Linearizable hierarchies of evolution equations in (1+1) dimensions

Linearisable Hierarchies of Evolution Equations in (1+1) Dimensions

Marianna EULER, Norbert EULER

and Niclas PETERSSON Department of Mathematics

Lule˚ a Universityof Technology SE-971 87 Lule˚ a, Sweden

∗

Corresponding author’s e-mail: Norbert@sm.luth.se

Copyright c
2002 byM Euler, N Euler and N Petersson

M. Euler, N. Euler and N Petersson, Linearisable Hierarchies of Evolution Equations in (1+1) Dimensions, Lulea University of Technology, Department of Mathematics Research Report 5 (2002).

Abstract:

In our article [5], “A tree of linearisable second-order evolution equations bygeneralised hodograph transformations [J. Nonlin. Math. Phys. 8 (2001), 342-362] we presented a tree of linearisable (C-integrable) second-order evolution equations in (1+1) dimensions.

Expanding this result we report here the complete set of recursion operators for this tree and present several linearisable (C-integrable) hierarchies in (1+1) dimensions.

Subject Classification (AMS 2000): 37K35, 37K10, 35Q58.

Key words and phrases: Nonlinear partial differential equations, Linearisation, Inte- grability, Transformations.

Note: This report has been submitted for publication elsewhere.

ISSN: 1400–4003

Lulea University of Technology

Department of Mathematics

S-97187 Lulea, SWEDEN

1 Introduction

In [5] we presented a tree of linearisable (that is, C-integrable) second-order evolution equations which can be transformed to linear partial differential equations. The transfor- mation under which the classification was performed in [5] is the so-called x-generalised hodograph transformation defined as follows:

n

H :

 

 



 

 

dX(x, t) = f

(x, u)dx + f

(x, u, u

, u

, . . . , u

)dt dT (x, t) = dt

U (X, T ) = g(x),

(1.1)

with n = 2, 3, . . . and u

∂f

∂u = ∂f

∂x + u

∂f

∂u + u

∂f

∂u

+ · · · + u

∂f

∂u

. (1.2)

Applying this transforms on an (1 + 1)-dimensional nth-order autonomous evolution equa- tion

U

= P (U, U

, U

, . . . , U

) (1.3)

leads in general to a (1 + 1)-dimensional nth-order x-dependent evolution equation of the form

u

= Q(x, u, u

, u

, . . . , u

). (1.4)

Obviously P and Q are related via (1.1) and (1.2).

The prolongations of (1.1) are U

= − f

f

dg

dx , U

= ˙g

f

(1.5)

U

= 1 f

 D

 1 f



D

 ˙g f



, n = 2, 3, . . . , where ˙g = dg/dx and D

is the total derivative operator

D

= ∂

∂x + u

∂

∂u + u

∂

∂u

+ · · · with

(D

a)

= D

(aD

a), (D

a)

= D

(aD

(aD

a)), . . . Following (1.5), the relation between f

and f

for (1.3) is

f

(x, u, u

, . . . , u

) = − f

˙g [P (U, U

, . . . U

)] 



, (1.6)

where Ω =



U = g(x), U

= ˙g

f

, . . . , U

= 1 f

 D

 1 f



D

 ˙g f



(1.7)

Equation (1.4) then follows from condition (1.2).

It should be pointed out that the x-generalised hodograph transformation (1.1) is a generalisation of the extended hodograph transformation introduced in [4], namely

X(x, t) =

f (u(ξ, t))dξ

T (x, t) = t (1.8)

U (X, T ) = x.

