Although nonlinear wave equations were known in the nineteenth century,
only a few solutions were known and their properties were difficult to
study. The most famous of these equations models a water wave
propagating in a canal. It was published in 1895 by Korteweg and de
Vries and is now known as the KdeV equation. Korteweg and de Vries
showed that this equation has pulse-shaped solutions which propagate
without changing shape. Such solutions were called solitary waves; they
were considered mathematical curiosities until fairly recently. Before
the advent of the computer, it was assumed that nonlinearities would
cause solutions of nonlinear PDEs to interact in such a way that their
identity would be lost if they overlapped. After all, 
would
probably not be a solution even though 
and 
were. However,
computer experiments showed that solutions existed which would
emerge from a collision with their shape and velocity unchanged.
Such special solitary waves were named solitons in
recognition of these particle-like properties. Many other nonlinear
wave equations that exhibit solitary wave and soliton-like behavior
have since been discovered.
We model the three nonlinear Klein-Gordon (NKG) equations
listed in table 
. Using appropriate units, these equations are
of the form
Since eq:NLKG is an extension of the linear KG equation, it is not surprising that it can be solved by the same numerical technique.
The linear Klein-Gordon equation is the equation for a relativistic
quantum-mechanical scalar (spin-zero) particle of mass m. We have
already shown that if 
, a plane wave,
where p = k= momentum and 
, then you get 
,
which is the correct relativistic relationship between
rest energy, momentum, and energy in units where 
and
c=1. The linear KG equation can also be obtained by
using the Euler-Lagrange equations from the Lagrangian density
The function, 
, now represents a quantized field describing the particle.
Again, this is a free particle without interactions. Now consider how
these particles might interact among themselves. The simplest term that
can be added to the Lagrangian is a quadratic term of the form
where b is a constant. The resulting wave equation is called the phi-four equation.
What does this mean? Classically, this is the same as adding a term
to the potential energy, which in the original formalism was just
where the minimum energy state is at 
.
This is still true when we add the quadratic term. (A cubic
term would have put the minimum energy state at
.) From the quantum field theoretic perspective, this term in the
Lagrangian means that processes occur in which two scalar particles can scatter
off each other at a point. The constant b is then a measure of the
strength of the interaction between particles (i. e., a coupling constant) and has been set
equal to one in our program.
The phi-four theory as described above is a ``toy'' model, in that there don't appear to be real, physical particles that interact in this simple way. But it is a good way to begin understanding the overall character of field theories of interacting particles.
The sine-Gordon equation is another nonlinear extension of the KG equation. It also originated in particle field theory, although it also models a Josephson junction and is the continuum limit for a chain of pendula coupled by springs. It is the only common nonlinear NKG equation that is completely integrable; i.e., it has exact analytical solutions. However, the solutions must be written in terms of Jacobi elliptic functions. The double sine-Gordon equation is similar and has applications in nonlinear optics. All of the above NKG equations will have what are called kink solutions that take the system from one asymptotically stable state to another.
