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ON SOLITON INSTABILITIES FOR THE NONLINEAR STRING EQUATION
F. Lambert, M. Musette
To cite this version:
F. Lambert, M. Musette. ON SOLITON INSTABILITIES FOR THE NONLINEAR STRING EQUA- TION. Journal de Physique Colloques, 1989, 50 (C3), pp.C3-33-C3-38. �10.1051/jphyscol:1989305�.
�jpa-00229444�
JOURNAL DE PHYSIQUE
Colloque C3, supplbment au n 0 3 , Tome 50, mars 1989
ON S O L I T O N I N S T A B I L I T I E S FOR THE NONLINEAR STRING EQUATION
F. LAMBERT and M. MUSETTE
Theoretische Natuurkunde, Vrije Universiteit Brussel, Pleinlaan 2 , B-1050 Brussels, Belgium
Resume - L'instabilite de solitons subsoniques dans certaines chaines atomiques est discutee dans le cadre de I'equation nonlineaire des cordes (equ. de type Boussinesq)
modelisant une interaction competitive (cubique) entre premiers et seconds voisins.
Le caractere "a sens unique" des processus resonants observes sur la chaine est
explique a I'aide du comportement analytique des solutions a deux solitons au voisinage de la resonance. L'importance des ondes solitaires "intermediaires" est soulignee.
Abstract
-
Soliton instabilities in atomic nonlinear chains with competitive first and second neighbour interactions of cubic type are discussed on the basis of the nonlinear string equation. The puzzling one-way character of the observed resonance processesis explained in terms of the analytic behaviour of two-soliton solutions near resonance. The importance of s.c. "intermediate solitary waves" is underlined.
1. Introduction
Several examples of onedimensional soliton equations with two-soliton resonances have recently been found 11-31, suggesting that onedimensional resonances might also be observed in actual systems. More recently, simulations on a nonlinear atomic chain 141 have shown that resonant soliton decay does take place for subsonic lattice solitons below a critical velocity. The chain, with competitive first and second neighbour
interactions of cubic type, can be described in the continuum limit by the nonlinear string equation:
utt
-
Uxx + Uxxxx + (u2)xx = 0, (1 )which is also called /5,6/ the good Boussinesq equation (NSgB).
This equation is known to posses N-soliton solutions 171, as well as degenerate vertex solutions 181 describing the resonant decay of a large soliton into two smaller counter- moving solitons, in the presence of N-2 "spectator solitons", or the reversed merger process.
The elementary vertex solutions at N=2 account for the soliton decays observed on the chain 14,9/. They explain some of the earlier numerical results I61 obtained for the NSgB equation and can also be used to interpret the exceptionally large phaseshifts produced by head-on collisions of two lattice solitons near resonance.
Yet, these vertex solutions do not explain why a lasting merger of two counter-moving solitons was never observed on the chain. Nor does the existence of decay vertices by itself explain the observed instability of solitary lattice excitations above a critical amplitude (i.e. below a critical velocity).
The one-way character of the resonance processes on the chain has either been interpreted physically 191, or has been ascribed 141 to the existence of a linear
instability mode I101 for the NSgB equation. However, it is our view that there is no real understanding of a matter so closely connected with the interaction properties of NSgB solitons without a clear insight into the analytic behaviour of the two-soliton solutions near resonance. Furthermore, it seems important that a phenomenon as unusual as the splitting of a one-soliton profile should be explained in terms of the existing solutions, without referring to a linear stability analysis I101 which requires the explicit
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989305
C3-34 JOURNAL DE PHYSIQUE
knowledge of a Lax-pair.
It is the aim of this note to provide new elements in the analysis of the NSgB
two-soliton solutions that could account for what is observed on the chain. The crucial point is the existence near resonance of "intermediate solitary waves" through which large NSgB-solitons (i.e. slow solitons with amplitude above a critical value) are coupled to a decay channel. These intermediate solitons correspond to the "virtual solitons" of
121. The present discussion should also apply to other onedimensional equations (they
need not be integrable) with sech2-solitary waves and with resonant two-soliton solutions, examples of which have been reported elsewhere /3,11/.
2. NSaB two-soliton
solution^
The equ. (1) has sech2-solitons which may travel in both directions:
us0l = 6k2 sech2 [k(x-st)+ T], s =
*d
1 -4k2, Ikl I 1with mass (area below the sech2-cu~e) mSol = 121kl I 6.