For second-order evolution equations we showed [5] that

The most general ( 1 + 1)-dimensional second-order evolution equation which maybe con- structed to be linearisable in

U

= U

+ λ

U

+ λ

U, λ

, λ

∈  (1.9)

via the x-generalised hodograph transformation (1.1) is necessarilyof the form

u

= F

(x, u)u

+ F

(x, u)u

+ F

(x, u)u

+ F

(x, u). (1.10)

This led to sixteen linearisable second-order evolution equations, eight of which are autonomous (by autonomous equations we mean equations which do not depend explicitly on their independent variables x and t). These equations are listed in [5] together with their linearising transformations. Only one equation of this class is autohodograph invariant, i.e., invariant under an x-generalised hodograph transformation, namely the equation

u

= h(u)u

+ {h}

u

, (1.11)

where h ∈ C

( ), dh/du = 0 and {h}

:= − 1

2 dh du + h

 dh du



d

h

du

. (1.12)

One can show the following:

The most general nth-order evolution equation which is linearisable in U

= λ

U +

λ

U

, λ

∈  (1.13)

byrepeatedlyapplying the transformation (1.1), is necessarilyof the form

u

= 

F

(x, u)u

u

· · · u

+ F

(x, u), (1.14)

where F

∈ C

( ) are functions of x and u, and 

j=1

jk

= r, k

∈ {0, 1, . . . , n}, 1 ≤ l ≤ r, 1 ≤ r ≤ n.

The objective in the present paper is to construct higher-order linearisable autonomous evolution equations by the use of the tree of second-order linearisable equations given in [5].

This is achieved by calculating the recursion operators for the second-order equations. The hierarchies obtained by the recursion operators are all linearisable via the x-generalised hodograph transformations and are of the general quasi-linear form (1.14). We do, how- ever, restrict ourselves to the autonomous case. We furthermore write the equations in potential form and use the pure hodograph transformation and corresponding recursion operators to construct hierarchies of autonomous nonlinear evolution equations and their linearising transformations.

We point out that the x-generalised hodograph transformation can be applied only on autonomous evolution equations and may produce nonautonomous evolution equation of the same order. However, it is easy to show that any nonautonomous evolution equation which has been generated by such a transformation can always be made autonomous.

This is achieved by writing the obtained equation in potential form followed by the pure hodograph transformation.

2 Second-order linearisable equations and their potential forms

The following eight second-order evolution equations were constructed to be linearisable via the x-generalised hodograph transformation (1.1) [5]:

u

= h

u

+ {h

}

u

, ˙h

(u) = 0 ( I) u

= h

u

+ λh

u

+ {h

}

u

, ˙h

(u) = 0, λ = 0 ( II) u

= h

u

+ {h

}

u

+ 2λ

h

˙h

, ˙h

(u) = 0, λ

= 0 (III) u

= u

+ λ

u

+ h

(λ

− ˙h

)u

+ h

, h

(u) = 0, λ

= 0 (IV.1) u

= u

+ λ

u

− h

˙h

u

+ h

, h

(u) = 0 (IV.2) u

= h

u

+ 

λh

− λ

λ

u

+ {h

}

u

, ˙h

(u) = 0, λ = 0, λ

= 0 ( V) u

= u

+ h

u

+ ¨ h

˙h

u

, h

(u) = 0 (VI) u

= h

u

+ λ

u

+ {h

}

u

, ˙h

(u) = 0, λ

= 0 ( VII)

u

= u

+ λ

u

+ h

u

(VIII)

Here h

are arbitrary C

-functions depending on u (with the indicated restrictions). The bracket {h

}

is defined by (1.12). All λ’s are arbitrary constants, unless otherwise stated.

Here and in the rest of this paper the overdot on h

denotes differentiation with respect to u.

The equations (I - VIII) listed above may be linearised to (1.9), i.e., U

= U

+ λ

U

+ λ

U − λ

, λ

∈ 

by the following transformations, respectively [5]:

(Trans-I) x = U (X, T ), dt = dT, h

(u) = U

. (Trans-II) x = 1

λ ln |λU|, dt = dT, h

(u) = 1 λ

 U

U



.

(Trans-III) x = 2 λ

 U

U



, dt = dT, h

(u) = 4 λ

 ∂

∂X

 U

U



.

