37
ANALYSIS OF A FOURTH-ORDER NONLINEAR EVOLUTION EQUATION BY CLASSICAL AND NON-CLASSICAL SYMMETRY
METHODS
Alfred Huber
Institute of Theoretical Physics,
Technical UniversityGraz, Austria.
ABSTRACT
In this paper the classical Lie group formalism is applied to deduce new classes of solutions of a nonlinear partial differential equation (nPDE) of the fourth order, the so called Derrida-Lebowitz-Speer-Spohn equation (DLSS) importantly in several physical applications.
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 new nonlinear transformations could derived.
It is further shown that algebraic solution techniques fail so a symmetry analysis justifies the application. In addition, we discuss approximate symmetries to the first time and moreover we shall see that the DLSS equation admits a new symmetry, the so called potential symmetry.
Broadly speaking, the analysis allows one to deduce wider classes of new unknown solutions either of practical and theoretical usage.
Keywords : Nonlinear partial differential equations, Evolution equations, DLSS equation, Similarity solutions, classical symmetries, non-classical symmetries.
PACS-Code : 02.30Jr, 02.20Qs, 02.30Hq, 03.40Kf.
MSC : 35K55, 35D35
1. INTRODUCTION
We summarize some known results and give a short overview whereby the remarks are far from being complete.
However some important notes should indicate:
The following nPDE of the fourth order with a logarithmic nonlinearity is under consideration
u u
xx
xxt
u ln
, (1)for which
u u ( t x , )
,u ( t x , )
: R
R
R, u , u
x, u
xx, ... 0
,x
R,x 0
,t
R+,t 0
.The equation plays an important role in theoretical physics derived firstly by the above named authors [1], [2].
These authors studied the interface fluctuations in a two-dimensional spin system, the so-called time discrete Toom model. In a suitable scaling limit a random variable
u
related to the deviation of the interface from a straight line satisfies the one-dimensional form of eq.(1). The multi-dimensional DLSS equation appears in quantum semiconductor modelling the zero-temperature as the zero-field limit of the quantum drift-diffusion model [3].The variable
u
describes the electron density in a microelectronic device or otherwise in a quantum plasma. In both applications the variableu
is a nonnegative quantity. In fact to prove the preservation of the positivity or the non- negativity of solutions seems the main analytical difficulty in studying of eq.(1). There is generally no maximum principle available for fourth-order equations which allow one to conclude fromu
0 0
thatu (., t ) 0
at later timest 0
. Consequently, one has to rely on suitable regularization techniques and a priori estimates. The latter are difficult to obtain because of the highly nonlinear structure of the equation.We remark that similar difficulties appear in studies of the Thin-Film Equation (TFE), [4] and other types of nPDEs.
The first mathematically treatment of eq.(1) was done in [5]. There, the local in-time existence of classical solutions for strictly positive initial data
W
1,p() withp d
were proven.The existence result is obtain by means of a classical semigroup theory applied to the equation
38
0
2
2
u
u u
t
u
foru ( x , 0 ) u
0( x ) 0
,x
d ; (2) which is equivalent to eq.(1) as long asu
remains bounded away from zero.The global existence for the one-dimensional eq.(1) was also shown. So the one-dimensional DLSS equation with periodic boundary conditions was extensively studied in the context of entropy production methods and the exponentially fast decay of the solutions to the steady state has been proved [5], [6], [7], [8], [9] and [10]. In addition a numerical study of the long-time asymptotic for various boundary conditions can found in [11].
It is further proven that eq. (1) constitutes the gradient flow for the Fisher information with respect to the Wasserstein measure. The resulting existence theorem is more general formulated as it holds in the non-physical dimensions
d 4
and on unbounded domains.The treatment of the DLSS equation as a gradient flow promotes a deeper understanding of its nature. A set of explicit solutions of eq. (1) are known in particular in one spatial dimension [12].
