We begin by describing the nonlinear string in the FPU model in more detail.
When the string is at rest it lies along the
x-axis. When the string moves it only moves in the y
direction. The string is modeled by considering a series of particles along
the x-axis with fixed x coordinates that are separated by a distance,
(so it follows that the mass of each particle is 
, where
is the constant density of the string). Each particle's motion in the
y direction is modeled by considering the effect of springs connecting each
particle to its two neighbors. In Rheology, long polymeric molecules
are often modeled by considering these ``bead-spring" type models, where each
bead/particle represents 10-30 monomer units. When the springs are
assumed to follow Hooke's law, these macromolecular models are called Rouse
chains; when the springs are not Hookean, they are called Rouse-Zimm
chains. [6]
Let 
represent the vertical displacement (i.e., the value of y) at time
of the particle whose horizontal location is 
, and let 
represent
the difference in the vertical displacement between two neighboring particles. We
can express the distance,
, between two neighboring particles as 
. The
magnitude of the force for most springs will usually be some function
G of this distance (divided by
so that the force is normalized), i.e.,
Since it is assumed that the particles do not move in the x direction, we are only concerned with the vertical component of the spring force. The absolute value of the force's vertical component is given by
which we can express strictly as a function of 
:
If the spring is linear (i.e., follows Hooke's law) and has an equilibrium length of
z=0, as is the case for Rouse chains, then
, where
k is the positive-valued spring constant, and therefore
. For linear springs with
nonzero equilibrium lengths or nonlinear springs where
is an analytic function near z=1, we can express
as a series of even powers of 
when 
is small, which leads to the following expression for 
:
If the spring force is an analytic function of 
instead of an analytic function of distance, we may also have even powers in the
series:
Now we consider the effect of the forces on a generic particle,
located at
. If we define 
and 
then we can express Newton's law by using eq (4b) to describe
the effect of each of the two springs acting on the particle:
As 
approaches 0 in eq (5), we obtain the nonlinear partial differential
equation corresponding to the FPU model:
We consider some special cases of our model. If the springs follow Hooke's law and have an equilibrium length of 0, then the nonlinear terms disappear and we are left with the standard linear wave equation
where 
, the spring constant. If 
stays small in the nonlinear model
we can ignore the effects of higher order terms. This leads to the quadratic FPU
equation
which FPU's scheme (ii) simulated.
When the spring force is strictly a function of distance, we work with eq (4a)
instead of eq (4b), so 
. For this common
case, we obtain the cubic FPU equation when 
stays small
which FPU's scheme (iii) simulated. Eqs (7)-(9) will be the three equations of interest in this paper.