan Ž c 0 s q c 7 . y c10 .
Ž 95 .
If we invert this relation and insert s into a, q as given by Ž94., Ž91. we finally obtain explicit formulae for the original variables U2 , U3 by means of the transformation Ž83., viz. U2 s
U3 s
) (
a20
a
q
c 02
a a02
c8 cosh
2 c0
2 Ž s q c10 . cosh Ž at .
'
c13 arctan
ž
c0 a20
Ž 96 . Ž s q c10 . q c6 q c9 U2
/
s s tanh Ž at . , where c8 s
)
c52 y c 4 c13
,
c9 s
c5
'c
.
Ž 97 .
13
It is readily seen that this class of solutions possesses the correct number of arbitrary constants, namely a0 , c6 , c9 , and c10 . The remaining constants are given in terms of the coefficients c1 , c 3 , c 4 contained in the 2-component system Ž93.. There remains the problem of solving the linear equation Ž81.. It is evident that, in general, the potential in this Schrodinger equation is ¨
MULTI-COMPONENT ERMAKOV SYSTEMS
213
complicated. Accordingly, we here restrict our attention to a subcase which allows the solution procedure to be completed explicitly. Thus, we set c 0 s a02 , c6 s ipr2, c 3 s c5 s c10 s 0, thereby decoupling the 2-component system into two Pinney equations. The original 3-component Ermakov system Ž78. is then symmetric in the dependent variables a i due to the special structure of the matrix
a 11 Ž a i j . s a 21 a 21
a 12 a 22 a 12
a 13 a 13 . a 33
0
Ž 98 .
The expressions for U2 and U3 reduce to U2 s a0 cosh Ž 2 at . ,
'
U3 s a0 c8 sinh Ž 2 at . ,
'
a0 s a0r'a , Ž 99 .
and the linear equation Ž81. assumes the form of the second Poschl]Teller ¨ equation w52x 1
ž / a1
q 4a 2 tt
m Ž m y 1. cosh Ž 2 at . 2
y
n Ž n y 1. sinh Ž 2 at . 2
y p2
1
ž / a1
s 0 Ž 100 .
on appropriate absorption of constants into m, n, and p. It is interesting to note that the trigonometric version of this equation was discussed by Darboux w53x. It may be solved via the change of variables 1ra1 s h Ž j . cosh m Ž 2 at . sinh n Ž 2 at . ,
j s ysinh 2 Ž 2 at . , Ž 101 .
leading to the hypergeometric equation
j Ž j y 1 . hY q Ž A q B q 1 . j y C hX q ABh s 0
Ž 102.
with A, B s 12 Ž m q n . " 12 p,
C s n q 12 .
Ž 103.
Hence, the general solution of the second Poschl]Teller equation is given ¨ by 1ra1 s g 1cosh m Ž 2 at . sinh n Ž 2 at . F A, B, C, ysinh 2 Ž 2 at . q g 2 cosh m Ž 2 at . sinhŽ1yn. Ž 2 at . =F A y C q 1, B y C q 1, 2 y C, ysinh 2 Ž 2 at . , Ž 104 . where g i are arbitrary constants and F w x are the usual hypergeometric functions.
214
ROGERS AND SCHIEF
The independent variable z is now given in terms of 1ra1 and a second solution 1raU1 with unit Wronskian, viz. z s aU1 ra1 ,
Ž 105.
whence we obtain the solution of the canonical 3-component system Ž78. parametrically via a1 s a1 Ž t . a2 s a1 Ž t . a0 cosh Ž 2 at . a3
' s a Ž t . a 'c sinh Ž 2 at . 1
0
Ž 106.
8
z s a1 Ž t . raU1 Ž t . . Finally, the solution of the original 3-component Ermakov system Ž76. is obtained by means of the transformation Ž77..
