Examples of the Painlev ´e Test Applications

S.9.3. Examples of the Painlev ´e Test Applications

In this section, we consider some examples of equations of mathematical physics. For their analysis, we first resort to the simplest and then the general scheme of the Painlev ´e test application based on series (1) and (2) from Section S.9.2.

Example 1. Consider the Burgers equation

1 ◦ . Substituting the leading term of the expansion (1) into this equation, we obtain

w ′ 0 + 0 αw x 0 ′

αw 0 2 = να(α + 1)w 0 ,

where x 0 = x 0 ( t) and w 0 = w 0 ( t); the prime denotes a derivative with respect to t. Retaining the leading singular terms (omitting the first two terms on the left), we find that

The Burgers equation, upon the insertion of series (1) in it and the collection of terms with the same powers of ε=x−x 0 ( t), takes the form

w t + ww x − νw xx =

E n ( t)ε n−3 = 0,

where E n ( t) = −(n + 1)(n − 2)νw n + ···.

n=0

Here, in the expression for E n ( t), the terms containing w 0 , ...,w n−1 and x 0 ( t) are omitted. It is clear that there is a single resonance, n = 2; the compatibility condition holds only in this case (the sum of the terms with lowest-subscript coefficients in the recurrence relation vanishes) and the function w 2 ( t) remains arbitrary. This can be seen from the structure of the following recurrence relations:

The relation for n = 2 is a consequence of the preceding relations and does not contain w 2 .

Thus, the Burgers equation satisfies the conditions of the Painlev´e test, and its solution contains two arbitrary functions, as required. Collecting terms with like powers of x−x 0 ( t), we can write out the solution in the form

w(x, t) = − 2 ν

0 + x 0 ′ ( t) + w 2 ( t)[x − x 0 ( t)] 2 x−x + ( t) ···,

where x 0 ( t) and w 2 ( t) are arbitrary functions. 2 ◦ . For the purpose of subsequent analysis of the Burgers equation, let us take advantage of the general expansion (2), where

w n = w n ( x, t) and ε = ε(x, t). From the condition of balance of the leading terms, we obtain

The recurrence relations for the next three terms of the expansion have the form w 1 ε x − νε xx + ε t =0 ( n = 1),

( w 1 ε x − νε xx + ε t ) x =0 ( n = 2),

( w 1 ) t − ν(w 1 ) xx + w 1 ( w 1 ) x +( w 0 w 2 ) x +( ε t − νε xx ) w 2

x 2 ( w 2 ) x + ε x ( w 1 w 2 + w 0 w 3 )−2 νε x w 3 =0 ( n = 3).

( w k = 0 for k ≥ 2). The remaining relations allow us to represent the solution in the form

These relations represent a B¨acklund transformation and allow us to use solutions w 1 = w 1 ( x, t) of the Burgers equation for the construction of its other solutions w = w(x, t). Taking, for example, w 1 = 0 to be the initial solution, we obtain the well-known Cole–Hopf transformation

ε w = −2ν x ,

which reduces the nonlinear Burgers equation to the linear heat equation

ε t = νε xx .

Example 2. Consider the Korteweg–de Vries equation

1 ◦ . Substituting the leading term of the expansion (1) into this equation yields

w ′ 0 αw 0 x ′

αw 0 α(α + 1)(α + 2)w 0

= 0, where x 0 = x 0 ( t) and w 0 = w 0 ( t). From the condition of balance of the leading terms, we find that

Upon the insertion of the expansion (1), the Korteweg–de Vries equation can be represented in the form

w t + ww x + w xxx =

E n ( t)ε n−5 = 0,

where E n ( t) = (n + 1)(n − 4)(n − 6)w n + ···.

n=0

From the expression for E n ( t), it follows that there are two resonances, n = 4 and n = 6. Writing out explicitly the first seven equations for the coefficients in the expansion (1), we see that the compatibility condition holds for the resonances,

The relations for n = 4 and n = 6 are consequences of the preceding ones and do not contain w 4 and w 6 . Therefore, the Korteweg–de Vries equation satisfies the conditions of the Painlev´e test. The three arbitrary functions w 4 ( t), w 6 ( t), and x 0 ( t) ensure the required generality of the solution of the third-order equation.

2 ◦ . Now, let us obtain a consequence of the general expansion by truncating series (2). Inserting the truncated series with w 3 = w 4 = · · · = 0 into the Korteweg–de Vries equation, we arrive at the B¨acklund transformation

w= w 0 + w 2 1 + w 2 = 12(ln ε)

Eliminating w 2 from the second and the third equations, we obtain an equation for the function ε, which can be reduced to a ◆✂❖ system of linear equations by means of several transformations. References : J. Weiss, M. Tabor, and G. Carnevalle (1983), M. Tabor (1989), J. Weiss (1993).

Example 3. Consider the Kadomtsev–Petviashvili equation

which can be regarded as an integrable generalization of the Korteweg–de Vries equation of a higher dimension and a higher order.

w(x, y, t) = α

w n ( y, t)ε n , ε=x−x 0 ( y, t).