It also has two-soliton solutions:
u12 = 6a,ln 2 F , F = 1+ epl + e m +Al2 e 91+%2
with qi = 2ki(x-sit)
+
z i , Si = q i d I - 4 k i , 2 qi = * I ,and with the two-soliton coupling function (phaseshift parameter):
2 2
A =
-
S2) -l (k1- k2)12 2 2
-
S2) - l2 (kl + k2)Regularity of uI2(x,t) at all x and t imposes the condition: A12 2 0.
This condition is only met if:
i) is1
-
s2)2 I 1 2(k1-
k2)2 which implies 0 I A12 < 1 ii) (sl- s2)2 2 12(kl+
k2)2 which implies 0 64;
.i1We shall see that regular non-trivial solutions u12 can also be defined at points ( k l ,k2) where the equality (7) is satisfied and A12 is unbounded.
3. Elastic collisions of counter-movina solitons.
When 0 < AI2 < co and s, # s2 the two-soliton solutions ~ ~ ~ ( ~ , t ; k i , ~ ~ ) , i= 1,2, describe an elastic two-soliton collision process. This property is well-known and follows directly from the asymptotic behaviour of F, as t
+ * - , in an arbitrary reference frame 181.
Without any loss of generality (invariance of the NSgB equation under space reflection and time reversal) we assume that:
O < k l s k 2 s & and ql=l.
As we are interested in the behaviour of u12 near resonance (where A12 gets unbounded) (8) we consider the case of counter-moving solitons in which q 2 = -1, so that:
s1 > 0 > s 2
(9) On account of the equalities (6) and (7) it is easy to check that u12 describes an elastic
2 2 3
head-on collision when (kl ,k2) lies outside the ellipse k,+ klk2+ k2 = ,,(open domain ADB
2 2 3
where 0 < A12 < I ) , or inside the ellipse k,+ klk2+ k2 =
16
(open domain AOC, plus the open segment OC, where 1 < A12 < CO) as shown on fig.1.Considering the solutions uI2(x,t ; 21 = 22 = 0) it is instructive to examine the emergence of the isolated (seemingly stationary) soliton-peaks at finite times.
We first consider u12 in the restfrarne of the larger soliton "2". Thus, setting x = s2t+
5
we rewrite u12(5,t) as follows:
uI2(5,t) = 635 In F 2
F = l+e2k2c
+
e(alt+2klc)
(I+ ~ ~ ~ e ~ ~ 2 5 ) , al= 2kl (s2-sI) < 0 (1 0) The first two terms of F are time-independent. They dominate the whole of F near5
= 0 at1+A12
sufficiently large positive times: t >> t(>)
=
t21n(T),
(1 1) where t2 = [2kl (sl-s2 )]-I> 0 denotes a characteristic time associated with soliton "2".The last two terms of F have the same time-dependence. Thev dominate the whole of F 2A12 k l
near
5
= -(2k2)-lln A,, at times: t cc=
t2[
In(--- -
In A~.].1+A12 k2
It follows that the larger (asymptotic) soliton is bound to emerge as an approximate sech2-peak of type (2) with k = k2 in two circumstances:
i) near
5
= 0 at times t >> t (+),
(13)(- )
ii) near
5
= -(2k2)-l In AI2 at times t c< t 2 (14)Comparison of t 2 and (+)
$2
confirms that this larger soliton-peak can 6nly appear once at a time.In the same way we find that the smaller soliton "1" will appear in its restframe (i.e.
when x = slt+
6 )
as an isolated peak:i) near
5
= 0 at times t << t(;)=
tl In(-)
21 +A1 2 (1 5 )
1 +A1 2 k2
ii) near
5
= -(2kl)-1 In A12 at times t >>f:-' =
tl[
In( -)+
In A, ,I,2 4 2 (1 6)
where tl
=
[2k2(s1-s2)]-1 > 0 is a characteristic time associated with soliton "1".Comparison of t (+) and
6;)
shows that for values of All close to zero there may be a significant period of time during which two soliton-peaks with k = kl appear simultaneously in u12, at a relative distance of the order of (2kl)-'1 In AI21. This particular feature tells us that, for some values of the soliton parameters (such that A,,= 0) the two solitons do not just collide
....