(Trans-IV.1) λ

= 0 : dx = dX, dt = dT,

1

h

(ξ) dξ = 1

λ

ln |λU|.

(Trans-IV.2) λ

= 0 : dx = dX, dt = dT, u

h

(u) = − 1 λ

U.

(Trans-V) x = 1

λ ln |λU|, dt = dT, h

(u) = 1 λ

 U

U



. (Trans-VI) dx = dX, dt = dT, h

(u) = 2 U

U . (Trans-VII) x = U, dt = dT, h

(u) = U

.

(Trans-VIII) dx = dX, dt = dT,

exp





h

(ξ

)dξ



 dξ = U

.

We now write equations (I) - (VIII) in potential form and introduce new arbitrary functions φ

(x), which then lead to new linearisable autonomous evolution equations after transforming the corresponding potential equations by the pure hodograph transformation.

The linearising transformations for the equations so obtained result by composing the potentials and hodograph tranformations with the corresponding transformations (Trans- I)–(Trans-VIII) listed above.

Equation (I): We give two possibilities for writing (I) in potential form:

(I.i): Let

v

(x, t) = h

(u) + φ

(x) v

(x, t) = − 1

2 h

(u) ˙h

(u)u

− λ

,

where φ

is an arbitrary function of x and λ

∈ . Then v

= v

leads to (I). The potential equation takes the form

v

= v

− φ

(x)

(v

− φ

(x))

− λ

. (2.1)

Transforming (2.1) by the pure hodograph transformation

v(x, t) = χ, t = τ, x = V (χ, τ ) (2.2)

leads to the autonomous equation V

= V

+ φ

(V )V

(1 − φ

(V )V

)

+ λ

V

, (2.3)

where φ

= dφ

/dV . By the given change of variables and (Trans-I), it follows that (2.3) linearises to

U

= U

+ λ

U

by the transformation

V (χ, τ ) = U (X, T ), dτ = dT, V

− φ

(V ) = U

. (2.4) (I.ii): Let

v

(x, t) = xh

(u) + φ

(x) v

(x, t) = h

(u) − x

2 h

(u) ˙h

(u)u

− λ

.

Here φ

is an arbitrary function of x and λ

is an arbitrary constant. Clearly v

= v

leads to (I). The potential equation takes the form

v

= v

− φ

(x)

(v

− φ

(x))

x

− λ

. (2.5)

Transforming (2.5) by the pure hodograph transformation (2.2) leads to the autonomous equation

V

= V

+ φ

(V )V

(1 − φ

(V )V

)

V + λ

V

, (2.6)

where φ

= dφ

/dV . By the given change of variables and (Trans-I), it follows that (2.6) linearises to

U

= U

+ λ

U

by the transformation

V (χ, τ ) = U (X, T ), dτ = dT, V

− φ

(V ) = U U

. (2.7)

The same procedure can be followed for equations (II) - (VIII). We list the results below:

Equation (II): Two cases are given.

(II.i): Let

v

(x, t) = h

(u) + φ

(x) v

(x, t) = − 1

2 h

(u) ˙h

(u)u

− λh

(u) − λ

, λ = 0.

The potential equation is then v

= v

− φ

(x)

(v

− φ

(x))

− λ

v

− φ

− λ

. (2.8)

By the pure hodograph transformation (2.2), (2.8) leads to V

= V

+ φ

(V )V

(1 − φ

(V )V

)

+ λV

1 − φ

(V )V

+ λ

V

, (2.9)

which linearises to U

= U

+ λ

U

by the transformation

V (χ, τ ) = 1

λ ln |U(X, T )|, dτ = dT, V

− φ

(V ) = λU U

. (2.10) (II.ii): Let

v

(x, t) = e

h

(u) + φ

(x) v

(x, t) = − 1

2 e

h

(u) ˙h

(u)u

− λ

, λ = 0.

