NUMERICAL ANALYSIS OF THE REGULAR LONG WAVE (RLW) EQUATION

(1)

HAL Id: jpa-00229449

https://hal.archives-ouvertes.fr/jpa-00229449

Submitted on 1 Jan 1989

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

NUMERICAL ANALYSIS OF THE REGULAR LONG WAVE (RLW) EQUATION

J. Bruyndonckx, W. Malfliet

To cite this version:

J. Bruyndonckx, W. Malfliet. NUMERICAL ANALYSIS OF THE REGULAR LONG WAVE (RLW) EQUATION. Journal de Physique Colloques, 1989, 50 (C3), pp.C3-71-C3-75.

�10.1051/jphyscol:1989310�. �jpa-00229449�

(2)

J. BRUYNDONCKX and W. MALFLIET

Department of Physics,University of Antwerp ( U I A ) , B-2610 Wilrijk, Be1 gium

RBsume

-

L'equation RLW est examinee numeriquement dans le cas d'une interaction entre des ondes solitaires positives et/ou negatives. Pour chaque interaction specifique une methode numerique, qui est la plus exacte, est selectionnee. Finalement, une explication qualitative est donnee pour comprendre le fait que cette equation n'a pas la propriete des ondes dits solitons.

Abstract

-

The RLW equation is investigated numerically in the case of an interaction between positive and/or negative solitary-waves. For each specific problem ths most accurate numerical method is selected. Finally, a qualitative explanation is given of the non-soliton behaviour of this equation.

1

-

INTRODUCTION

The RLW or BBM equation /1,2/ is a nonlinear wave equation closely connected with the well-known KdV eguation. It is written as

Solitary-wave solutions are easily found to be

Because no other analytical solutions are known, one has to perform numerical computations.

To investigate the interaction of these waves, we notice that for 4k2<l we deal with positive waves (p), while in the other case negative waves (n) arise. Hence three different kinds of interactions can be discerned: p

-

^p,

n

-

n and n

-

^p.

The NL wave equation under study can be rewritten into the following form:

(ut

+

^Ux

+

^2uux

+

^Uxxx)

-

^utxx

-

^Utxxxx⁼^O ( 3 ) by adding together ( 1 ) with

7 a

( 1 )

.

Apart from the last two terms, eq. ax (3) coincides with the KdV equation. The relationship with the latter equation can also be observed in the limit of small k-values: the amplitude

A(k) = - 6k2 approaches 6k2 and the velocity -

'

becomes nearly 1+4k2,

1-4k2 1-4k2

which transforms (2) into the KdV solitary-wave solution.

But the wanted property of soliton behaviour, so typical for the KdV equation, is not present. This was revealed earlier by numerical calculations /3/.

In the first part of this paper the different interactions, mentioned above, are studied numerically. Although most of these interactions already have been studied in literature /4,5/, our aim was to compare different numerical methods to acquire the most accurate one for the problem at hand.

In the second part, we shall give some qualitative explanation of the results Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989310

(3)

C3-72 JOURNAL DE PHYSIQUE

of p-p and n-n interactions, by means of a decomposition technique originally studied for the interaction of KdV solitary-waves.

2 - NUMERICAL ANALYSIS 1. p-p interaction.

For this 'problem the Fourier method of Chan and Kerkhoven /6/ was the most suited one among the methods frequently used in literature.

As a numerical example for this case, we have taken k l = 0.3 and k a . = 0.2. To measure the accuracy of the method, we checked the mass density de ~ n e d by

I1 = JU dx = constant. As a result we got for the error AI1 < 2.l0-~ % during the time period understudy.

After the interaction (see figure 1)

,

a dispersion wave train developed, with a maximum amplitude of about 6

This corresponds to

-

^0.2^% of the amplitude A(k ) of the smallest solitary wave. Moreover a (very small) change of amplitu8e of the largest solitary- wave was recorded:

+

0.005 %. This positive change was calculated indirectly by the numerical computation of fu dx before and after the interaction. On the other hand, the mass density o$ the smallest solitary-wave could not be

Fiq. 1

-

The numerical solution u of e-q. (1) after the interaction of two (positive) solitary-waves with original amplitudes of A(kl)=0.84 and A(k )=0.28 2 determined accurately due to the presence of the wave train. Our experience in a similar case / 7 / (a KdV-soliton with a dispersive tail) indicates a very slow convergence towards the final amplitude of the solitary wave: the dispersive wave train remains sticked to the back of the former.

2. n-n interaction.

Aqain, the Fourier method /6/ used in the first case appeared to be the best choice. We now started with kl=0.6 and k2 = 1.0

. ere,

the error on the mass density A11<5.10%. The same results occurred as in the previous case, butmore pronounced. A wave train, quite larger now appeared at the back of the smallest solitary-wave, with a maximum amplitude of about 10-3 (1,3% of A(k2),

(see fig. 2)

.

Fig. 2

-

The numerical solution u of eq. (1) after the interaction of two (negative) solitary-waves with original amplitudes of A(k2)=-4.9 and A(k2)e-2.0

(4)

almost negligeable.

-

3. p-n interaction.