We mention both the travelling wave solution
u ( x , t ) Ai
2( x t )
where Ai denotes the Airy function and the stationary solutionu ( x , t ) sin( 2 x )
.Now however, we bridge the gap between some known results and new derived by using an alternative approach.
For solving nPDE many powerful algebraic solution tools are well-know, e.g. the hyperbolic tangent method [13], several new algebraic methods derived by the author [14], [15] and the use of classical and non-classical Lie group analysis, e.g. [16], [17] .
Firstly, we will show that algebraic methods cannot apply to handle eq.(1): Differentiating out eq.(1) and using the a frame of reference so that
u ( x , t ) f ( )
,f C
1, x t
and
R is a constant to be determined.Eq.(1) becomes a nonlinear ordinary differential equation (nODE) of the fourth order. The balancing parameter n (which is necessary to determine a series) computes by balancing the highest order nonlinear term with the highest order partial derivative term in the relevant nODE of the fourth order.
In this case we get the result:
3 0
. This contradiction means that algebraic methods fail in general. Since we are not successful by the use of algebraic methods we need a symmetry analysis to deduce further classes of solutions.Further we arrange that we suppress in the following the item ‘classes of solutions’ i.e. we always mean simple
‘solutions’.
2. CLASSICAL SYMMETRY ANALYSIS - ALGEBRAIC GROUP PROPERTIES
This part is devoted to the classical Lie group analysis. A similarity reduction of a differential equation (DE) is closely connected with the invariance of the equation. We take up now the developments given in [18], [19], [20]
omitting all technical details. To use symmetry groups in any application we first deduce the symmetries of eq.(1).
The result is a well defined system of 15 linear homogeneous PDEs for the infinitesimals
i
i( u x , )
and) , ( u x
i i
.
1 stands for the first independent variablex
,
2 for second variablet
and the dependent variableu
is related to
.These constitutes the so-called determining equations for the symmetries of eq. (1) generated by the derivative due to Fréchet [21], [22], [23].
The related infinitesimals could obtain by solving the system of determining equations to give
1 k
2 k
3x ,
2 k
1 4 k
3t , k
4u ( x , t )
. (2.1) The result shows that the symmetry group of eq.(1) constitutes a finite four-dimensional point group where the group parameters are denoted byk
i,i 1 , 2 , 3 , 4
. So eq.(1) admits the four-dimensional Lie algebra L of its classical infinitesimal point symmetries related to the following vector fields:
V
1
x, V
2
t, V
3 4 t
t x
x, V
4 u
u.
(2.2) This group of four vector fields contains translations in time and space so that t ' t , x ' x
holds for V
1,V
2
and the associated differential operators V
3,V
4
are related to scaling and/or stretching operations.Therefore, to form a Lie algebra L we have:
V
1, V
3 4 V
1, V
2, V
3 V
2, V
3, V
1 4 V
1, V
3, V
2 V
2 . (2.3)39
For this four-dimensional Lie algebra the commutator table for
V
i is a4 4
table whose( i , j ) th
entries expresses the Lie Bracket
V ,
iV
j
given in (2.3).The table is skew-symmetric and the diagonal elements vanish. The coefficient
C
i,j,k is the coefficient ofV
i of theth ) j , i
(
entry of Tab.1. The structure constants can be read from Tab.1:
C
1,3,1 4 , C
3,2,2 1 , C
3,1,1 4 , C
3,3,2 1
. (2.4) Theorem: The given algebra is solvable.Proof: A Lie algebra L is called solvable if
V
(n) 0
for somen 0
.V
(1) represents an ideal V
1, V
2, V
3
,V
(2) an ideal with V
1,V
2
; this can be reduced toV
(4) 0
forn 4
.Other useful algebraic group properties are mentioned: For the eq.(1),
V
4 is the Casimir operator, the group order is four containing 15 subgroups. These subgroups are important later to perform asimilarity reduction deducing new solutions.
