TRANSVERSAL DYNAMICS OF ONE-DIMENSIONAL CHAIN ON
NONLINEAR ASYMMETRIC SUBSTRATE.
L.
Manevitch, V. Pervouchine
Institute of Chemical Physics, Kosygin str.4, Moscow, 117977, Russia
ABSTRACT. We present a study of localized transversal excitation in a system of
weakly nonlinear oscillators coupled by weightless elastic beam. The equations of
motion are written in a complex form and then the multi-scale expansion is used.
Short wavelength asymptotic have been considered. We have shown that in the case
Nonlinear Schrodinger Equation (NSE) corresponds to the main approximation. This
equation, in particular, possess soliton-like solutions (breathers).
1. Introduction.
Investigation of the longitudinal dynamics of quasi-one-dimensional oscillatory chains
on a nonlinear substrate goes back to the work [1] and attracts a lot of interest today in
connection with localized excitations (solitons and breathers) analysis. But for all that
a substrate with symmetric anharmonicity is usually considered. This is well taken in
the case of longitudinal dynamics investigation, when this substrate simulates shear
elastic interaction (e.g. between the neighboring chains in polymer crystals).
Meanwhile, transversal dynamics of a quasi-one-dimensional asymmetric chain on a
nonlinear substrate has apparently not been considered yet. Moreover, even nonlinear
dynamics of an isolated oscillatory chain was considered without taking its bending
flexibility into account. Difficulty in transversal dynamics analysis of chains with
regard to the bending properties is caused by rising of the equation order (when
compared with the case of longitudinal dynamics). This is the reason for disuse of
conventional procedures for localized waves search. Besides, in case of transversal
dynamics of polymer chains taking into account only symmetric anharmonicity
remains no longer correct, because the interaction between chains is usually described
by the Lehnard-Jones potential.
In the present paper, multiple-scale procedure is used to overcome these problems.
The procedure seemed to be the most effective when complex presentation of
equations of motion is used [2]. In linear approximation all asymptotics were obtained
in work [3], where beam inertia was also taken into consideration.
2. Model
An infinite chain of particles on a substrate is considered. Let us assume the
interaction between the particles obeys a parabolic potential interaction, which
depends upon an alteration of the angle between two adjacent elements of the chain.
The interaction between the particles and the substrate is described by a weekly
anharmonic asymmetric fourth-power potential. Lagrangian of the system under
consideration is
L

  2w  w
m  2
j
j 1  w j 1
w j  C
2 
r02


  j  

2   1 C   w2   3 / 2 C   w3   2 C   w4
1 j
2 j
3 j
2
3
4


where r0 is the distance between the adjacent particles,
dimensionless form are:
4 - 31
(1)

ε1 .
Equations of motion in


 j   6w j  4w j 1  4w j 1  w j  2  w j  2  w j   3 / 2 21w2j   231w3j  0
w
 
2C
C1 r02
(2)
,  21  C 2 / C1 ,  31  C 3 / C1
Let us divide all particles into two groups («odd» and «even» indexes) and consider
each two adjacent particles as a subsystem. Then we introduce the following change
of variables
(3)
w j ,1  u j , w j ,2  v j
Now there are two equations of motion




3/ 2
2
2
3

u j   6u j  4v j  4v j 1  u j 1  u i 1  u j    21u j    31u j  0

3/ 2
2
2
3

v j   6v j  4u j 1  4u j  v j 1  u i 1  v j    21v j    31v j  0
(4)
Combining these two equations we obtain the new system of equations. The first
equation describes the motion of the center of inertia of each subsystem while the
second one describes the relative motion in each subsystem.







