ON A NONLINEAR THIRD-ORDER EVOLUTION EQUATION - PRESENT DEVELOPMENTS

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ON A NONLINEAR THIRD-ORDER EVOLUTION EQUATION - PRESENT

DEVELOPMENTS

Alfred Huber E-mail: soliton.alf@web.de

Address constantly: A-8062 Kumberg, Prottesweg 2a

ABSTRACT

In this paper the classical Lie group formalism is applied to deduce new classes of solutions of a less studied nonlinear partial differential equation (nPDE) of the third order. The nPDE under consideration is closely related to motions of plane curves.

Up to now no carefully performed symmetry analysis is available. Therefore we determine the classical Lie point symmetries including algebraic properties. Similarity solutions are given as well as nonlinear transformations could derived and periodic wave trains are obtained.

Since algebraic solution techniques fail symmetry analysis justifies the application yielding a deeper insight into the solution-manifold.

In addition, we shall see that the nPDE admits a new symmetry, the so called potential symmetry.

For some nonlinear ordinary differential equations (nODEs) created by similarity reductions the Painlevé property is discussed.

KEYWORDS: Nonlinear partial differential equations, evolution equations, similarity solutions, classical symmetries, potential symmetries.

PACS-CODE: 02.30Jr, 02.20Qs, 02.30Hq, 03.40Kf. MSC: 35K55, 35D35.

1. INTRODUCTION - OUTLINE THE PROBLEM

Although the origin of nonlinear partial differential equations (nPDEs) is very old, they have undergone remarkable new developments during the last half of the twenty century. One of the main impulses for developing nPDEs has been the study of nonlinear wave propagation problems. These problems arise in different areas of applied mathematics, physics, engineering, including fluid dynamics, nonlinear optics, solid mechanics, plasma physics, field theories, and condensed matter theories to mention some practical examples.

However, also pure mathematics like geometry, differential geometry including curved surfaces are subjects to nPDEs.

The paper deals with the following less studied nPDE of the third order [1] relating to motions of plane curves

    

  

    

  

   

 

2

2 2

2 1

u x

u_t , (1)

for which u u(x,t), uR2(,),



u,ux,ut



0, (x,t)R,  R

t , where the function u(x,t) is related to a variation of a physical quantity depending upon the positive time t. Also we have to require that uxxu0 preventing singularities at the critical point.

We seek for classes of solutions for whichuF(x,t), where FR2 and DR2 is an open set and further we excludeD:



(u,(x,t))D~:u(x,t)0



. Suitable classes of solutions are uI, I an interval so that ID and

2

:I R

u  holds.

Note:In what follows we suppress the item ‘classes’, so that ‘classes of solutions’ simply mean ‘solutions’.

Since the r.h.s of eq.(1) is a continuous functions we ensure at least local existence and due to the lemmas both from Peano and Picard-Lindelöf we assume uniqueness (also at least locally) in a given domain.

We also note that it may necessary to expand the domain so that we admit complex-valued solutions. For the nPDE, eq.(1) we require therefore:

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 

_ _

:C C C so we ensure complex-valued solutions if necessary.

The equivalence between integrable equations for the curvature and the invariant motion leads to new integrable equations. Since we often can express a motion law as a single evolution equation for some quantity in view of this equivalence; this evolution equation should also be integrable. In general there are many ways to reduce the motion, more precisely, the Euclidean motion

f n gt t

 

  

 

, (1.1)

where f and g suitable chosen and n and tare the Euclidean tangent and normal to a single equation [2]. Let us consider for a moment the mKdV flow [3]. Suppose the flow  can be expressed as the graph (x,u(x,t)) of some

function u on the xaxis. Using the fact that the normal speed of , u_t /(1u_x2)3/2 is given by ks, one finds that the function usatisfies

    

  

 

 

2 / 3 2

) 1

( x

xx t

u u x

u . (1.2)

Then, if we set vux we have

_

  

 

 



 2₂ ₂ ₃_/₂

) 1 ( v

v x

t . (1.3)

Both the eq.(1.2) and eq.(1.3) should be integrable. In fact, it turns out the integrability of eq.(1.3) established in [4] showing that it is the compatibility condition of a certain WKI-Scheme of inverse scattering transformation. This WKI-Scheme for the function v is connected to the AKNS-Scheme for kby a gauge transformation explicitly performed in [5].

