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to generate families of exact solutions of linear and nonlinear Schr?dinger equations.

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1 Introduction

Higher order symmetry operators (SOs) have many important applications in modern

mathematical physics. These operators correspond to hidden symmetries of partial

di?erential equations, including Lie–B?cklund symmetries [1, 2], as well as super- and

a

parasupersymmetries [3–7].

Higher order SOs can be used to construct new conservation laws which cannot

be found in the classical Lie approach [3, 8]. These operators are applied to separate

variables [9]. Moreover, one should use SOs whose order is higher than the order of

the equation whose variables are separated [10].

In the present paper we investigate higher order SOs of the Schr?dinger equation,

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which are “non-Lie symmetries” [8, 11]. The simplest non-Lie symmetries are consi-

dered in detail and all related SOs are explicitly calculated. The potentials admitting

these symmetries are found as solutions of the corresponding nonlinear compatibility

conditions. It is shown that the higher order SOs extend the class of potentials which

were previously obtained in the Lie symmetry analysis.

Algebraic properties of higher order SOs are investigated and used to construct

exact solutions of the linear and related nonlinear Schr?dinger equations. We propose

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a new method to generate extended families of exact solutions by using both the

conditional symmetries [8, 12–14] and higher order SOs.

The Schr?dinger equation with a time-independent potential V = V (x) is studied

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mainly. Time-dependent potentials V = V (t, x) are discussed brie?y in Section 6.

By this, we recover the old result [15] connected with the Lax representation for

the Boussinesq equation, and generate some other nonlinear equations admitting this

representation.

The distinguishing feature of our approach is that coe?cients of symmetry opera-

tors and the corresponding potentials are de?ned as solutions of di?erential equations

which can be easily generalized to the case of multidimensional Schr?dinger equation

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contrary to the method of inverse scattering problem.

J. Math. Phys., 1997, 38, ¹ 11, P. 5944–5959; Preprint ASI-TPA/9/96, Arnold-Sommerfeld-

Institute for Mathematical Physics, Germany, 1996, 23 p.

Higher symmetries and exact solutions of Schr?dinger equation

o 143

This paper continues (and in some sense completes) our works [16–18] where non-

Lie symmetries of the Schr?dinger equation were considered. A detailed analysis of

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higher symmetries of multidimensional Schr?dinger equations will be a subject of our

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subsequent paper.

2 Symmetry operators of the Schr?dinger equation

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Let us formulate the concept of higher order SO for the Schr?dinger equation

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L?(t, x) = 0, L = i?t ? H,

(2.1)

1 ? ?

??x + U (x) , ?t ? , ?x ?

2

H= .

2 ?t ?x

In every sense of the word, a SO of equation (2.1) is any (linear, nonlinear, di-

?erential, integro-di?erential, etc.) operator Q transforming solutions into solutions.

Restricting ourselves to linear di?erential operators of ?nite order n we represent Q

in the form

n

(hi · p)i , (hi · p)i = {(hi · p)i?1 , p}, (hi · p)0 = hi , (2.2)

Q=

i=0

where hi are unknown functions of (t, x), {A, B} = AB + BA, p = ?i?x .

1

p2 + U

Operator (2.2) includes no derivatives w.r.t. t which can be expressed as 2

on the set of solutions of equation (2.1).

De?nition [8]. Operator (2.2) is a SO of order n of equation (2.1) if

(2.3)

[Q, L] = 0.

Remark. The more general invariance condition [3] [Q, L] = ?Q L, where ?Q is

a linear operator, reduces to relation (2.3) if L and Q are operators de?ned in (2.1),

?

(2.2). Terms proportional to i ?t cannot appear as a result of commutation of Q and L;

hence, without loss of generality, ?Q = 0.

For n = 1, 2 SOs (2.2) reduce to di?erential operators of the ?rst order and can

be interpreted as generators of the invariance group of the equation in question. For

n > 2 these operators (which we call higher order SO) correspond to non-Lie [8, 11]

symmetries.

The Lie symmetries of equation (2.1) were described in Refs. [19–21]. The general

form of potentials admitting nontrivial (i.e., distinct from time displacements) sym-

metries is as follows

a3

U = a0 + a1 x + a2 x2 + (2.4)

,

(x + a4 )2

where a0 , . . . , a4 are arbitrary constants. No other potentials admitting local invari-

ance groups exist.

Group properties of equation (2.1) with potentials (2.4) were used to solve the

equation exactly, to establish connections between equations with di?erent potentials,

to separate variables, etc. [9]. Unfortunately, all these applications are valid for a very

restricted class of potentials given by formula (2.4).

144 W.I. Fushchych, A.G. Nikitin

The class of admissible potentials can be essentially extended if we require that

equation (2.1) admits higher order SOs [17]. The problem of describing such potentials

(and the corresponding SOs) reduces to solving operator equations (2.2), (2.3). Eva-

luating the commutators and equating the coe?cients for linearly independent diffe-

rentials we arrive at the following system of determining equations (which is valid for

arbitrary n) [5]:

?x hn = 0, ?x hn?1 + 2?t hn = 0,

?x hn?m + 2?t hn?m+1 ?

[ m?2 ]

2

2(n ? m + 2 + 2k)!

? (?1)k 2k+1

hn?m+2k+2 ?x U = 0, (2.5)

(2k + 1)!(n ? m + 1)!

k=0

[ n?1 ]

2

(?1)p+1 h2p+1 ?x U = 0,

2p+1

?t h 0 +

p=0

where m = 2, 3, . . . , n, and [y] is the entire part of y.

Formulae (2.5) de?ne a system of nonlinear equations in hi and U . For n = 2 the

general solution for U is given by formula (2.4).