JOURNAL DE PHYSIQUE
4. Exchanae processes.
By looking closer into the interaction process near the boundary AB of the regularity domain one finds that u12(x,t ; AI2 = 0) describes a particle-like "exchange process" in which two counter-moving solitons exchange their mass difference 12(k1- k2) through an intermediate soliton of the form:
6kE sech2 [kE(x-sEt)] 2 with kE= k2- kl, k,s,-k,s, S ~ = k2-k,,
which -in contrast with the asymptotic solitons- has a finite lifetime of the order of:
A careful analysis shows that the two incoming solitons do not collide: the larger soliton
"2" decays, before it reaches the smaller soliton "I", into a replica of this soliton
(outgoing soliton "1 ") and the intermediate soliton "En. The intermediate soliton collides with the incoming soliton "1" to form the outgoing soliton "2". This exchange process may be represented by a world-line diagram (zero-width approximation of a contour-map I21 of uI2) reminiscent of a particle exchange diagram characterized by a three-particle intermediate state 131. As (k1,k2) reaches the boundary AB on which A12 = 0 the intermediate soliton becomes an asymptotic soliton, and the two-soliton solution degenerates into a vertex-solution EI2 = uI2(x,t ; AI2=0)
which describes the following merger process: (1 9)
soliton (kl ,sl >O)
+
soliton (kE,sE <O)+
soliton (k2 = kl+kE, s2 <O) (20) The parameters of the three solitons in the vertex are related by S.C. "resonanceconditions":
k2 = kl+ kE , Q2 = Q1+ Q, with Qi = kisi, i = 1,2, (21) Since exchange processes, as those described by u12 near the boundary AB, are
characterized by a three-soliton intermediate state they have no direct bearing on the propagation properties of a solitary excitation.
5. Resonance Drocesses.
As (kl,k2) approaches the other elliptic segment AC on fig.1, from below, i.e. as A,,
+
W,there is another exceptional reference frame which (as the restframe of the above intermediate soliton "EN) does 'not correspond to the restframe of a soliton, but which is nevertheless the restframe of an intermediate solitary wave. To see this it suffices to consider two-soliton solutions with the following choice of the z-parameters:
u12(x,t ; xi=O, ~ ~ , ~ = - l n A ~ ~ ) with i = l or 2: (22) Setting x = sRt+g with SR = k l ~ l + k 2 ~ 2
kl+k2
we find that these solutions correspond to a function:
(i) (i)
F(i)(c,t)= 1 + e2(k1+k2)c+ el-% t + Z k l i ) + e(aR t+2k2c) A1 2
s2-s1 with (-)ia:' I 2k k
(-)<
kl+k2 0.The first two terms in ~(i)(x,t) are time-independent and dominate the whole of ~ ( i ) near x=O at intermediate times t which are such that:
1 a(0 t
c< e R << 2A1 (26) By choosing (kl,k2) close enough to the boundary AC one can make A,, large enough so that the conditions (26) are satisfied during a significant period of time of the order of:
tRln(4A12) with t R = k l +k2
> 0 2 k l k 2 ( ~ 1 -s2)
During this time, which increases logarithmically with A12, the solutions (22) coincide almost with a solitary wave of the form:
+
-
which carries all the available soliton-mass: m(u12)= jdx ~ 1 2 = 12(kl+k2).
