Clarkson–Kruskal Direct Method: a Special Form for Similarity Reduction

S.6.1. Clarkson–Kruskal Direct Method: a Special Form for Similarity Reduction

S.6.1-1. Simplified scheme. Examples of constructing exact solutions. Prior to giving a description of the Clarkson–Kruskal direct method in the general case, consider a

simplified scheme. The basic idea of the method is the following: for an equation with the unknown function w = w(x, t), an exact solution is sought in the form

(1) The functions f (t), g(x, t), ϕ(t), and ψ(t) are found in the subsequent analysis and are chosen in

w = f (t)u(z) + g(x, t),

z = ϕ(t)x + ψ(t).

such a way that, ultimately, the function u(z) would satisfy a single ordinary differential equation. Below we consider some cases in which it is possible to construct exact solutions of nonlinear equations of the form (1).

Example 1. Consider the generalized Burgers–Korteweg–de Vries equation

We seek its exact solution in the form (1). Inserting (1) into (2), we obtain

af ϕ n u ( z n) + bf 2 ϕuu ′ z +

f (bgϕ − ϕ ′ t x−ψ t ′ ) u ′ z +(

x bf g − f t ( ′ ) u + ag x n) + bgg x − g t = 0.

Equating the functional coefficients of u ( z n) and uu ′ z in (3), we get

f=ϕ n−1 .

Further, equating the coefficient of u ′ z to zero, we obtain

Inserting the expressions (4) and (5) into (3), we arrive at the relation ϕ 1 2 n−1

( au ( n) z + buu ′ z ) + (2 − n)ϕ n−2 ϕ ′ t u+

(2 ϕ 2 − ϕϕ ) x + 2ϕ ψ − ϕψ bϕ = 0. 2 t tt t t tt

Let us require that the functional coefficient of u and the last term be constant,

ϕ −2 n−2 ( ϕψϕ tt − ϕ 2 ψ tt +2 ϕϕ t ψ t −2 ψϕ 2 t )= B, where

ϕ − n−1 ϕ ′ t =− A,

A and B are arbitrary. As a result, we arrive at the following system of ordinary differential equations for ϕ and ψ:

Using (6) and (7), we obtain an equation for u(z),

For A ≠ 0, the general solution of equations (7) has the form

ϕ(t) = (Ant + C 1 ) − 1 n ,

n−1

A 2 ( , n − 1) where C 1 , C 2 , and C 3 are arbitrary constants.

ψ(t) = C 2 (

Ant + C

C Ant + C

Formulas (1), (4), (5), and (9), together with equation (8), describe an exact solution of the generalized Burgers– Korteweg–de Vries equation (2). In the special case of n = 3 and a = b = −1, the solution constructed above turns into the solution obtained by Clarkson and Kruskal (1989).

Example 2. Consider the Boussinesq equation

Just as in Example 1, we seek its solutions in the form (1), where the functions f (t), g(x, t), ϕ(t), and ψ(t) are found in the subsequent analysis. Substituting (1) into (10) yields

4 ′′′′ + 2 af ϕ 2 u f ϕ uu ′′ +

Equating the functional coefficients of u ′′′′ and uu ′′ , we get

f=ϕ 2 .

Equating the functional coefficient of u ′′ to zero and taking into account (12), we obtain

Substituting the expressions (12) and (13) into (11), we arrive at the relation 6 ( ′′′′ +

ϕ ϕ t ϕ t x+ψ t ) = 0. Let us perform the double differentiation of the expression in square brackets and then divide all terms by ϕ 6 . Excluding x

with the help of the relation x = (z − ψ)/ϕ, we get au ′′′′ + uu ′′ +( u ′ ) 2 + ϕ −5 ( ϕ tt z + ϕψ tt − ψϕ tt ) u ′ +2 ϕ −5 ϕ tt u + · · · = 0.

Let us require that the functional coefficient of u ′

be a function of only one variable, z, i.e.,

ϕ −5 ( ϕ tt z + ϕψ tt − ψϕ tt )= ϕ −5 ϕ tt z+ϕ −5 ( ϕψ tt − ψϕ tt )≡ Az + B, where

A and B are arbitrary constants. Hence, we obtain the following system of ordinary differential equations for the functions ϕ and ψ:

Let us eliminate the second and the third derivatives of the functions ϕ and ψ from (14). As a result, we arrive at the following ordinary differential equation for the function u(z):

au ′′′′ + uu ′′ +( u ′ ) 2 +( Az + B)u ′ +2 Au − 2(Az + B) 2 = 0.

Formulas (1), (12), and (13), together with equations (15)–(16), describe an exact solution of the Boussinesq equation (10).

S.6.1-2. Description of the Clarkson–Kruskal method. A special form for similarity reduction.