The potential equation is then v

= e

v

− φ

(x)

(v

− φ

(x))

− λe

v

− φ

(x) − λ

. (2.11)

By the pure hodograph transformation (2.2), (2.11) leads to V

= e

V

+ φ

(V )V

(1 − φ

(V )V

)

+ e

λV

1 − φ

(V )V

+ λ

V

, (2.12) which linearises to

U

= U

+ λ

U

by the transformation

V (χ, τ ) = 1

λ ln |U(X, T )|, dτ = dT, V

− φ

(V ) = λU

U

. (2.13) Equation (III): Two cases are given.

(III.i): Let

v

(x, t) = h

(u) + φ

(x) v

(x, t) = − 1

2 h

(u) ˙h

(u)u

− λ

x − λ

, λ

= 0.

The potential equation is then v

= v

− φ

(x)

(v

− φ

(x))

− λ

x − λ

. (2.14)

By the pure hodograph transformation (2.2), (2.14) leads to V

= V

+ φ

(V )V

(1 − φ

(V )V

)

+ λ

V V

+ λ

V

, (2.15) which linearises to

U

= U

+ λ

U

by the transformation

V (χ, τ ) = 2 λ

U

U , dτ = dT, V

− φ

(V ) = λ

2

 ∂

∂X

 U

U



. (2.16)

(III.ii): Let

v

(x, t) = xh

(u) + φ

(x) v

(x, t) = h

(u) − 1

2 xh

(u) ˙h

(u)u

− λ

x

2 − λ

, λ = 0, λ

= 0.

The potential equation then takes the form v

=

 v

− φ

(x) (v

− φ

(x))



x

− λ

x

2 − λ

. (2.17)

By the pure hodograph transformation (2.2), (2.17) leads to V

= V

 V

+ φ

(V )V

(1 − φ

(V )V

)

 + λ

2 V

V

+ λ

V

, (2.18)

which linearises to

U

= U

+ λ

U

+ λ

U by the transformation

V (χ, τ ) = 2 λ

U

U , dτ = dT, V

− φ

(V ) = U

U

 ∂

∂X

 U

U



. (2.19)

Equation (IV.1): Let v

(x, t) = exp



λ

1

h

(ξ) dξ + rx



 + φ

(x)

v

(x, t) =

 λ

h

(u) u

+ λ

r

 exp



λ

1

h

(ξ) dξ + rx



 − λ

, λ

= 0,

where r = λ

2 ±

 λ

4 − λ



. (2.20)

The potential equation then becomes

v

= v

− φ

(x) + (λ

− 2r)(v

− φ

(x)) − λ

. (2.21) By the pure hodograph transformation (2.2), (2.21) leads to

V

= V

V

+ φ

(V )V

+ ( 2r − λ

)(1 − φ

(V )V

) + λ

V

(2.22) which linearises to

U

= U

+ λ

U

+ λ

U by the transformation

V (χ, τ ) = X, dτ = dT, V

− φ

(V ) = e

U , (2.23) where r is given by (2.20).

Equation (IV.2): Let v

(x, t) = e

1

h

(ξ) dξ + φ

(x) v

(x, t) = e

u

h

(u) + 1

λ

e

− λ

, λ

= 0.

The potential equation then becomes

v

= v

− λ

(v

− φ

(x) − φ

(x)) + 1

λ

e

− λ

. (2.24)

By the pure hodograph transformation (2.2), (2.24) leads to V

= V

V

+ φ

(V )V

+ λ

(1 − φ

(V )V

) − 1

λ

e

V

+ λ

V

(2.25) which linearises to

U

= U

+ λ

U

by the transformation

V (χ, τ ) = X, dτ = dT, V

− φ

(V ) = e

U (ξ, T )dξ. (2.26) Equation (V): Let

v

(x, t) = h

(u) + φ

(x) v

(x, t) = − 1

2 h

˙h

(u)u

− λh

(u) − λ

λ h

(u) − λ

, λ = 0.

