B ¨acklund Transformations. Differential Substitutions

S.2.4. B ¨acklund Transformations. Differential Substitutions

S.2.4-1. B¨acklund transformations.

1 ◦ . Let w = w(x, y) be a solution of the equation

∂w ∂w ∂ 2 w ∂ 2 w ∂ 2 w

(30) and let u = u(x, y) be a solution of another equation

∂x∂y ∂y 2

∂u ∂u ∂ 2 u ∂ 2 u ∂ 2 u

(31) Equations (30) and (31) are said to be related by the B¨acklund transformation

F 2 x, y, u,

∂x ∂y ∂x ∂x∂y ∂y 2

∂w ∂w

∂u ∂u

1 x, y, w,

, u,

∂x ∂y

∂x ∂y

∂w ∂w

∂u ∂u

2 x, y, w,

, u,

∂x ∂y

∂x ∂y

(31), (32) implies (30). If, for some specific solution u = u(x, y) of equation (31), one succeeds in solving equations (32) for w = w(x, y), then this function w = w(x, y) will be a solution of equation (30). Relations (32) are also called differential constraints.

B¨acklund transformations may preserve the form of equations* (such transformations are used for obtaining new solutions) or establish relations between solutions of different equations (such transformations are used for obtaining solutions of one equation from solutions of another equation).

2 ◦ . For two nth-order evolution equations of the forms

a B¨acklund transformation is often sought in the form of a differential constraint

containing derivatives in only one variable x (the second variable, t, is present implicitly through the functions w, u). This constraint can be regarded as an ordinary differential equation in one of the dependent variables.

S.2.4-2. Differential substitutions. In mathematical physics, apart from the B¨acklund transformations, one often resorts to the so-called

differential substitutions. For second-order differential equations, differential substitutions have the form

A differential substitution increases the order of an equation (if it is inserted into an equation for w) and allows us to obtain solutions of one equation from those of another. The relationship between the solutions of the two equations is generally not invertible and is, in a sense, unilateral. A differential substitution may be obtained as a consequence of a B¨acklund transformation (although this is not always the case).

S.2.4-3. Examples of B¨acklund transformations and differential substitutions.

Example 9. The Burgers equation

is related to the heat equation

by the B¨acklund transformation

Eliminating w from (35), we obtain equation (34). * In such cases, these are referred to as auto-B¨acklund transformations.

equation with respect to x, and taking into account that (u t /u) x =( u x /u) t , we obtain

u xx

Hence, taking into account the relations that follow from the first equation in (35),

2 4 we obtain the Burgers equation (34).

Remark. The first relation in (35) can be rewritten as the differential substitution (the Hopf–Cole transformation)

w=

Substituting (36) into (33), we obtain the equation

which can be converted to

Thus, using formula (36), one can transform each solution of the linear heat equation (34) into a solution of the Burgers equation (33). The converse is not generally true. Indeed, a solution of equation (33) generates a solution of the more general equation

∂u

f (t)u,

∂t

∂x

where f (t) is a function of t. Example 10. The nonlinear Schr ¨odinger equation with a cubic nonlinearity

2 i ∂w + ∂ w

∂x 2 +| w| 2 w = 0, where w is a complex-valued function of real variables x and t (i 2 = −1), is invariant under the B¨acklund transformation

2 g ag 4 f 1 | f 1 | +| f 2 | ∂t . ∂t ∂x ∂x Here, we have used the notation f 1

i af 1 − 1 2 f 2 g 1 , where

1 = w−e w, f 2 = w+e w,

g 1 = iε b − 2|f 1 | 2 /2 , g 2 =

a and b are arbitrary real constants, ε = ✆ 1. Example 11. The Korteweg–de Vries equation

and the modified Korteweg–de Vries equation

are related by the B¨acklund transformation

The first relation in (37) is a Miura transformation which can be rewritten as a differential substitution by solving (37) for w.

S.2.4-4. B¨acklund transformations based on conservation laws. Consider a differential equation written as a conservation law,

The B¨acklund transformation dz = F (w, w x , w y , . . .) dy − G(w, w x , w y , . . .) dx, dη = dy

determines the passage from the variables x and y to the new independent variables z and η according to the rule

F and G are the same as in (38). The transformation (39) preserves the order of the equation under consideration.

Remark. Often one may encounter transformations (39) that are supplemented with a transfor- mation of the unknown function in the form u = ϕ(w).

Example 12. Consider the third-order nonlinear equation

which represents a special case of equation (38) for y = t, F = [f (w)w x ] x , and

G = −w.

In this case, transformation (39) has the form

dz = w dx + [f (w)w x ] x dt,

dη = dt

and determines a transformation from the variables x and y to the new independent variables z and η according to the rule

∂ = ∂ +[ f (w)w ] ∂ .

Applying transformation (41) to equation (40), we obtain

The substitution w = 1/u reduces (42) to an equation of the form (40),

In the special case of f (w) = aw −3 , the nonlinear equation (40) is reduced to the linear equation u η = au zzz by the transformation (41). ✝✂✞

References for Subsection S.2.4: G. L. Lamb (1974), R. M. Miura (1976), R. L. Anderson and N. H. Ibragimov (1979), A. S. Fokas and R. L. Anderson (1979), A. S. Fokas and B. Fuchssteiner (1981), M. J. Ablowitz and H. Segur (1981), N. H. Ibragimov (1985, 1994), H. Stephani (1989), B. J. Cantwell (2002).