1 ◦ . The basic idea of the method is the following: for an equation with the unknown function w = w(x, t), an exact solution is sought in the form

(17) The functions f (x, t), g(x, t), and z(x, t) are determined in the subsequent analysis, so that ultimately

w(x, t) = f (x, t)u(z) + g(x, t),

z = z(x, t).

one obtains a single ordinary differential equation for the function u(z).

2 ◦ . Inserting (17) into a nonlinear partial differential equation with a quadratic or a power nonlin- earity, we obtain

Φ 1 ( x, t)Π 1 [ u] + Φ 2 ( x, t)Π 2 [ u] + · · · + Φ m ( x, t)Π m [ u] = 0. (18) Here, the Π k [ u] are differential forms that are the products of nonnegative integer powers of the

function u and its derivatives u ′ z , u ′′ zz , etc., and the Φ k ( x, t) depend on the functions f (x, t), g(x, t), and z(x, t) and their partial derivatives with respect to x and t. Suppose that the differential form

Π 1 [ u] contains the highest-order derivative with respect to z. Then the function Φ 1 ( x, t) is used as

a normalizing factor. This means that the following relations should hold:

(19) where the Γ k ( z) are functions to be determined.

Φ k ( x, t) = Γ k ( z)Φ 1 ( x, t),

k = 1, . . . , m,

3 ◦ . The representation of a solution in the form (17) has “redundant” generality and the functions

f , g, u, and z are ambiguously determined. In order to remove the ambiguity, we use the following three degrees of freedom in the determination of the above functions: (a) if f = f (x, t) has the form f = f 0 ( x, t)Ω(z), then we can take Ω ≡ 1, which corresponds to the replacement u(z) → u(z)/Ω(z); (b) if g = g(x, t) has the form g = g 0 ( x, t)+f (x, t)Ω(z), then we can take Ω ≡ 0, which corresponds to the replacement u(z) → u(z) − Ω(z); (c) if z = z(x, t) is determined by an equation of the form Ω(z) = h(x, y), where Ω(z) is any invertible function, then we can take Ω(

z) = z, which corresponds to the replacement z → Ω −1 ( z).

4 ◦ . Having determined the functions Γ k ( z), we substitute (19) into (18) to obtain an ordinary differential equation for u(z),

(20) Below we illustrate the main points of the Clarkson–Kruskal direct method by an example.

Π 1 [ u] + Γ 2 ( z)Π 2 [ u] + · · · + Γ m ( z)Π m [ u] = 0.

Example 3. We seek a solution of the Boussinesq equation (10) in the form (17). We have

4 x af z u ′′′′ +

2 a(6f z 3 x z xx +4 f x z x ) u ′′′ + 2 f 2 z x uu ′′ + · · · = 0.

Here, we have written out only the first three terms and have omitted the arguments of the functions f and z. The functional coefficients of u ′′′′ and uu ′′ should satisfy the condition [see (19)]:

af z x 4 Γ 3 ( z),

where Γ 3 ( z) is a function to be determined. Hence, using the degree of freedom mentioned in Item 3 ◦ (a), we choose

f=z 2 x ,

Γ 3 ( z) = 1/a.

Similarly, the functional coefficients of u ′′′′ and u ′′′ must satisfy the condition

6 2 3 fz 4 x z xx +4 f x z x = fz x Γ 2 ( z),

where Γ 2 ( z) is another function to be determined. Hence, with (22), we find

14 z xx /z x =Γ 2 ( z)z x .

Integrating with respect to x yields

ln

x = I(z) + ln e ϕ(t),

I(z) =

14 z) dz,

Z e − I(z) dz = e ϕ(t)x + e ψ(t),

where e ψ(t) is another arbitrary function. We have a function of z on the left, and therefore, using the degree of freedom mentioned in Item 3 ◦ (c), we obtain

z = xϕ(t) + ψ(t),

where ϕ(t) and ψ(t) are to be determined. From formulas (22)–(24) it follows that

Substituting (24) and (25) into (17), we obtain a solution of the form (1) with the function f defined by (12). Thus, the general approach based on the representation of a solution in the form (17) ultimately leads us to the same result as the approach based on the more simple formula (1).

Remark 1. In a similar way, it can be shown that formulas (1) and (17) used for the construction of an exact solution of the generalized Burgers–Korteweg–de Vries equation (2) lead us to the same result.

Remark 2. The above examples clearly show that it is more reasonable to perform the ini- tial analysis of specific equations on the basis of the simpler formula (1) rather than the general formula (17). ✴✂✵

References for Subsection S.6.1: P. A. Clarkson and M. D. Kruskal (1989), D. Arrigo, P. Broadbridge, and J. M. Hill (1993), P. A. Clarkson, D. K. Ludlow, and T. J. Priestley (1997), D. K. Ludlow, P. A. Clarkson, and A. P. Bassom (1999, 2000).