The potential equation then becomes v

= v

− φ

(x)

(v

− φ

(x))

− λ

v

− φ

(x) − λ

λ (v

− φ

(x)) − λ

. (2.27) By the pure hodograph transformation (2.2), (2.27) leads to

V

= V

+ φ

(V )V

(1 − φ

(V )V

)

+ λV

1 − φ

(V )V

+ λ

λ (1 − φ

(V )V

) + λ

V

(2.28) which linearises to

U

= U

+ λ

U

+ λ

U by the transformation

V (χ, τ ) = 1

λ ln |λU(X, T )|, dτ = dT, V

− φ

(V ) = λ U

U

. (2.29)

Equation (VI): Let

v

(x, t) = h

(u) + φ

(x) v

(x, t) = ˙h

(u)u

+ 1

2 h

(u).

The potential equation then takes the form v

= v

− φ

(x) + 1

2 (v

− φ

(x))

. (2.30)

By the pure hodograph transformation (2.2), (2.30) leads to V

= V

V

+ φ

(V )V

− 1

2 (1 − φ

(V )V

)

V

(2.31) which linearises to

U

= U

(2.32)

by the transformation

V (χ, τ ) = X, dτ = dT, V

− φ

(V ) = 2 U

U . (2.33)

Equation (VII): Let

v

(x, t) = h

(u) + φ

(x) v

(x, t) = − 1

2 h

˙h

(u)u

− λ

h

(u) − λ

. The potential equation then becomes

v

= v

− φ

(x)

(v

− φ

(x))

− λ

(v

− φ

(x)) − λ

. (2.34)

By the pure hodograph transformation (2.2), (2.34) leads to V

= V

+ φ

(V )V

(1 − φ

(V )V

)

+ λ

(1 − φ

(V )V

) + λ

V χ (2.35) which linearises to

U

= U

+ λ

U

+ λ

by the transformation

V (χ, τ ) = U (X, T ), dτ = dT, V

− φ

(V ) = U

. (2.36) Equation (VIII): Let

v

(x, t) = e

exp





h

(ξ

)dξ



 dξ + φ

(x)

v

(x, t) = e

exp





h

(ξ)dξ



 u

− λ

.

The potential equation then takes the form

v

= v

− λ

(v

− φ

(x)) − φ

(x) − λ

(2.37) By the pure hodograph transformation (2.37) leads to

V

= V

V

+ φ

(V )V

+ λ

(1 − φ

(V )V

) + λ

V

(2.38) which linearises to

U

= U

+ λ

U

by the transformation

V (χ, τ ) = X, dτ = dT, V

− φ

(V ) = e

U

. (2.39)

The same procedure could now, in principle, be applied again on these new equations (2.3), (2.6), (2.9), (2.12), (2.15), (2.18), (2.22), (2.25), (2.28), (2.31), (2.35), and (2.35), to construct new chains of linearisable equations. We present here only one example.

An example to further extend (2.3): Let W

(χ, τ ) = V + Ω(χ)

W

(χ, τ ) = V χ

(1 − φ

(V )V

) ,

where Ω is an arbitrary function of χ. We set λ

= 0 in (2.3). The potential equation then takes the form

W

= W

− Ω

1 − (W

− Ω

)φ

(W

− Ω) . (2.40)

Performing the pure hodograph transformation W (χ, τ ) = ξ, χ = ω(ξ, η), τ = η on (2.40) leads to

ω

= ω

ω

+ ω

Ω

ω

+ (ω

+ ω

Ω

)φ

(ω

− Ω) (2.41) which may be linearised to

U

= U

by the transformation

ω

− Ω(ω) = U(X, T )

dη = dT (2.42)

ω

ω

+ Ω

(ω) = U

U

φ

(U ) − 1 where Ω

= dΩ/dω.

A detailed analysis of this type of extensions will be considered elsewhere.

3 Recursion operators and hierarchies of linearisable equa- tions

In this section we give the recursion operators for the linearisable equations listed in Section 2. Before we list our results, we mention some relevant facts concerning recursion operators of evolution equations.