As a last algebraic property we point out that the metric (
4 4
Cartanian tensor) satisfies:
0 0 0 0
0 17 0 0
0 0 0 0
0 0 0 0
g
ij with det(g) = 0 , (2.5)and, since the condition
det( g ) 0
holds, the given algebra is therefore degenerate.Note: Alternatively one can write with eq.(2.5) in a convenient form
nk i
k mi i lk
im
c c
g
1 ,
. 2.1 Classes of similarity solutions
All relevant 15 classes are listed in Tab.2 . Let us now discuss important similarity solutions for special subgroups.
If we set the group parameters
k
1 k
2 k
4 0
andk
3 1
, (the Case M1); otherwisek
1 k
2 1
and4
0
3
k
k
, (Case J1) respectively, the traveling-wave transformation results. The similarity functionS ( )
itself has to satisfy the following nODEs of the fourth order:0 '
' ' 2 ) ' ' ' 2 ' ' ' ' 2 ' ' ( 2 ' ' ' 5 ' ' '
2 ' 2 '
'
2 2 3 4 3 2 2 2 3 3 (4)3
S S S S S S S S S S S S S S S S S S S S S
S
, (2.6)for the case J1 (the prime means derivation w.r.t the independent variable
) and
5 2 2
30
2 4 2 2 3
3 2
2 2
2 4
4
3
d
S S d d
dS d
S d d
S d d S dS d
dS d
S S d d
S
S d
, (2.6a)for the case M1. We require that
S
: R
R
R ,S 0
,
R, a a
holds.Such nODEs can only solved numerically and in Fig.1 we show some solution curves for different suitable initial values.
Further, for practical calculations some series representations up to order four are useful to give:
54 3
0
3 2
1 3 0 3 2 1 2 2 1 2 0 2 1 2 1 0 4 1
3 3
2 2
1 0
) 1 ( ) 1 24 (
)) 12 2
( )) 2
( 4 8 ( ) 5 ( 2 2 (
) 1 ( ) 1 ( ) 1 ( )
(
a O
a a
a a a a a a a a a a a a a
a a
a a S
and for the eq.(2.6) we found (2.6b)
40
5 4
3 0
3 1 2 2 2 0 2 2 1 0 4
1 1 3 0
3 3
2 2
1 0
)]
1 [(
) 1 ( 24
)) 3 2 ( 4 10
2 (
) 1 ( )
1 ( ) 1 ( )
(
O a
a a a a a a a a
a a
a a
a a S
, (2.6c)
with arbitrary chosen
a
i,i 0 , 1 , 2 , 3
buta
0 0
.In Fig.2 we simulated the above given series as three-dimensional plots. We start by eq.(2.6b) with
x t
for the similarity variable by 20
. By increasing the wave number the functionu ( t x , )
increases fast but the outer form of the solution surface does not change.For the series (2.6c) a similar surface is obtain but here we use a negative value for the wave number, that is
t x 4
. We remark that by choosinga
i 0 S ( ) 1
and fora
0 a
3 1 a
1 a
2 0
,S ( )
becomes a cubic polynomial.
2.2 The non-classical case I: Potential symmetries
For more technical details we refer to [18], [19], [20] and [24]. For the DLSS equation we found the following: The equation admits two possible potential systems
1,
2.Both systems can be formulated for the four dependent variables
V
i,i 1 , 2 , 3 , 4
and both variables are treated in their derivations w.r.t. the independent variables and are denoted by subscripts:Potential system
1 Potential system
2
. 0
0 1 0
1 1 3
2 2 3 2
x u V
x V V
x u t
V x
u u
(2.7)
. 0 0
1 0 1
2 2 2 4
2 2 4
2
x x u V
x V x V
x u x t V x
u x u
(2.7a)
The given systems are related to new symmetries which differ from the symmetry group (2.1) completely: In opposite to the symmetry group (2.1), here, we are confronted with a finite four dimensional PT that means that the difference exists in the dimension of the group as well as the number of elements.