Let us consider the case n = 3, which corresponds to the simplest non-Lie sym-

metry, in more detail. The corresponding system (2.5) reduces to

? ?

2h2 + h1 ? 6h3 U = 0, (2.6a)

h3 = 0, h2 + 2h3 = 0,

? ?

2h1 + h0 ? 4h2 U = 0, h0 ? h1 U + h3 U (2.6b)

= 0,

where the dots and primes denote derivatives w.r.t. t and x respectively.

Excluding h0 from (2.6b) and using (2.6a) we arrive at the following equation:

?

F (a, b, c; U, x) ? aU

? (2?x2 + 6aU + c ? 2bx)U ?

a

(2.7)

...

?

? 6(2?x + aU ? b)U ? 12?U ? 2(2?t ax2 ? 2 b x + c) = 0,

4

a a ?

where a, b, c are arbitrary functions of t.

Equation (2.7) is nothing but the compatibility condition for system (2.6). If the

potential U satis?es (2.7) then the corresponding coe?cients of the SO have the form

h2 = ?2ax + b,

h3 = a, ? h1 = g1 + 6aU,

(2.8)

4 ... 3

a x + 2? 2 ? 2cx ? 4a? + 4(b ? 2ax)U + d,

h0 = ? bx ? ? ?

3

where

?

g1 = 2?x2 ? 2bx + c, (2.9)

a ?= U dx, u=?, d = d(t).

3 Equations for potential

Equation (2.7) was obtained earlier [17] (see Ref. [22]) and, moreover, particular

solutions for U were found [17]. Here we analyze this equation in detail.

Higher symmetries and exact solutions of Schr?dinger equation

o 145

First of all, let us reduce the order of equation (2.7). Integrating it twice w.r.t. x

and choosing the new dependent variable ? de?ned in (2.9) we obtain

1 4 4 2 ... 3

a[? ? 3(? )2 ] ? (g1 ?) = ?t ax ? b x + cx2 + dx + e. (3.1)

?

3 3

Using the fact that ? depends on x only while a, b, c, d, e are functions of t, it is

possible to separate variables in (3.1). Indeed, dividing any term of (3.1) by a = 0,

di?erentiating w.r.t. t and integrating over x we obtain the following consequence

g1 a ? g1 a

? ? 1 1 4 5 1 ... 4 1 2 1 2

? ax ? b x + cx + dx + ex + f (3.2)

? = ?t ? .

15 t

a2 a 6 3 2

Consider equation (3.2) separately in two following cases:

g1 a ? g1 a = 0, (3.3a)

? ?

g1 a ? g1 a = 0. (3.3b)

? ?

Let condition (3.3a) be valid. Then dividing the l.h.s. and r.h.s. of (3.2) by ?t (g1 /a)

we come to the following general expression for ?

?4 ?1 x + ?2

? = ?3 x3 + ?2 x2 + ?1 x + ?0 + (3.4)

+2 ,

x + ?5 x + ?3 x + ?4

where ?0 , . . . , ?5 , ?1 , . . . , ?4 are constants.

It is possible to verify by a straightforward but cumbersome calculation that relati-

on (3.4) is compatible with (3.1) only for ?1 = ?2 = 0. We will not analyze solutions

(3.4) inasmuch as they correspond to potentials (2.4) and to SOs which are products

of the usual Lie symmetries [19–21].

If condition (3.3a) is valid, we obtain from equation (3.2)

? (3.5)

a = ak1 ,

? b = k2 a, c = k3 a,

where k1 , k2 , k3 are arbitrary constants. The corresponding equation (3.1) reduces to

? ? 3(? )2 ? (G ?) = 2k1 G + k4 x + k5 , (3.6)

where

1 1 1

k1 x4 ? k2 x3 + k3 x2 , G = g1 = 2k1 x2 ? 2k2 x + k3 , (3.7)

G=

6 3 2

k4 and k5 are constants.

Let us prove that, up to equivalence, equation (3.6) can be reduced to one of the

following forms:

U ? 3U 2 + 3?1 = 0, (3.8a)

U ? 3U 2 ? 8?2 x = 0, (3.8b)

(U ? 3U 2 ) ? 2?3 (xU + 2U ) = 0, (3.8c)

124

? ? 3(? )2 ? 2?4 (x2 ?) = (3.8d)

? x + ?5 , U =?,

34

146 W.I. Fushchych, A.G. Nikitin

where ?1 , . . . ?5 are arbitrary constants. Indeed, by using invertible transformations

? > ? + C1 x + C2 , x > x + C3 , (3.9)

where Ck (k = 1, 2, 3) are constants, it is possible to simplify the r.h.s. of (3.6). These

transformations cannot change the order of polynomial G, and so there exist four

nonequivalent possibilities:

(3.10a)

k1 = 0, k2 = 0, k4 = 0,

(3.10b)

k1 = 0, k2 = 0, k4 = 0,

(3.10c)

k1 = 0, k2 = 0,

(3.10d)

k1 = 0.

Setting in (3.9)

1 12

C1 = ? k3 , k5 ? (3.11a)

C2 = C3 = 0, k = ?1 ,

12 3

6

2

1 k5 k3

C1 = ? k3 , C3 = ? + (3.11b)

C2 = 0, , k4 = 8?2 ,

6 k4 12k4

2

k4 k5 3k4 k3 k4 k3 3k4

k2 = ??3 , (3.11c)

C1 = , C2 = + 3 + 8k 2 , C3 = + 2,

4k2 2k2 32k2 2k2 4k2

2

2 3

1 k2 k4 k2 k3 k2

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