Here the three-level scheme, introduced by Eilbeck and Mc Guire /8/ gave the best results. Two cases are considered.

a) In a first example, with k = 0.4 and k 2 = 0.6 we got, after the interaction of such positive an& negative wave, 3 negative and 2 positive waves together with a dispersive wave train. This latter represents

some 12 % of the initial mass density I = Judx. To check the accuracy of the numerical method, we calculated &he errors on the three known conservation laws /1/ of the RLW equation; it yielded the following results: ~ 1 ~ = 3 . 1 0 - ~ % , A12=0.01% and A13=0.03%. Remark that this case was extensively studied earlier /4,5/.

b) In a second example we simulated the so-called "collider experiments"

(named after proton and anti-proton collisions). We take: I1 = ju dx =

-Jundx, so that we are dealing with waves with the same mass densiey but opposite sign.

We studied I =2, 3.5 and 4.5. Numerically, these examples were diffi- cult to handle: the error AI1 was < I % , substantially larger than usual.

No creation of new solitary waves were found now; at least for the examplesunder consideration. We got the familiar pictureof a dispersive wave train coupled to the original positive and negative solitary-wave.

Fig. 3

-

"Collider" simulation of a positive and negative solitary-wave with the same mass density I1 = 3.5, before and after the interaction.

Fig. 4

-

"Collider" simulation of a positive and negative solitary-wave with the same mass density I1 = 2, before and after the interaction.

In the case of 11=2, the negative wave clearly disappears after the interaction: those waves have a forbidden amplitude range in the intervallo, -1.51.

The positive wave seems to travel (unscathed?) to the right, imbedded in the middle of the dispersive tail.

3 - ANALYTICAL STUDY

As already mentioned no analytical solutions, except solitary-wave solutions, are known for the RLW equation. From the numerical calculations it turns out that a dispersive wave train always appears after the interaction.

Except for the p-n case, the interaction looks very much KdV-like: the disper-

(5)

C3-74 JOURNAL DE PHYSIQUE

sive wave trains are relatively very small and the change of wave number k negligeable. Moreover the wave train remains just behind the slowest solitary wave. This fact reminds us of the same feature occurring in the KdV problem choosing an arbitrary initial condition

(not

a reflectionless potential in terms of the IST): solitons come out separately, apart from the last one which is connected with a dispersive tail (see for instance ref. /7/).

A qualitative explanation of the p-p and n-n behaviour can be found asfollows.

We start with the familiar KdV equation

vt

-

6vvx

+

vxxx = 0

.

( 4 )

As a solution we take a two-soliton solution, represented through

v = soliton 1

+

^{soliton 2}

.

⁽⁵

we decompose the KdV equation into associated equations:

(vl)t

-

6m v ( v ~ ) ~

-

6(1-m) vlvx

+

(vl),,, = 0 (6) and iv2)

-

6m v (v2)

-

6 (1-m) v2vx

+

(v2)

, , ,

= 0

.

⁽⁷⁾

Note that vl

+

v obeys the KdV equation.

This set of equa$ions (6) and (7) is numerically investigated for several values of m, in the following way:

At t = -T (before the interaction) we start with v1 = soliton 1 and v2 = soliton 2

.

As a result we get for t = +T (after the interaction) v1 = A(m) .soliton 1

+

B(m). soliton 2

+

tail

v2 = (1-A(m) ) .soliton 1

+

(1-B(m) )

.

^soliton²

-

tail

The sum of these quantities is still the two-soliton solution (as it shouldbe).

Special cases are:

m = 1 : A(m) = 1 , B(m) = O r t a i l = O

kl-k2 with R =

-

kl+k2

In the first case (m=l) we deal with the perfect soliton-interaction picture:

a soliton comes in and the very same soliton comes out (which is also proved analytically). In the other cases (apart for m = 1/2) the interaction causes tail formation which is clearly dispersive.

Fig. 5

-

Numerical simulation of eqs. (6) and (7) for m = 1/6, before and after the interaction. The solid curve and dashed curve representrespectively u1 and u2.

This picture can be extended to all KdV equations. Now, for small wave- numbers k,we know that the solitary waves of both KdV and RLW are connected with each other:

u RLW - l v K d V a n d ~ 2 RLW - --- 1 KdV 1 1-4k12 1 1-4k2 ²v2

Assuming that the RLW interaction is similar to the interaction of KdVsolitons with m-1 (but # l), we are able to simulate this problem if we multiply an

(6)

tails do not vanish anymore by adding them together.

This qualitative picture is supported by the following decomposition of the RLW equation which we have investigated numerically (n-n interaction):

with u = ul

+

u2

.

⁽¹⁵⁾

Fig. 6

-

Numerical simulation of eq. (14) for ul (solid curve) and u2 (dashed curve) before and after the interaction.

The interaction is XdV like: both quantities ul and u2 have developed a tail different in size, causing the non-soliton character of the interaction. A remarkable fact is that the zeros of the tail in both cases coincide, as was proposed in the description above.

REFERENCES

/1/ Benjamin T., Bona J. and Mahony J., Phil. Trans. Roy. Soc. London 272 A (1972) 47.

/2/ Peregrine D., J. Fluid Mech.

20

(1966) 321.

/3/ Abdoellev K., Bogolubsky I., Makhankov V. Phys. Lett.

56

(1976) 427.

/4/ Lewis J. and Tjon J., Phys. Lett. 73 A (1979) 275.

/ 5 / Santorelli A., ~ u o v o Cimento

9

B 8 ) 179.

/6/ Chan T. and Kerkhoven T., SIAM J. Num. Analy. 22 (1985) 441.

/7/ Malfliet M. and Bruyndonckx J.,J. Phys. Soc.

JE

56 (1987) 1675.

/ 8 / Eilbeck J. and Mc Guire G., J. Comp. Phys.

2

( 1 9 m ) 63 and J. Comp. Phys 19 (1975) 43.

-