The most important fact however is that a new symmetry appears in form of the new infinitesimal
3 related to the dependent variableV
i. The point symmetry group is also a finite-dimensional group forming a six-dimensional Lie algebra.Therefore we conclude that the eq.(1) admits a new symmetry, the potential symmetry which represent a new contribution:
. 5 4
) 2 , (
4
3 3 1 2 3
1 6 1 1 3 2
6 1 1
6 5 2
6 4 1
x k V k k
V V k
k k
k k t x u
t k k
x k k
(2.8)
Since the symmetries for the potential system (2.7a) does not differ extremely we restrict to point out only the symmetries of the potential system
1. These infinitesimals can further be use to generate new solutions.41 2.3 Approximate symmetries
In this section we follow [24], [25] and [26], respectively and our intension is to deduce new results without referring too much theory. Let us introduce
as a small parameter0 1
determining the strength of the nonlinearity of the DLSS equation. So that we can rewrite eq.(1) as
u u
xx
xxt
u ln
,u ( t x , )
: R
R
R, a x a
,t
R+,t 0
, 0
(2.9)Then, approximate symmetries follow by the infinitesimals
1 k
3( 1 ) ,
2 k
4 k
1 , k
2( 1 )
(2.10) representing a four-dimensional infinite approximate point symmetry group in first order approximation depending on the perturbation parameter
. The generating vector fields are
V
1
t, V
2
t, V
3 ( 1 )
u, V
4 ( 1 )
x.
(2.11) The group of four vector fields contains translations depending on the perturbation parameter
sothat the collected coefficients of the generating vector fields are deduced to :
0 , , 0 , 0 , 0 , 1 1 , 0 0 , 1 , 0
. (2.11a) Possible reductions can be calculated by combining several sub-groups of (2.11a).That is in detail
V
l V
m V
n with l , m , n 1 , 2 , 3 , 4
.In total three suitable combinations for the above given vector fields exist. They allow one to investigate new useful approximate symmetry solutions.
We refer to Tab.3 where all relevant transformations are listed. We have a special interest in the case of traveling waves, that means especially the combination
V
1 V
3 V
4, Case C2 .Surprisingly this leads to the same nODE as for the classical case, that is in detail analogues to eq.(2.6a) for the case
1
. For the case A2 the following complicate nODE with non-constant coefficients of the fourth order is given to, 0 )) ) 2 ( ' ' ' 4 ( ) 2 ( ) 2 ( ' ' 2 ' ' 20 ( ) 2 ( ' ' ' ) 2 ( 2 ) 2 ( ' ' 20
) 16 8 ( 8 ) 12 ) 6 ( ( 16 ( ' ) ' ' ) 2 ( 5 48 ( ' ' 2 ' 16 32
) 4 ( 2
2
3 2
2 4 2
S S
S S
S S S S
S S
S S
S S S S S S
S S S
(2.11b) and for the case B2 we get the following nODE:
5 2 2
30
2 4 2 2 3 3 2
2 2
2 4
4
3
S
d dS d
S d d
S d d S dS d
dS d
S S d d
S
S d
, (2.11c)which differs from eq.(2.6a) by a first derivative term. For both equations a series is calculated to
1 [( 1 )] .
) 2 ( 3 ) 2 ( ) 2 ( 5 ( 4 )) 6 (
)) 6 ( 12 ( 4 ( ) 3 ( 16 ( 2 ) 2 ( 5 24 ( ) 8
2 ( 12
1
) 1 ( )
1 ( ) 1 ( )
(
3 4 2
0 3 0
2 2 0
0 0
0 0 0
2 1
2 0
2 1 4 1 3 3 1
0
3 3
2 2
1 0
O a
a a
a a
a a
a a a
a a
a a
a a a a
a a
a a S
(2.11d) For the case A2, the nODE (2.11c) we have
5 4
3 0
3 1 2
2 0 0 2 2 1 0
4 1
3 3
2 2
1 0
)]
1 [(
) 1 ( 24
)) 12 8
( 10
( 2
(
) 1 ( )
1 ( ) 1 ( )
(
O a
a a a
a a a a a
a
a a
a a S
, (2.11e)
with arbitrarily chosen
a
i,i 0 , 1 , 2 , 3
anda
0 0
. The convergence suitably has to be proven.The simulations in the figures Fig.3 and Fig.4 show planar as well as three-dimensional plots.
