GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 3, 1995, 237-240

ON THE DARBOUX TRANSFORMATION. I

VERONIKA CHRASTINOVÁ

Abstract. Automorphisms of the family of all Sturm-Liouville equations y′′ = qy are investigated. The classical Darboux transformation arises as a particular case of a general result.

1. The Darboux theorem. Let nonvanishing functions y = y(x), z = z(x) be solutions of the Sturm-Liouville equations

y′′ = p(x)y, z′′ = q(x)z,  (1)

where the potentials p = p(x), q = q(x) differ by a constant: p − q = λ ∈ ℝ. Then the function ȳ = y′ − yz′/z satisfies the Sturm-Liouville equation

ȳ′′ = p̄(x)ȳ (p̄ = p − 2(\frac{z′}{z})′).  (2)

For a direct proof, various modifications and numerous applications on the soliton theory we refer to [1]. For a very striking proof based on factorization of differential operators see [2]. Our aim is to propose a more systematic approach within the framework of Monge systems (underdetermined systems of ordinary differential equations).

2. Generalization. Retaining the notation and all assumptions as above, let us introduce new variables

u = \frac{y′}{y} + \frac{z′}{z} , v = \frac{y′}{y} − \frac{z′}{z}  (3)

and choose quite arbitrary functions f = f(λ), g = g(λ, x, v) ≠ 0. Then the functions

ỹ = e^{\frac{1}{2} ∫ (ũ+ṽ)dx},  z̃ = e^{\frac{1}{2} ∫ (ũ−ṽ)dx},  (4)

237 where

1991 Mathematics Subject Classification. 34C20.

Key words and phrases. Sturm–Liouville equation, Darboux transformation.

1072-947X/95/0500-0237$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

ũ == \frac{1}{g}(f − \frac{∂g}{∂x} − v′ \frac{∂g}{∂v}), ṽ = g,  (5)

satisfy certain Sturm-Liouville equations

ỹ′′ = p̃(x)ỹ, z̃′′ = q̃(x)z̃  (6)

such that the difference of new potentials p̃ = p̃(x), q̃ = q̃(x) is equal to the constant p̃ − q̃ = f ∈ ℝ.

Proof. Clearly

\frac{y′′}{y} − \frac{z′′}{z} = p − q = λ = const.  (7)

In terms of variables (3), this second-order relation turns into the first-order Monge system

v′ + uv = λ, λ′ = 0.  (8)

We shall look for transformations of (8) into itself. Assuming that the independent variable is preserved, x̃ = x, we shall quite arbitrarily choose the transformed λ̃ = f(λ) and ṽ = g(λ, x, v). Then clearly

(v′)^{}˜ = (\frac{dv}{dx})^{}˜ = \frac{dṽ}{dx} = (ṽ)′ = \frac{∂g}{∂x} + v′ \frac{∂g}{∂v} ,

hence ṽ′ + ũṽ = λ̃ (see (8)) gives (5):

ũ′ = (λ̃ − ṽ′)/ṽ = \frac{1}{g}(f − \frac{∂g}{∂x} − v′ \frac{∂g}{∂v}).

Finally, using the counterparts of (3) for the transformed variables, namely

ũ = \frac{ỹ′}{ỹ} + \frac{z̃′}{z̃} ,  ṽ = \frac{ỹ′}{ỹ} − \frac{z̃′}{z̃} ,

the sought result (4) follows at once. According to the construction, the difference of new potentials is

p̃ − q̃ = \frac{ỹ′′}{ỹ} − \frac{z̃′′}{z̃} = ṽ′ + ũṽ = λ̃ = f(λ).

3. Some particular cases. For the special choice f = Aλ, g = Bv, where A, B ∈ ℝ are constants, one can easily find

ũ = (\frac{A}{B} − 1)\frac{v′}{v} + \frac{A}{B} u,

whence

ỹ = v^{\frac{1}{2} ( \frac{A}{B} −1)}y^{\frac{1}{2} ( \frac{A}{B} +B)}z^{\frac{1}{2} ( \frac{A}{B} −B)},
z̃ = v^{\frac{1}{2} ( \frac{A}{B} −1)}y^{\frac{1}{2} ( \frac{A}{B} −B)}z^{\frac{1}{2} ( \frac{A}{B} +B)}.

These results simplify if, for example, A = B or A = B^{2}. In the latter case, it follows in particular that the potentials of the couple ỹ = vy^{3}, z̃ = vz^{3} (the choice B = 3) or the potentials of ỹ = 1/vy, z̃ = 1/vz (the choice B = −1), differ by a constant.