It is of interest to find the inverse scattering transform for other equations as for the Euclidean and other geometries. In particular we mention that when the curve is locally convex a convenient way to express eq.(1.1) as a single equation is in terms of its support function h(,t), [1].

For the mKdV flow the equation becomes the form

  2 _ 



( _ )2



_ 2

1 ) ( 2 1

h h k

k

ht s , (1.4)

which represents exactly the nPDE, eq.(1) under consideration.

2. CLASSICAL SYMMETRY ANALYSIS - ALGEBRAIC GROUP PROPERTIES

Applied to nPDG, Lie’s classical method leads to group invariant solutions. Physically significant solutions arising from symmetry methods allow one to investigate the physical behavior of general classes of solutions. Group invariant solutions obtained via Lie’s approach provide insight into the physical models themselves.

Note: Well known algebraic solution techniques like the tanh-approach and other special function methods cannot be applied since the balancing parameter vanishes.

Explicit solutions also serve as benchmarks in the design, accuracy testing and comparison of numerical algorithms. In general one can say that a solution of any nPDE in two independent variables can be constructed by two invariants of the group.

One of these two invariants becomes the new independent variable 

 

x,t the so-called similarity variable and the other invariant plays the role of a dependent variableS

 

 .

A similarity reduction of a DE is closely connected with the invariance of the equation. We take up now the developments given in [9], [10], [11] omitting all technical details. To use symmetry groups in any applications we first deduce the symmetries of eq.(1).

The result is a well defined system of linear homogeneous PDEs for the infinitesimals i i(x,u) and )

, (xu i i 

 . ₁ stands for the first independent variable x, ₂ for second variable t and the dependent variable u is related to .

These constitutes the so-called determining equations for the symmetries of eq.(1) generated by Fréchet’s derivative [12], [13], [14]. Solving the coupled system of the linear homogeneous equations we found the symmetries to

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The result shows that the symmetry group of eq.(1) constitutes a finite five-dimensional point group (the group parameters are denoted by

k

_i, i1,2,3,4,5) generated by the vector fields

V1 t, V2 x,

3

3 t u

u t

V     , V4sinxu, V5cosxu. (2.1)

V5 /3. (2.2)

Tab.1

Commutator table of the nPDE, eq. (1)

where the Lie Brackets satisfy (i) bilinearity, (ii) skew symmetry and (iii) Jacobi identity. For this five-dimensional Lie algebra the commutator table for Vi is a skew-symmetric (55)-table whose(i,j)th entry expresses the Lie Bracket [V_i,V_j] given in eq.(2.2).

The coefficient Ci,j,k is the coefficient of Vi of the (i,j)thentry of Table 1 and the related structure constants can be read from Table 1 to give

C₁_,₂_,₁ 1, C₂_,₁_,₁1, C₂_,₄_,₄ 1/3,

C₂_,₅_,₅1/3, C₄_,₂_,₄ 1/3, C₄_,₃_,₅1, C₅_,₂_,₅ 1/3, C₅_,₃_,₄ 1. (2.3) Theorem:The Lie algebra of eq.(1) is solvable.

Proof: A Lie algebra L is called solvable if V(n) 0 for some n0. V(1) represents an ideal containing



V1,V2,V3,V4,V5



,

) 2 (

V is an ideal containing



V₁,V₄,V₅



and so on. This can be reduced to

V

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

0 

. Other useful algebraic group properties are mentioned: V₃ is the Casimir operator; the group order is five containing 31 subgroups. These subgroups are important later to perform a similarity reduction deducing new solutions. The metric (with the symmetric55second-rank tensor as the Cartanian) satisfies

     

 

     

 



0 0 0 0 0

0 0 0 9 / 11 0

0 0 0 0 0

j i

g with det(g)0 , (2.4)

and since the condition det(g)0 holds, the given algebra is therefore degenerate.

Note: Alternatively one can write with eq.(2.4) in a more convenient form



 

n

k i

k mi i lk im c c g

1 ,

.

In the Figure 1 we show the symmetry plots for a specific given choice of the group parameters ki. One observes a movement of the intersection both of the curves towards the negative xaxes.