42 2.4 The non-classical case II: Generalized symmetries (GS)
We find it advisable mentioning some basic notes. It is obvious from Lie theory that point symmetries are subsets of generalized symmetries [27] and [28].
The determination of the characteristics for the general case follows by a similar algorithm as in the case of point transformations (PT) in the classical case.
Classical symmetries of a (n)PDE (assumed in a general form
0
) are PT which guarantee the invariance of the solution space and so PT are created by infinitesimal transformations.The determining equations for the characteristics
GS
are consequences of the following relation:
pr
GS
0 0
. (2.12) Here,
pr
GSdenotes the prolongation of the vector field
GS and ‘GS’ means ‘Generalize(d) Symmetry’. Let D be an open set in R2 andu
2(D). The functionu
(2): D U
(2) U U
1 U
2,) , , , , ,
)
(
2 (
tt xt xx t
x
u u u u
u u
u
as an example is called the prolongation of order two of the function u which is used next.If the order is equal to identity we arrive at the well-known contact transformations. By increasing the order of derivatives
n 1
we shall find higher order GSs (difficulties relating to computer time appear in practical calculations).The total space
D U
(2) is called jet space of the order two of the base spaceD U
.The main difference however is the fact that in general the characteristics depend on derivatives of an infinite order.
In case of the DLSS equation we found GSs of the first order depending linearly on the first x-derivative:
GS
1 x , t , u , u
x, u
t k
2u k
1u
x. (2.13) This symmetry also changes from the symmetries given in (2.1), (2.8) and (2.10).Here we are confronted with a two-dimensional finite group of transformations where the second part
u / x
is related to scaling and/or stretching operations.For the case
n 2
, assuming second partial derivatives as defined above we interestingly found the same result as in the casen 1
:
GS
2 x , t , u , u
x, u
xx, u
tt, u
xt k
2u k
1u
x. (2.13a)That means that generalized symmetries exist for the eq.(1) but only for the lower cases
n 1 , 2
these symmetries can be calculated in a reasonable computer time.3. SUMMARY
Let us now summarize some new important results for the DLSS equation, 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 method are suitable. Algebraic group properties are discussed in detail in the chapters 2 and 2.1.
We refer to the nODEs, eq.(2.6) and eq.(2.6a) where a graphical plot is seen in Fig.1 because a closed-form solution is not possible. Therefore, some series representations for practical calculations are given as well as some numerical studies are performed.
Next we showed the existence of a new symmetry; the potential symmetry in chapter 2.2, especially created by the infinitesimals (2.8).
The dimension of this symmetry also changes comparing by (2.1).
In case of approximate symmetries nonlinear transformations are summarized in table form and some interesting cases are studied. Here, the same situation appears so that the related nODEs could not be solved in a closed-form.
In addition, for practical calculations we gave some series representation in terms of the similarity function explicitly.
In the chapter 2.4 we found to the first time that generalized symmetries with reduced order arise.
As a conclusion one can say that the given paper is suitable to improve the understanding of the DLSS equation considerably and we showed that we can improve the solution-manifold successfully by an alternative approach.