U j   2U j  U j 1  U j 1  2V j 1  2V j 1  U j   3 / 2 21 U 2j  V j2   231U j U 2j  3V j2  0


(5)
3/ 2
2
2
2
Vj   10V j  3V j 1  3V j 1  2U j 1  2U j 1  V j  2  21U jV j   31V j V j  3U j  0



3. Short waves consideration
Let us consider the short wave approximation for the system. We assume that we
measure a distance between particles in terms  1r0 and introduce an appropriate space
coordinate  . Representing the difference expressions in the equations by their Taylor
expansions and changing the time variable as   t 16   we obtain


2 
  2V
 U


2  V 
 4


V



3

  3 / 2 21 UV   2 31 V V 2  3U 2  0
2
2



8
16

 
 


2
2

 U
V
 U 


2
  3 / 2 21 U 2  V 2   2 31 U U 2  3V 2  0
 2  U     4
2


16
16
 


 




(6)
here the last term in the second equation is of order of  3 and is omitted.
Now we use complex functions for the relative motion coordinate
V 
      
,V 
,   ei 
2i
(7)
2
We introduce further the «slow» times along with «fast» time  0  
1   ,  2   2 ,...
(8)
and present  , U as series
1/ 2
3/ 2
2

   0    1   2    3    4  ...

1/ 2

U  U 0   U 1  U 2  ...
(9)
where each function depends upon all times, «fast» and «slow», as well as upon  .
After substitution (9) and (7) in (6) taking into account (8) we obtain a system of
equations in which the coefficients at corresponding power of  should be equal to
zero.
1)
0 :
 0
 2U 0
 0,
0
 0
 02
(10)
4 - 32
therefore 0  0  ,1, 2 ,...,
2)  1 / 2 :
 1
 0,
 0
U0  U0  ,1, 2 ,...
U 1
0
 0
 0  2

0
 1  0
3)  1 :
 
 i  0
i 0  0 
e 0
(11)

U

2
i


e
0
0


 
 02

Since U0  U0 1,.. , U0  0 in order the function U to be regular. Furthermore, we obtain
 2U 2
an expression for U 2
 

 
U 2  2i  0 exp( i 0 )  0 exp( i 0 ) 
 




(12)
 1  3

0
 1  0
4)  3 / 2
Now we substitute U in the first equation for the expression above and equal to zero
coefficients at  2 .
  4
 2 : ei 0 

  0
5)

2 
 2  0 
3i
 2 0
i 0   0
    8i  (ei 0


e
)
1  2 
2
 2
 2






3
 21 i 0
 0
 0  i 31 i 0
e  0  e i 0  0  ei 0
 e i 0

e  0  ei 0  0  0

8

  128

(13)
Taking into account again the absence of secular terms we obtain the following
equation
 0
 2 0
2
 iA
 iB  0  0  0
2
 2

Here
A  (8 2 
3

 ), B  31
2
128
(14)
.
This equation coincides exactly with corresponding equation for a chain of oscillators
but the sign of A depends upon the value of  . The soliton-like solutions, e.g.
breathers, can propagate along the chain in case of A>0. For this case we have

 2   Ak 2
0  

B

where
k
v
.
2A

1/ 2


1/ 2


2
   Ak 
  v 2 
expi (k   2 sec h
 A 





(15)
Amplitude and velocity of soliton are independent parameters here.
6. Conclusions
We have considered firstly the short wavelength asymptotic for the homogenous chain
of oscillators coupled by elastic beam. Corresponding small parameter characterizes a
strong coupling between weakly nonlinear oscillators. Preliminary, transition to
complex variables has allowed to utilize efficiently the multiple-scale expansion. As a
result we have shown that in short wavelength approximation a very complicated
problem can be greatly simplified being reduced to NSE whose localized solutions are
breathers.
4 - 33
6. References.
1. Frenkel J., T. Kontorova
“On the Theory of Plastic Deformations and Twisting” J. of Phys. 1(1993), pp. 137149.
2. L. I. Manevitch
“Complex Representation of Dynamics of Coupled Nonlinear Oscillarors”,
Mathematical Models of Non-Linear Excitations, Transfer Dynamics and Control in
Condensed Systems and Other Media, Edited by L. Uvarova, A. Arinstein, A.
Latyshev.
3. L. I. Manevitch, V. G. Oshmyan
“An Asymptotic Study of the Linear Vibrations of a Stretched Beam with
Concentrated Masses and Discrete Elastic Support”, J. of Sound and Vibration (1999)
223(5), pp. 679-691.
4 - 34