6. SEQUENCES OF CLASSICAL ERMAKOV SYSTEMS LINKED VIA DARBOUX TRANSFORMATIONS It has been established that the linearization procedure for N-component Ermakov system leads to linear Schrodinger equations. In general, ¨ the solutions of the latter cannot be given explicitly. It is therefore natural to enquire as to the form of N-component systems which reduce to ‘‘solvable’’ Schrodinger equations. Here, we consider sequences of classical ¨ Ermakov systems which are linked via Darboux transformations. The Darboux theorem w54x provides both potentials and solutions of Schro¨ dinger equations so that the solution of the corresponding Ermakov systems may, in principle, be given in terms of the solution of a ‘‘seed’’ Ermakov system. The first step in this process is to determine the equivalence class of Ermakov systems that is defined by a given linear Schro¨ dinger equation. 6.1. The In¨ erse Procedure Let w be a solution of the Schrodinger equation ¨
ws s s U Ž s . w ,
Ž 107.
where UŽ s . is a given ‘‘potential.’’ If we set s s SŽ q . and introduce f Ž q . and g Ž q . according to f Ž q . s UŽ SŽ q . . ,
g Ž q . s qf Ž q . y
1 2
1
ž / S q2
, q
Ž 108 .
MULTI-COMPONENT ERMAKOV SYSTEMS
215
then the equations Ž107. and Ž108. 2 assume the form of the decoupled system Ž88., namely
ws s s f Ž q . w
Ž 109.
q s s s qf Ž q . y g Ž q . . The change of independent variable dz s wy2 ds
Ž 110.
then takes the system Ž109. to the canonical Ermakov system aY q
f Ž bra. a
3
s 0,
bY q
g Ž bra. a3
s 0,
Ž 111.
where a s 1rw ,
b s qrw .
Ž 112.
It is emphasized that the variable z may be expressed explicitly in terms of w and a second solution with unit Wronskian w U of the Schrodinger ¨ equation Ž107., viz. z s w Urw .
Ž 113.
It is concluded that the Schrodinger equation Ž107. defines canonical ¨ Ermakov systems modulo the arbitrary function S. This implies that Ermakov systems possess a hidden gauge freedom which acts within the equivalence classes given by the underlying Schrodinger equation Ž107.. ¨ Classical Ermakov systems are now generated via the change of variables u Ž t . s aŽ z . f Ž t . ,
¨ Ž t . s bŽ z . f Ž t . ,
z s c Ž t . rf Ž t . , Ž 114 .
where c and f are solutions with unit Wronskian fc˙ y cf˙ of the linear base equation
f¨ q v Ž t . f s 0,
Ž 115.
which takes the canonical system Ž111. to u ¨ q vŽ t. u q ¨¨ q v Ž t . ¨ q
f Ž ¨ ru . u3 g Ž ¨ ru . u3
s0
Ž 116. s 0.
216
ROGERS AND SCHIEF
This inverse procedure ŽSchrodinger ª Ermakov. will now be exploited ¨ to derive sequences of classical Ermakov systems related via Darboux transformations. 6.2. The Darboux Theorem The application of the classical Darboux theorem and its extensions to construct multi-soliton solutions and induce auto-Backlund transforma¨ tions for integrable systems is well-established w55]57x. Here, by contrast, Darboux’s original result may be used to relate classical Ermakov systems. The Darboux theorem reads as follows: DARBOUX THEOREM. The linear Schrodinger equation ¨
lQ s Q x x q w Ž x . Q
Ž 117.
is form-in¨ ariant under the transformation
˜ s Qx y QªQ
Q0 x Q0
Q
Ž 118.
wªw ˜ s w q 2 Ž ln Q 0 . x x , where Q 0 is a solution of the Schrodinger equation Ž117. with parameter l0 . ¨ Crum w58x has shown that the above Darboux transformation can be iterated in a purely algebraic manner so that one has to solve only the Schrodinger equation Ž117. with the seed potential w in order to obtain ¨ ˜ The sequences of new potentials w ˜ and corresponding eigenfunctions Q. explicit compound Darboux transformation is given in: CRUM’S ITERATION FORMULA. n-fold application of the Darboux transformation Ž118. produces the in¨ ariance
˜s QªQ
W Ž Q, Q1 , . . . , Qn . W Ž Q 1 , . . . , Qn .
Ž 119.
wªw ˜ s w q 2 ln W Ž Q1 , . . . , Qn .
xx,
where W Ž . is the usual Wronskian of eigenfunctions Q and Qi associated with the parameters l i . In the present context, the Darboux theorem may be applied in two important and distinct ways. Firstly, if we set x s t,
w s v,
Q s f,
l s 0,
Ž 120.