A recursion operator in two independent variables x and t is a linear integro-differential operator of the form

R[u] = 

P

D

+ 

Q

D

, (3.1)

where D

denotes the total x-derivative and D

the j-fold product of the inverse of D

. The coefficients P

and Q

depend in general on x, t and u, as well as a finite number of x derivatives of u. These operators were introduced by Olver [10] to generate (infinite) sequences of Lie-B¨ acklund (also called Generalised) symmetry generators. For an nth- order evolution equation in u,

u

= F (x, t, u, u

, u

, . . . , u

), (3.2)

a Lie-B¨ acklund symmetry generator Z is of the form

Z = η(x, t, u, u

, u

, . . . , u

) ∂

∂u , q > n, (3.3)

if it exists. The recursion operator is such that when acting on the Lie-B¨ acklund symmetry generator, the result is still a Lie-B¨ acklund symmetry of the same evolution equation (3.2), i.e.,

η

= R[u]η

. (3.4)

The recursion operator then satisfy the commutation relation

[L[u] , R[u]] = D

R[u], (3.5)

where L[u] is the linear operator

L[u] = ∂F

∂u + ∂F

∂u

D

+ ∂F

∂u

D

+ · · · + ∂F

∂u

D

(3.6)

and D

R[u] calculates the explicit derivative with respect to t. For more details we refer to [11, 2, 8].

In this paper we consider only autonomous evolution equations, i.e., equations invariant under translation in t and x. Therefore the equations admit the point symmetry generators

u

∂

∂u , u

∂

∂u . (3.7)

A hierarchy of evolution equations can then be obtained by applying the recursion operator on the t-translation symmetry, or equivalently on F , i.e.,

u

= R

[u]F, m ∈ N . (3.8)

3.1 Recursion operators for (I) - (VIII)

Below we list the recursion operators for the eight linearisable equations (I - VIII) listed

in Section 2. Note that equations (II) and (V) share the same recursion operator, as do

equations (I) and (VII):

I R

[u] = h

D

+ {h

}

h

u

+ 1 2

 h

u

+ {h

}

u

D

h ˙

h

II R

[u] = h

D

+ {h

}

h

u

+ λh

+ 1 2

 h

u

+ {h

}

u

+ λh

u

D

h ˙

h

III R

[u] = h

D

+ {h

}

h

u

+ 1

2 λ

x + 1 2



h

u

+ {h

}

u

+ 2λ

h

h ˙

 D

h ˙

h

IV R

[u] = D

+ 1

h



λ

− ˙h

 u

V R

[u] = h

D

+ {h

}

h

u

+ λh

+ 1 2

 h

u

+ {h

}

u

+ λh

u

D

h ˙

h

VI R

[u] = D

+

h ¨

h ˙

u

+ 1

2 h

+ 1

2 u

D

h ˙

VII R

[u] = h

D

+ {h

}

h

u

+ 1 2

 h

u

+ {h

}u

D

h ˙

h

VIII R

[u] = D

+ h

u

The recursion operators R

and R

are also given in [7].

Below we list the first nontrivial Lie-B¨ acklund symmetries of the eight linearisable second-order evolution equations (I - VIII), with the same equation numbers. We note that all symmetries are of order three and only equation (III) has an x-dependent third- order symmetry.

I Z

= h



u

+ 3 h ¨

h ˙

u

u

+ h ...

h ˙

u



∂

∂u

II Z

= h



u

+ 3 h ¨

h ˙

u

u

+ h ...

h ˙

u

+ 3λu

+ 3λ h ¨

h ˙

u

+ 2λ

u

 ∂

∂u

III Z

=

 h



u

+ 3 h ¨

h ˙

u

u

+ h ...

h ˙

u



+ 3 2 λ

x



h

u

+ {h

}u

+ 2λ

h

h ˙



+ 3λ

h

u

∂

∂u