ε n=0

Equating the leading singular terms for the Kadomtsev–Petviashvili equation, we obtain the same result as that for the Korteweg–de Vries equation,

Substituting the expansion (3) into the original equation, we obtain

w tx +

ww 2 xx + w x + w xxxx + aw yy =

ε n−6 E n ( y, t) = 0,

n=0

E n ( y, t) = (n + 1)(n − 4)(n − 5)(n − 6)w n + ···.

It is apparent that there are three resonances: n = 4, 5, 6. In order to verify the conditions of the Painlev ´e test, let us write out recurrence relations for the first seven terms of the expansion,

E 0 = 10 w 0 ( w 0 + 12) = 0

( n = 0),

( n = 1), E 2 = 3[2(

+ 2[( ε t + aε 2 y + w 2 w 4 + 1 2 w 2 3 + w 5 w 1 +( w 0 + 12) w 6 ] =0 ( n = 6). The last three relations (corresponding to resonances), in view of the preceding relations, hold identically and do not contain

w 4 , w 5 , w 6 . There are four arbitrary functions ( ε, w 4 , w 5 , w 6 ) in the solution of the forth-order equation under consideration, which indicates that the Painlev´e property holds.

2 ◦ . The utilization of the general expansion, with the series truncated so that w n = 0 for n > 2, leads us to the B¨acklund transformation (for simplicity, we set

a = 1)

w = 12(ln ε) xx + w 2 , 2 2 ε 2 t ε x +4 ε x ε xxx −3 ε xx + ε y + w 2 ε x = 0,

ε xt + ε xxxx + ε yy − w 2 ε xx = 0, ( w 2 ) tx + w 2 ( w 2 ) xx +( w 2 ) 2 x +( w 2 ) xxxx +( w 2 ) yy = 0.

Eliminating w 2 from the second and the third equations, we obtain an equation for the function ε, which allows us to pass to a solution of a system of linear equations.

Example 4. Consider the model system of equations (Gorodtsov, 1998, 2000)

∂(wc) ∂ + 2 = c

that describes convective mass transfer of an active substance in a viscous fluid in the case where the flow is affected by the substance through the pressure quadratically dependent on its concentration. Such equations are used for describing one-dimensional flows of electrically conducting fluids in a magnetic field with high magnetic pressure.

1 ◦ . By analogy with the expansion (1), let us represent the desired quantities in the form

w(x, t) =

w n ( t)ε n ,

c(x, t) =

c n ( t)ε n ,

ε n=0 β n=0

ε≡x−x 0 ( t).

Equating the leading singular terms of the equations, we find that

α = β = 1, 2 w 0 =− χ, c

Let us write the recurrence relations for the series terms in matrix form

−( n − 2)[χ + ν(n − 1)]

( n − 2)c 0 w n

f n−1

n − 2)c . 0 −( n − 2)nχ c

g n−1 g n−1

νχ(n + 1)(n − 2) 2 ( n − 2 + χ/ν) = 0.

All these resonances are positive integers (except for the special resonance n = −1) only if the Prandtl number is equal to unity, Pr ≡ ν/χ = 1. One resonance, n = 1, is simple, and the other, n = 2, is multiple, so that the overall number of resonances is equal to four.

Writing out the first three recurrence relations

we see that the compatibility condition holds for the resonance n = 1, since the two relations for n = 1 coincide by virtue of the expressions for n = 0 (w 0 = P c 0 ). The multiple resonance n = 2 also satisfies the compatibility condition, since both coefficients w 0 , c 0 are constant. Therefore, the Painlev´e property takes place for the equations of a fluid with an active substance (for ν/χ = 1).

2 ◦ . Using the general expansion with the series truncated so that w 2 = w 3 = · · · = 0 and c 2 = c 3 = · · · = 0, we obtain a B¨acklund transformation for the equations of a fluid with an active substance

( c 1 ) t +( w 1 c 1 ) x = ν(c 1 ) xx . Comparing this with the B¨acklund transformation for the Burgers equation, we see that if, in the above transformation, we

( w 1 ) t + w 1 ( w 1 ) x =− c 1 ( c 1 ) x + ν(w 1 ) xx ,

pass to the new variables equal to the sum and the difference of the original variables, we obtain identical equations. Indeed, passing to such variables in the original equations with unit Prandtl number, we obtain a pair of identical Burgers equations,

each of which reduces to the linear heat equation (see Example 1).

Numerous investigations show that many known integrable nonlinear equations of mathematical physics possess the Painlev´e property. Some new equations with this property have also been found. During the verification of the conditions of the Painlev ´e test for more complex equations and systems, resonances with higher n may arise. In such situations, analytical solution becomes more and more difficult. However, the Painlev´e test is highly adapted for algorithmization and allows for the utilization of symbolic computation methods. For example, the Maple software has been successfully used to obtain a complete classification of integrable cases of the equations of shallow water with dissipation and dispersion of lower orders [see Klimov, Baydulov, and Gorodtsov (2001)]. ❘✂❙

References for Subsection S.9.3: M. Jimbo, M. D. Kruskal, and T. Miwa (1982), J. Weiss, M. Tabor, and G. Carnevalle (1983), J. Weiss (1983, 1984, 1985), W.-H. Steeb and N. Euler (1988), R. Conte (1989, 1999), R. Conte and M. Musette (1989, 1993), M. Tabor (1989), M. Musette (1998).