MULTI-COMPONENT ERMAKOV SYSTEMS
217
the Darboux theorem produces an invariance of the linear base equation Ž115.. Consequently, the Darboux transformation maps the Ermakov system Ž116. to Ermakov systems of the form u ¨ q v˜ Ž t . u q ¨¨ q v ˜Ž t.¨ q
f Ž ¨ ru . u3 g Ž ¨ ru . u3
s0
Ž 121. s 0,
where the functions f and g remain unchanged since the canonical Ermakov system Ž111. is not affected by this transformation. Secondly, the identifications x s s,
Q s w,
w s yU,
l s 0,
Ž 122.
allow an application to the Schrodinger equation Ž107.. Hence, the ¨ Darboux transformation maps the Ermakov system Ž116. to associated Ermakov systems u ¨ q vŽ t. u q ¨¨ q v Ž t . ¨ q
f˜Ž ¨ ru . u3
˜g Ž ¨ ru . u3
s0
Ž 123. s 0.
In this case, only the canonical Ermakov system is transformed so that the base equation Ž115. remains the same. As far as praticability is concerned, an important feature of the Darboux theorem is that its application to the constant coefficient Schrodinger ¨ equation leads to a nontrivial sequence of potentials and corresponding eigenfunctions. Thus, we illustrate the second application of the Darboux theorem by starting with the seed solution U Ž s . s n s const.
Ž 124.
The inverse procedure as given in the previous section then generates the corresponding decoupled classical Ermakov system u ¨ q vŽ t. u q ¨¨ q v Ž t . ¨ q
n u3
s0
g Ž ¨ ru . u3
Ž 125. s 0,
where g is an arbitrary function of its argument. By construction, this Pinney-type system is gauge-equivalent Žin the sense of subsection 6.1. to
218
ROGERS AND SCHIEF
the case g s 0 which is exactly the classical system originally introduced by Ermakov w1x. In order to generate a sequence of related Ermakov systems it is necessary to solve the Schrodinger equations ¨
l i w s ws s y nw
Ž 126.
according to Crum’s iteration formula. Here, we indicate the procedure only for the single Darboux transformation as given in the Darboux theorem. Thus, on inserting the general solution of the above equation for i s 0 into the transformation formula Ž118. 2 we obtain the new potential U˜ s n y
2 m2 cosh 2 Ž m s .
,
m2 s l0 q n ,
Ž 127.
where we have suppressed an irrelevant integration constant. The new functions f˜ and ˜ g in the transformed Ermakov system Ž123. are given by the tilde version of Ž108., namely f˜Ž q . s n y
2 m2 cosh 2 Ž m S Ž q . .
˜g Ž q . s n q y
2 m2 q cosh 2 Ž m S Ž q . .
y
1 2
Ž 128.
1
ž / S q2
. q
The Darboux transformation therefore maps the functions f s n and g to the functions f˜ and ˜ g as given above. Different choices of the arbitrary function S correspond to different ‘‘gauges’’ of the same Ermakov system. It is natural to ask whether the above class of Ermakov system contains a known system for a particular gauge. In this connection, we recall the formula Ž91. associated with the 2-component canonical Ermakov system Ž84., Ž85.. This suggests the ansatz q 2 s r cosh Ž m S Ž q . .
Ž 129.
and in this particular gauge it is seen that f˜Ž q . s n y
2 m2r 2 q4
,
˜g Ž q . s Ž n y m2r4 . q y
9m2r 2r4 q3
, Ž 130 .
MULTI-COMPONENT ERMAKOV SYSTEMS
219
corresponding to the Ermakov system u ¨ q vŽ t. u q ¨¨ q v Ž t . ¨ q
ž
ž
n y m2r4 u4
n u4 y
y
2 m 2r 2 ¨4
9m2r 2r4 ¨4
/ /
us0
Ž 131. ¨ s 0.
We therefore note that the Darboux transformation links the equivalence class of Ermakov systems represented by the original Ermakov system Ž125. gs 0 to an equivalence class of Ermakov systems represented by the three-parameter class of Pinney-type equations Ž131.. Interestingly, for constant v the latter is exactly of the form Ž80. derived in connection with a three-layer fluid system.
ACKNOWLEDGMENTS One of the authors ŽC.R.. acknowledges with gratitude support under an ARC grant.
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