]

[, V₁ V₂ V₃ V₄ V₅

1

V 0 V1 0 0 0

2

V 0 0 0 V4 V5/3

3

V ₀ 0 0 0 0

4

V 4V₁ V₄/3 V₅ 0 0

5

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8

- > 0, k2 - > 0, k3 - > 0, k4 - > 0, k5 - >

-2 -1

0 1

2 x

-2 -1

0 1

2

t 0.5

0.75 1 .25 1.5

u

-2 -1

0 1

2 x

8

-2

-1

0

1

2 x

-2

-1

0

1

2 t

0.5

1

1.5 u

-2

-1

0

1

2 x

Fig.1 Specific symmetry plots for the nPDE, eq.(1) generated by a special choice of the group parameters: Top: 1

5 

k , all other ki 0, below: k4 1, all other ki 0. 2.1 Classes of similarity solutions (invariant solutions)

Lie symmetries act on solutions of DE by mapping the set of solutions to itself. One set is of particular interest those which remain invariant under the action of a Lie symmetry. Given any one-parameter Lie group acting on a DE one is able to find all invariant solutions.

If a DE admits a r-parameter Lie group, each of the infinitesimal generators may have associated invariant solutions. Our main intension is to discuss solvable similarity solutions.

Therefore we are interested in two cases:

Case A: We set the group parameters k1k2 1 and k3 k4 k5 0 and the similarity variable



together with

the relevant transformation reads as x0 and

3 / 1

) 1 ( t

u S



 .

The related nODE of the third order is derived to

3 ₃ 0

3 3

2 2

     

  

        

  

 

d S d d dS d

S d S

S . (2.5)

Case B: If we set the group parameters k₁k₃ 1 and k₂ k₄ k₅ 0 we get the important case of traveling waves by the transformation tx and S u for the nODE

3 3 3

2 2

1 dS

S d dS

S d S d

dS _

   

   

     

  



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Transformations fail as well as the related system of the nODEs of the first order can not be solved explicitly. Therefore, a numerical procedure is of advantage. First we take up a series representation up to order four at the regular point 0 for the DE, eq.(2.5)

₀ ₁ ₂ 2



₀4 3 ₁ 6 ₀3 ₂ 12 ₀2 ₂2 8 ₀ ₂3



3

 

4 18

1 )

( a a a   a  a  a a  a a  a a  O 

S (2.6)

with suitable chosen parameters ai, i0,1,2 and for the nODE (2.5a) we calculate

0 1 2 2 1



8 23 6 02 2 12 0 22 03 1



3

 

4

6 )

( a aa a a  a a  a a a   O 

S . (2.6a)

Theorem I: Eq.(2.5) does not possess the Painlevé property. To see this we firstly expand eq.(2.5):

3 3 3 3 4 0

2 2 3 2

2 2 2 3

2 2 3

3

             

 

       

 

 d S

dS d

S d S d

S d

. (2.7)

Proof:We assume the solution of the resulting nODE, eq.(2.7) in the form f ~(₀)p where Re(p)0, ₀ arbitrary and substitute it into eq.(2.7). Different terms may balance to provide the values of p and . We found several possibilities for a suitable balance to give the following values:p₁ 1, p₂ 1/3, p₃ 5/3. At this stage the analysis terminates since we found a broken value indicating an algebraic branch point. That means that the eq.(2.7) is not of P-type. However, the case p1might allow a suitable Laurent expansion.

For the pair (p,) the nODE is truncated to retain only the leading terms to derive the truncated ODE

3 4 0

3 3

   S

d S d

. (2.8)

which allows to calculate the parameterdirectly. We found a polynomial of the third order for the quantitywhich has solutions

₁3 18,

3 6 3 / 2 3

, 2

4 3 3 ) 2 / 3

(  i

 

 , i2 1 (2.8a)

Note: The general form of Eq.(2.8), say, y'''ayn, for y y(x)and a3, n4 can be reduced to an Emden-Fowler-type equation with known solutions. To see this, introduce w(y)(y')2 to derive w''(y)2 ay''w1/2. Then a two-term expansion in the form f ~(₀)p(₀)pr is sought. Substituting this expansion into the leading terms of the nODE and retaining terms of order, the characteristic polynomial Q(r)0 is found to

0 18 11 6 2

3_ _ _ _

r r

r . The roots of the equation Q(r)0 determine the resonances given by

r

1





1 ,



7 23



2 1

3 ,

2 i

r   . (2.8b)

The root r₁1 represents the arbitrariness of the position of the singularity ₀ and, since r₂C this root indicates an algebraic branch point. Therefore, the given equation is not of P-type ■

Note: Physically speaking the occurrence of complex-valued roots in eq.(2.8b) are closely related with chaotic behaviour of the nPDE eq.(1) under consideration.