43 4. REFERENCES
[1] B. Derrida, J. Lebowitz, E. Speer, H. Spohn, Dynamics of an anchored Toom interface, J. Phys. A 24, p. 4805, 1991 [2] B. Derrida, J. Lebowitz, E. Speer, H. Spohn, Fluctuations of a stationary nonequilibrium interface,
Phys. Rev. Lett. 67, p.165, 1991
[3] A. Jüngel, R. Pinnau, A positivity preserving numerical scheme for a nonlinear fourth order parabolic equation, SIAM J. Numer. Anal. 39, p.385, 2001
[4] A. Huber, A classical and non-classical symmetry analysis of variant Thin Film Equations, Int. J. Engineering Science and Technology, (in press), 2010,
[5] A. Jüngel, R. Pinnau, Global non-negative solutions of a nonlinear fourth-oder parabolic equation for quantum systems, SIAM J. Math. Anal. 32, p.760, 2000
[6] M. Caceres, J. Carrillo, G. Toscani, Long-time behavior for a nonlinear fourth order parabolic equation, Trans. Amer. Math. Soc. 357, p.1161, 2004
[7] J. A. Carrillo, J. Dolbeault, I. Gentil, A. Jüngel, Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations, Discrete Contin. Dyn. Syst. B 6, p.1027, 2006
[8] J. Dolbeault, I. Gentil, A. Jüngel, A logarithmic fourth-order parabolic equation and related logarithmic Sobolev inequalities, Commun. Math. Sci. 4, p.275, 2006
[9] M. P. Gualdani, A. Jüngel, G. Toscani, A nonlinear fourth-order parabolic equation with non-homogeneous boundary conditions, SIAM J. Math. Anal. 37, p.1761, 2006
[10] A. Jüngel, G. Toscani, Exponential decay in time of solutions to a nonlinear fourth order parabolic equation, Z. Angew. Math. Phys. 54, p.377, 2003
[11] J. A. Carrillo, A. Jüngel, S. Tang, Positive entropic schemes for a nonlinear fourth order equation, Discrete Contin. Dynam. Syst. B 3, p.1, 2003
[12] P. Bleher, J. Lebowitz, E. Speer, Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations, Comm. Pure Appl. Math. 47, p.923, 1994
[13] A. Huber, Solitary solutions of some non-linear evolution equations, Applied Mathematics and Computation 166/2, p.464, 2005
[14] A. Huber, A novel algebraic procedure for solving evolution equations of higher order, Chaos, Solitons and Fractals 34/3, p.765, 2007
[15] A. Huber, On non-linear evolution equations of higher order – the introduction and application of a novel computational approach, Appl. Math. and Comp. 215, p.2337, 2009
[16] A. Huber, A note on new solitary and similarity class of solutions of a fourth order non-linear evolution equation, Appl. Math. and Comp. 202, p.787, 2008
[17] A. Huber, A note on class of travelling wave solutions of a non-linear third order system generated by Lie’s approach, Chaos, Solitons and Fractals 32/4, p.1357, 2007
[18] N. Ibragimov, Lie Group Analysis, Vol. III, CRC Press, Inc., 1994
[19] P. Olver, Applications of Lie Groups to Differential Equations, Springer, 1986 [20] G. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, 1989
[21] A. Huber, Int. J. Differential Equations and Dynamical Systems, Vol.15, Nos.1&2, p.27, 2007
[22] A. Huber, A note on class of similarity solutions of a non-linear second order field equation, Electronic Journal of Theoretical Physics (EJTP), (in press), 2009
[23] A. Huber, A note on class of solitary and solitary-like solutions of the Tzitzéica-equation, Physica D 237, p.1079, 2008
[24] A. Huber, The Cavalcante -Tenenblat equation – Does the equation admit a physical significance?, Appl. Math. and Comp. 212, p.14, 2009
[25] N. Ibragimov, Transformation Groups Applied to Mathematical Physics, Reidel Publ., Dortrecht, 1985 [26] N. Ibragimov, Sophus Lie and harmony in mathematical physics, On the 150th anniversary of his birth, Math.
Intel. 16, p.20, 1994
[27] E. Noether, Transport Theory Stat. Phys. 1, p.186, 1971
[28] F. Klein, Über Differentialgesetzte für die Erhaltung von Impuls und Energie in der Einsteinschen Gravitationstheorie, Nachr. Ges. Wiss. Göttingen Math. Phys.2, p.171, 1918
44 5. TABLES
V1 V2 V3 V4
V1 0 0 - 4V1 0
V2 0 0 -V2 0
V3 4V1 V2 0 0
V4 0 0 0 0
Tab.1 Commutator table of the DLSS equation, eq.(1).