-
mThus, near the boundary AC the NSgB equation has regular two-soliton solutions which describe a particle-like "resonance process" in which two counter-moving solitons collide and fuse to form a larger intermediate soliton (kR,sR) which, eventually, decays into the same two outgoing solitons. These solutions can be represented (in the zero-width approximation) by a world-line diagram which is reminiscent of a particle resonance diagram, characterized by a one-particle intermediate state (fig.2)
A particular resonance process, produced by the collision of two counter-moving solitons
1 1
of the same size is observed as (kl,k2) approaches the endpoint C(;i;$ along the line
OC
( f i g l ) Near C the equation has two-soliton solutions of type (22) which may be written as follows:
uj2 = 68, In [l+e2k(x:st) 2
+
ce2k(x*st)+
e4kx]"
and c = (1 -4k2)-1 .(I-1 6k2).with k =
i,
s =d m
=y
(29)1 1
As k varies within a small interval, say [,.. E,
q
[, there is a finite period of time~+(E)+ln(') - during which u i 2 is indistinguishable from a solitary, stationary, pulse of the
E
form:
2 3
68, In (l+e4kx)
I
k = (1 14) =5
sech2 (): (30)This intermediate pulse keeps appearing (for very small values of E) as one and the same soliton, with an amplitude of about four times the amplitude of the incoming solitons but with a variable lifetime T+(E). AS E
+
0 it is the increasing lifetime of the intermediate solitary wave which accounts for the exceptionally large phaseshifts observed on the chain /4,9/. In the limit E = 0, the intermediate pulse becomes an asymptotic soliton as the solutions (29) degenerate into the following vertex solutions:*
2 x43
u12= 68, In
[I+
ex+
e G $ ~ t ) ] . (31C3-38 JOURNAL DE PHYSIQUE
The solution u;,describes a merger process of type (20) with kl = kE =
;;,
1 whereas IJ,, describes the reversed decay process.Simulations on the nonlinear chain 191 have shown that intermediate solitary waves can actually be produced by appropriate two-soliton collisions. The fact that a lasting merger of two counter-moving solitons, as described by the vertex solution uf,, was never
observed reflects the simple fact that simulations cannot be performed with infinite precision on the parameters of the colliding solitons so as to satisfy the conditions (21) in an exact manner. Finally, we see that as (k, ,k2) reaches the boundary AC (where A;~=o) the two-soliton solutions (22) degenerate into vertex solutions describing a merger process of type (20) if i = l or the corresponding decay process if i=2.
Though resonance conditions of the form (21) are satisfied for the vertex solutions occuring on both the boundaries AB and AC, it is worth emphasizing that true resonances, which correspond to the resonant breakdown of uT2 as an iterated solution 17,121
generated by a solution uo = exp cpl+ exp cp2 of the linearized NSgB equation, do only occur on the boundary AC , and not on AB.
6. Soliton instabilitv above the critical amplitude.
The resonant processes of sect. 5 are characterized by the appearance of an intermediate
43
1 1solitary wave (kR,sR) with 4 ckR
l z
and 0 l sR c2.
The amplitude of this sech2-pulse is determined by kR= kl+ k2, the soliton parameters (k1,k2) being restricted to a narrow strip (of width E) along the inner side of the boundary AC. Its shape remains practicallyunchanged when (k, ,k2) is varied within a portion of the strip near the endpoint C, where the boundary is almost parallel lo the line kl+ k2=
5
1 (fig.l). The details, however, of the profile (which changes slightly but continuously as t evolves during the lifetime T(E)) depend on the particular values of kl and k2.43 -
The instability of the NSgB-solitons above the critical k-value: kc= ,(below the critical velocity s,=
5)
is related to the ability of an intermediate solitary pulse to fit initial data which are an approximation to exact soliton-data with k > kc. Though perfect solitonic data should propagate uniformly for any value of k E ]o,$], it is clear that under the slightest perturbation such data should evolve as those corresponding to an intermediate solitary wave which is bound to decay.References,
111 Tajiri M., Nishitani T., J. Phys. Soc. Japan
fi
(1982) 3720 I21 Hirota R., Ito M., J. Phys. Soc. Japan52
(1983) 744I31 Musette M., Lambert F., Decuyper J.C., J. Phys. A (1987) 6223 I41 Flytzanis N., Pnevmatikos S., Remoissenet M., Physica
26
D (1987) 31 1 I51 Mc Kean H.P., Physica3
D (1981) 294I61 Manoranjan V.S., Mitchell A.R., Morris J.LI., Siam J. Sci. Stat. Comput.
5
(1984) 946 171 Rosales R., Stud. Appl. Math.59
(1 978) 1 17I81 Lambert F., Musette M., Kesteloot E., Inverse Problems
3
(1987) 275 191 Pnevmatikos S., Springer Math. Studies103
(1985) 307I101 Fal'kovich G.E., Spector M.D., Turitsyn S.K., Phys. Lett. 9 9 A (1983) 271 11 11 Lambert F., Musette M., J. Phys. Soc. Japan
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(1988) 2207.1121 Newell A.C., Redekopp L.G., Phys. Rev. Lett. % (1977) 377