Theorem II: The nODE, eq.(2.5a) also does not possess the Painlevé property.

Proof: To exclude algebraic branch points at the first stage we require Re(p)0 with pZ.

We foundp₁ 2, p₂ 4/3 and p₃ 2/3. This contradicts the given assumption and we conclude therefore that the nPDE is also not of P-type■

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-4 -2 0 2 4 -1 -0.5 0 0.5 1 1.5 2 2.5

-4 -2 0 2 4

-1 -0.5 0 0.5 1 1.5

Fig.2 Periodic wave trains of the nODE, eq.(2.5) left and the nODE, eq.(2.5a) right. The motions are performed by using different initial conditions, e.g. S(0)0, S'(0)1, S''(0)0. It can be shown that the motions are stable in

the considered domain.

Case C: Another case of interest is given by the choice of the parameters k₂ k₄ 1 and k₁k₃ k₅ 0 which transforms by x and S(u3sinx)t1/3. We chose this example to show the complexity of the underlying nODE whereby such types of DEs only can be solved by numerical standard procedures. Let the third-order nPDE be . 0 729 729 729 2430 243 1620 2700 1458 27 270 900 972 1000 270 3 3 3 / 11 3 2 2 3 / 20 2 2 2 2 3 / 17 2 2 2 3 / 14 2 2 3 3 / 14 2 2 2 3 / 11 2 2 2 3 / 8 2 2 3 / 8 4 3 / 11 3 3 / 8 2 2 3 / 5 3 / 5 3 3 / 2 3 / 2                                                                                                             d S d d S d S d S d d dS d S d d dS S d S d d dS d S d d dS S d S d d dS S d S d d dS d dS S d dS S d dS d dS S S (2.9)

In Fig.3 we show some planar solution curves to imagine the basic run of some special solutions.

Case D:The last case of interest is generated if we set the group parameters k₁k₄ 1and k₂ k₃ k₅ 0 with the transformations x and Stsinx. The relating nODE of the third order is given by

sin 0 3 2 2 3 3                 d S d S d dS d S d

. (2.10)

Again, a solution in terms of an analytical power series representation up to order six yields

₀ ₃ ₂ 2 ₃ 3



₀3 ₂ ₀2 ₂ ₀ ₂2 ₂3



4 3 5

 

6

20 8 12 6 2 24 1 6 )

( a  aa  a  a  a  a a  a a  a  a  O 

S ,

(2.10a) with suitable chosen parameters ai.

-7.5 -5 -2.5 2.5 5 7.5

x -10 -5 5 10 15

   

xx u u t V

, 1  0

 

u x V

. (2.11)

The infinitesimals of the symmetry are calculated to

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k k sinx k cosx 3

4

5 4

3

1  

 , k V k k cosx k sinx

3 3 4 5

1 6

2    

 . (2.11a)

The given systems are related to new symmetries which differ from the symmetry group, eq. (2) completely: In opposite to the symmetry group eq.(2), here, we are confronted with an infinite six dimensional PT. That means that the difference exists in the dimension (increasing one time) of the group as well as in the number of elements.

3. CONCLUSION

Let us now summarize some new important results for the less studied nPDE, eq.(1).

Due to the fact that algebraic solution techniques fail we have to use alternative methods if we are interested in calculating solutions explicitly. So, the classical and the non-classical symmetry methods are suitable. Algebraic group properties are discussed in detail in the Chapter 2.

The Chapter 2.1 introduces similarity reductions to derive highly nODEs if we assume a special choice for the group parameters. Such types of nODEs can not be solved analytically therefore numerical studies are performed.

For some of the given nODEs it was shown that they are not of P-type. A special nODE, the eq.(2.10) admits periodic wave trains importantly in physical applications.

Next we showed the existence of a new symmetry; the potential symmetry in Chapter 2.2, especially created by the infinitesimals (2.11a) where the dimension of this symmetry changes comparing by the classical symmetry, eq.(2). In addition, for practical calculations we gave some series representation in terms of the similarity function explicitly.

As a conclusion one can say that the given paper is suitable to improve the understanding of the underlying nPDE considerably and we showed that we can enlarge the solution-manifold successfully by an alternative approach.

4. REFERENCES

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