Case Choice of the group parameters Transformation for
Transformation for SA1
k
1 k
2 k
3 0 , k
4 1 t u S
B1
k
1 k
2 k
4 0 , k
3 1 t x
4 u S
C1
k
1 k
2 0 , k
3 k
4 1 t x
4 u x
1 S
D1
k
1 k
3 0 , k
2 k
4 1 t u e
x S
E1
k
1 k
4 0 , k
2 k
3 1 t ( 1 x )
4 u S
F1
k
2 k
3 0 , k
1 k
4 1 x u e
t S
G1
k
1 k
2 k
3 0 , k
2 1 t u S
H1
k
1 0 , k
2 k
3 k
4 1 t x
4 u x
1 S
I1
k
2 0 , k
1 k
3 k
4 1 ( 1 4 t ) / 4 x
4 u x
1 S
J1
k
3 0 , k
1 k
2 k
4 1 t x u e
x S
K1
k
4 0 , k
1 k
2 k
3 1 ( 1 4 t ) / 4 ( 1 x )
4 u S
L1
k
2 k
3 k
4 0 , k
1 1 x u x
1 S
M1
k
3 k
4 0 , k
1 k
2 1 t x u S
N1
k
1 k
2 k
3 k
4 1 ( 1 4 t ) / 4 ( 1 x )
4 u ( 1 x )
1 S
O1
k
2 k
4 0 , k
1 k
3 1 ( 1 4 t ) / 4 x
4 u S
Tab.2 The 15 nonlinear transformations for the DLSS equation, eq.(1), kimeans the group parameters, is the similarity variable and S means the similarity function according to the 15 classes of the group.
Note: For completeness all cases are listed, A1 is equal G1
45
Case Combinations of the vector fields Transformation for Transformation for S
A2
V
l V
2 V
3
1
t x u x S
B2
V
l V
2 V
4x u t S
C2
V
l V
3 V
4t x u S
Tab.3 Nonlinear transformations in case of approximate symmetries of the DLSS, eq. (1), the second column means suitable combinations of the vector fields and is the perturbation parameter.
6. GRAPHICS AND FIGURES
Fig.1 Numerically generated integral curves for the case of traveling-wave solutions of the nODEs eq.(2.6), (2.6a) for different initial conditions. Left: The case J1, Right: The case M1. Note, that the negative similarity function S is plotted against the similarity variable. In the considered domain the solution curves are stable.
Fig.2 Three-dimensional solution surfaces for the series representations eq.(2.6b) and (2.6c).
Left: The case J1 generated by the choice for the parameters a0 200, a140,a2 50and a38. For the wave number we use 20. Right: The case M1 for a0a1a2a31 and the wave number is a positive number so that the similarity variable is x4t. By increasing for both cases the outer form of the surfaces remain constant. One has to be in mind the high value of the solution functionu(x,t).
46
0.2 0.4 0.6 0.8 1
2 4 6 8 10 12 S
-3 -2 -1 1 2 3
10 20 30 40 50 S
Fig.3 Numerically generated integral curves for the case of traveling-wave solutions of the nODEs eq.(2.11b), (2.11c) by choosing different initial conditions for the case of approximate symmetries with 1.
Left: The case A2, Right: The case B2. Note, that the negative similarity function S is plotted against the similarity variable once again. Also, in the considered domain the solution curves are stable
.
-2 0
x 2 -2
0 2
t -10000
-5000 0 5000
u
-2 0 x 2
0 1
2
3 x
0 1
2 3 t
0 10 20
u
0 1
2
3 x
0 1
2 3 t
Fig.4 Three-dimensional solution surfaces for the series representations eq.(2.11d) and (2.11e) for the case of approximate symmetries. Left: The case A2 generated by the choice for the parameters a0 a31and a1a21, the similarity variable takes tx/2 and 1. Right: The case B2 is generated by the choicea0a1a2a31, , and remains equal as before.