Analogously, for the special choice f = Aλ, g = Bλ/v where A, B ∈ ℝ are constants, one can find ũ = Av/B + v′/v, whence

ỹ = v^{\frac{1}{2} (1+B)}y^{\frac{1}{2} ( \frac{A}{B} +B)}z^{− \frac{1}{2} ( \frac{A}{B} −B)}, z̃ = v^{\frac{1}{2} (1−B)}y^{\frac{1}{2} ( \frac{A}{B} −B)}z^{− \frac{1}{2} ( \frac{A}{B} +B)}.

This result simplifies if, for example, B = ±1, or A = ±B. In particular, if A = B = 1 (hence f = λ and g = λ/v) we obtain ũ = v + v′/v, hence ỹ = y′ − yz′/z = ȳ (the Darboux transformation), z̃ = 1/z. It follows that

q̃ = \frac{z̃′′}{z̃} = −q + 2(\frac{z′}{z})^{2} = q − 2(\frac{z′}{z})′

hence p̃ = q̃ + f = p − 2(z′/z)′ = p̄. This is the Darboux theorem.

4. Some comments. We shall seek the true sense of the above elementary proof. First of all, it is clear that the main result may still be generalized if, for example, either the function q is allowed to depend on some higher order derivatives or the variable x is not preserved. In full generality, let us deal with the infinite-dimensional space M with coordinates λ, x, v, v′, v′′, … (owing to (8_{1}), u can be expressed in terms of them, and owing to (8_{2}) the derivatives of λ are not needed). Then the infinite system

dλ = dv − v′dx = dv′ − v′′dx = ⋯ = 0  (9)

clearly involves all given data. Every mapping of (9) into itself which turns the above coordinates into new ones λ̃, x̃, ṽ, (v′)^{}˜ , (v′′)^{}˜ , … (and preserves the formula (8_{1})) can be as for a generalization of the classical Darboux result.

In Section (2) above, we have apriori postulated x̃ = x, λ̃ = f(λ), ṽ = g(λ, x, v), but this is not necessary. For instance, for the choice x̃ = x, λ̃ = λ, ṽ = v′/v, one can easily find the transformed variable ũ = λv/v′ − v′′/v′ − v′/v, whence ỹ = v^{λ}/v′, z̃ = v^{λ}/v′v^{2}. We shall not continue in this direction here, however.

It is worth mentioning that if we are interested in automorphism of (9), i.e., in bijective transformations, then all truncated finite dimensional systems

dλ = 0, dλ = dv − v′dx = 0, dλ = dv − v′dx =
= dv′ − v′′dx = 0, …,  (10)

are transformed into themselves (see [3]). It follows that such automorphisms are (prolonged) contact transformations depending on the parameter λ. If, moreover, the invariance x̃ = x is postulated, then necessarily ṽ = g(λ, x, v) is a certain function of λ, x, v (as follows from the invariance of the system (10_{2})), and the invertibility is guaranteed if

f′ ≠ 0, ∂g/∂v ≠ 0.  (11)

So we have the following result: Assuming (11), the generalization of Section 2 includes all transformations between the families of couples (y, z) → (ỹ, z̃) which are bijective.

Adopting the point of view of infinite-dimensional Pfaffian systems, various developments are at hand, but we briefly mention only the following one: if instead of (1) we deal with the equations

y′′′ = p(x)y, z′′′ = q(x)z  (12)

of third order, then the problem leads to the Monge system

v′′ + \frac{3}{2}(uv)′ + \frac{1}{4}u(u^{2} + 3v^{2}) = λ, λ′ = 0  (13)

for the same functions (3) as above. Such a system cannot be investigated by elementary methods. We postpone these problems to a future publication.

References

1. V. B. Matveev and M. A. Salli, Darboux Transformations and Solitons. Springer Series in Nonlinear Dynamics. Springer-Verlag, Berlin, 1991.

2. R. Beals, P. A. Deift, and C. Tomei, Direct and inverse Scattering on the Line. Mathematical Surveys and Monographs. 28. American Mathematical Society, Providence, RI, 1988.

3. J. Chrastina, On the equivalence of variational problems II. Arch. Math. (Brno) 29(1993), 197-220.

(Received 15.04.1994)

Author’s address:

Masaryk University Faculty of Science

Janáčkovo nám. 2a, 66295 Brno

Czech Republic